{ "schemaVersion": "2.0", "title": "Optimization Open Problems", "corpusBasis": "journal publication date", "publicationWindow": { "from": "2023-01-01", "through": "2026-07-13" }, "statusCheckedThrough": "2026-07-13", "caveat": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "counts": { "journalRecords": 1542, "researchArticles": 1507, "fullTextScreened": 1040, "fullTextUnavailable": 467, "confirmedProblems": 484, "sourcePapers": 357, "byJournal": { "Mathematical Programming": 186, "SIAM Journal on Optimization": 72, "Foundations of Computational Mathematics": 82, "Mathematics of Operations Research": 144 }, "byYear": { "2023": 110, "2024": 158, "2025": 122, "2026": 94 }, "byStatus": { "no_resolution_found": 446, "partial_progress": 25, "resolved": 13, "disputed": 0 } }, "coverage": [ { "journal": "Mathematical Programming", "dateFrom": "2023-01-01", "dateTo": "2026-07-13", "records": 469, "researchArticles": 454, "fullTextScreened": 454, "fullTextUnavailable": 0, "confirmedProblems": 186 }, { "journal": "SIAM Journal on Optimization", "dateFrom": "2023-01-01", "dateTo": "2026-07-13", "records": 407, "researchArticles": 405, "fullTextScreened": 266, "fullTextUnavailable": 139, "confirmedProblems": 72 }, { "journal": "Foundations of Computational Mathematics", "dateFrom": "2023-01-01", "dateTo": "2026-07-13", "records": 151, "researchArticles": 147, "fullTextScreened": 145, "fullTextUnavailable": 2, "confirmedProblems": 82 }, { "journal": "Mathematics of Operations Research", "dateFrom": "2023-01-01", "dateTo": "2026-07-13", "records": 515, "researchArticles": 501, "fullTextScreened": 175, "fullTextUnavailable": 326, "confirmedProblems": 144 } ], "problems": [ { "id": "mor-2023-manual-10.1287-moor.2022.1288-1", "journal": "Mathematics of Operations Research", "publicationYear": 2023, "title": "Asymptotic optimality of the Anderson–Weber rendezvous strategy", "topic": "Online Algorithms And Search", "publicationDate": "2023-01-06", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 3, Introduction", "sourceQuote": "However, in general it is conjectured that the Anderson-Weber strategy is asymptotically optimum", "problemStatement": "In the symmetric rendezvous game on a complete graph with n locations, two players use the same strategy and seek to minimize their expected meeting time. The Anderson–Weber strategy is optimal for n = 2 and n = 3 but not exactly optimal for n = 4. Determine whether it is nevertheless asymptotically optimal as n tends to infinity, with expected meeting time about 0.829n.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "rendezvous search", "Anderson–Weber strategy", "asymptotic optimality", "complete graph" ], "literature": [], "source": { "title": "On the Effect of Symmetry Requirement for Rendezvous on the Complete Graph", "authors": [ "Marthe Bonamy", "Michał Pilipczuk", "Jean-Sébastien Sereni", "Richard Weber" ], "doi": "10.1287/moor.2022.1288", "url": "https://doi.org/10.1287/moor.2022.1288", "preprintUrl": "https://arxiv.org/pdf/2002.03764", "volume": "48", "issue": "2", "pages": "942-953" }, "number": 1 }, { "id": "mor-2023-manual-10.1287-moor.2022.1336-2", "journal": "Mathematics of Operations Research", "publicationYear": 2023, "title": "Finite extension phases for complete dynamic-equilibrium flows", "topic": "Dynamic Network Flows", "publicationDate": "2023-01-06", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 34, Conclusions and Open Questions", "sourceQuote": "whether a finite number of such extensions is enough to compute a complete equilibrium flow is still open for dynamic equilibria", "problemStatement": "Dynamic-equilibrium flows can be constructed phase by phase through α-extensions that continue the equilibrium from its current labels and queues. A single phase is polynomial-time computable on series–parallel graphs, whereas the analogous instantaneous-dynamic-equilibrium construction is known to terminate after finitely many phases. Determine whether finitely many extensions always suffice to construct a complete full-information dynamic equilibrium.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "dynamic equilibrium", "alpha-extension", "finite termination", "flows over time" ], "literature": [], "source": { "title": "The Price of Anarchy for Instantaneous Dynamic Equilibria", "authors": [ "Lukas Graf", "Tobias Harks" ], "doi": "10.1287/moor.2022.1336", "url": "https://doi.org/10.1287/moor.2022.1336", "preprintUrl": "https://arxiv.org/pdf/2007.07794" }, "number": 2 }, { "id": "mor-2023-manual-10.1287-moor.2022.1336-1", "journal": "Mathematics of Operations Research", "publicationYear": 2023, "title": "Monotonicity of dynamic-equilibrium makespan under higher inflow", "topic": "Dynamic Network Flows", "publicationDate": "2023-01-06", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 7, discussion of the dynamic-equilibrium monotonicity conjecture", "sourceQuote": "provided that a certain monotonicity conjecture holds. While this conjecture is wildly believed to be true, it still remains unproven to this day.", "problemStatement": "A dynamic equilibrium routes a fixed amount of flow through a capacitated network over time. The monotonicity conjecture says that increasing a constant source inflow rate cannot increase the equilibrium makespan; if true, it yields the sharp e/(e−1) price-of-anarchy bound. Prove or refute this conjecture for general networks.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "dynamic equilibrium", "flows over time", "monotonicity conjecture", "price of anarchy" ], "literature": [], "source": { "title": "The Price of Anarchy for Instantaneous Dynamic Equilibria", "authors": [ "Lukas Graf", "Tobias Harks" ], "doi": "10.1287/moor.2022.1336", "url": "https://doi.org/10.1287/moor.2022.1336", "preprintUrl": "https://arxiv.org/pdf/2007.07794" }, "number": 3 }, { "id": "mp-2023-001", "journal": "Mathematical Programming", "publicationYear": 2023, "title": "Entropy regularized softmax policy gradient", "topic": "Machine learning and reinforcement learning", "publicationDate": "2023-01-23", "recordType": "reviewed", "status": "resolved", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Discussion", "sourceQuote": "How to understand the (in)-effectiveness of entropy-regularized PG methods is of fundamental importance in the theory of policy optimization.", "problemStatement": "The paper proves that vanilla softmax policy gradient can require exponentially many iterations on a family of discounted Markov decision processes. Establish a nonasymptotic global-convergence and sample-complexity analysis for softmax policy gradient when the objective includes entropy regularization. The target is a rigorous theory beyond results that assume another regularizer or an exact oracle.", "statusEvidence": "The current version proves global optimality convergence and near-quadratic sample complexity for stochastic entropy-regularized vanilla softmax policy gradient.", "verificationNote": "This status is tied to the cited primary-literature result and the precise scope stated above.", "keywords": [ "policy gradient", "entropy regularization", "reinforcement learning", "sample complexity" ], "literature": [ { "title": "Beyond Exact Gradients: Convergence of Stochastic Soft-Max Policy Gradient Methods with Entropy Regularization", "url": "https://arxiv.org/abs/2110.10117", "relation": "The current version proves global optimality convergence and near-quadratic sample complexity for stochastic entropy-regularized vanilla softmax policy gradient." } ], "source": { "title": "Softmax policy gradient methods can take exponential time to converge", "authors": [ "Gen Li", "Yuting Wei", "Yuejie Chi", "Yuxin Chen" ], "doi": "10.1007/s10107-022-01920-6", "url": "https://doi.org/10.1007/s10107-022-01920-6", "volume": "201", "issue": "1-2", "pages": "707-802" }, "number": 4 }, { "id": "focm-2023-lis-finite-size-expansion", "journal": "Foundations of Computational Mathematics", "publicationYear": 2023, "title": "LIS Finite Size Expansion", "topic": "Probability and asymptotics", "publicationDate": "2023-01-26", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 4.1, conjectured expansion (41)", "sourceQuote": "there is further a function F2,2(t) such that", "problemStatement": "Prove the uniform second finite-size correction expansion for the longest-increasing-subsequence distribution, including existence of F2,2. The remainder should be uniformly O(n^-1) over integer thresholds. Here L_n is the length of the longest increasing subsequence of a uniform random permutation, and F2 is the Tracy-Widom distribution appearing in the leading random-matrix limit.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "longest increasing subsequence", "Tracy-Widom", "finite-size correction" ], "literature": [], "source": { "title": "A Stirling-Type Formula for the Distribution of the Length of Longest Increasing Subsequences", "authors": [ "Folkmar Bornemann" ], "doi": "10.1007/s10208-023-09604-z", "url": "https://doi.org/10.1007/s10208-023-09604-z", "volume": "24", "issue": "3", "pages": "915-953" }, "number": 5 }, { "id": "mp-2023-003", "journal": "Mathematical Programming", "publicationYear": 2023, "title": "Rectangular matrix perspective reformulation", "topic": "Convex and nonlinear optimization", "publicationDate": "2023-02-18", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Appendix D, rectangular matrices", "sourceQuote": "Developing the theoretical tools necessary to extend the MPRT to rectangular matrices, is therefore a question for future research.", "problemStatement": "The paper rigorously extends its matrix-perspective reformulation technique to rectangular matrices only for the quadratic map X ↦ XᵀX. Develop the general theory for spectral penalties on rectangular matrices, where the perspective may depend on two positive-definite arguments. The goal is an exact and tractable counterpart of the symmetric-matrix construction.", "statusEvidence": "No primary paper explicitly claiming a proof or counterexample to this precise formulation was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "matrix perspective", "rectangular matrices", "low-rank optimization", "spectral penalties" ], "literature": [], "source": { "title": "A new perspective on low-rank optimization", "authors": [ "Dimitris Bertsimas", "Ryan Cory-Wright", "Jean Pauphilet" ], "doi": "10.1007/s10107-023-01933-9", "url": "https://doi.org/10.1007/s10107-023-01933-9", "volume": "202", "issue": "1-2", "pages": "47-92" }, "number": 6 }, { "id": "mp-2023-002", "journal": "Mathematical Programming", "publicationYear": 2023, "title": "Weaker condition for matrix perspective convexity", "topic": "Convex and nonlinear optimization", "publicationDate": "2023-02-18", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "The Matrix Perspective Function and its Applications", "sourceQuote": "It is, however, an open question whether a weaker notion than matrix convexity could ensure the joint convexity of tr(g_f).", "problemStatement": "The matrix-perspective reformulation is convex when the scalar spectral function is matrix convex, but the paper observes that this hypothesis may be stronger than necessary. Characterize a weaker property that still guarantees joint convexity of the trace of the matrix perspective. Such a characterization would enlarge the class of low-rank penalties covered by the convex framework.", "statusEvidence": "No primary paper explicitly claiming a proof or counterexample to this precise formulation was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "matrix perspective", "matrix convexity", "low-rank optimization", "spectral functions" ], "literature": [], "source": { "title": "A new perspective on low-rank optimization", "authors": [ "Dimitris Bertsimas", "Ryan Cory-Wright", "Jean Pauphilet" ], "doi": "10.1007/s10107-023-01933-9", "url": "https://doi.org/10.1007/s10107-023-01933-9", "volume": "202", "issue": "1-2", "pages": "47-92" }, "number": 7 }, { "id": "mp-2023-005", "journal": "Mathematical Programming", "publicationYear": 2023, "title": "Axiomatic characterizations of phragmen rules", "topic": "Combinatorial optimization and social choice", "publicationDate": "2023-03-06", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conclusion", "sourceQuote": "It would be of great interest to find axiomatic characterizations of Phragmén’s rules.", "problemStatement": "Find sets of axioms that uniquely characterize leximax-Phragmén, var-Phragmén, sequential Phragmén, and Eneström-Phragmén. The characterization should distinguish each rule from other approval-based committee rules. No direct complete characterization of all four named rules was located.", "statusEvidence": "No primary paper explicitly claiming a proof or counterexample to this precise formulation was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "Phragmén", "axiomatic characterization", "approval voting", "multiwinner elections" ], "literature": [], "source": { "title": "Phragmén’s voting methods and justified representation", "authors": [ "Markus Brill", "Rupert Freeman", "Svante Janson", "Martin Lackner" ], "doi": "10.1007/s10107-023-01926-8", "url": "https://doi.org/10.1007/s10107-023-01926-8", "volume": "203", "issue": "1-2", "pages": "47-76" }, "number": 8 }, { "id": "mp-2023-004", "journal": "Mathematical Programming", "publicationYear": 2023, "title": "EJR with committee monotonicity", "topic": "Combinatorial optimization and social choice", "publicationDate": "2023-03-06", "recordType": "reviewed", "status": "partial_progress", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conclusion", "sourceQuote": "Since seq-Phragmén violates EJR, it remains an open problem whether EJR is compatible with committee monotonicity.", "problemStatement": "Extended justified representation and committee monotonicity are two desirable axioms for approval-based multiwinner elections. Determine whether a voting rule can satisfy both on the unrestricted candidate-approval domain. A compatible rule is known on the narrower party-approval apportionment domain, but that does not settle the general question.", "statusEvidence": "It proves compatibility of EJR and committee monotonicity for party-approval apportionment, a structured special domain rather than unrestricted candidate-approval elections.", "verificationNote": "This status is tied to the cited primary-literature result and the precise scope stated above.", "keywords": [ "Phragmén", "extended justified representation", "committee monotonicity", "approval voting" ], "literature": [ { "title": "Approval-based apportionment", "url": "https://doi.org/10.1007/s10107-022-01852-1", "relation": "It proves compatibility of EJR and committee monotonicity for party-approval apportionment, a structured special domain rather than unrestricted candidate-approval elections." } ], "source": { "title": "Phragmén’s voting methods and justified representation", "authors": [ "Markus Brill", "Rupert Freeman", "Svante Janson", "Martin Lackner" ], "doi": "10.1007/s10107-023-01926-8", "url": "https://doi.org/10.1007/s10107-023-01926-8", "volume": "203", "issue": "1-2", "pages": "47-76" }, "number": 9 }, { "id": "mp-2023-006", "journal": "Mathematical Programming", "publicationYear": 2023, "title": "Generalized lax conjecture", "topic": "Convex algebraic geometry", "publicationDate": "2023-04-08", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Introduction", "sourceQuote": "The generalized Lax conjecture asserts that every hyperbolicity cone is a spectrahedral cone.", "problemStatement": "A hyperbolicity cone is defined by a hyperbolic polynomial, while a spectrahedral cone is a linear slice of the positive-semidefinite cone. Prove or disprove that every hyperbolicity cone has a spectrahedral representation. The conjecture is known in dimension three but remains open in general.", "statusEvidence": "No primary paper explicitly claiming a proof or counterexample to this precise formulation was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "generalized Lax conjecture", "hyperbolicity cone", "spectrahedral cone", "semidefinite programming" ], "literature": [], "source": { "title": "Hyperbolicity cones are amenable", "authors": [ "Bruno F. Lourenço", "Vera Roshchina", "James Saunderson" ], "doi": "10.1007/s10107-023-01958-0", "url": "https://doi.org/10.1007/s10107-023-01958-0", "volume": "204", "issue": "1-2", "pages": "753-764" }, "number": 10 }, { "id": "mp-2023-012", "journal": "Mathematical Programming", "publicationYear": 2023, "title": "Three diverse spanning trees complexity", "topic": "Combinatorial optimization and complexity", "publicationDate": "2023-04-15", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conclusion", "sourceQuote": "Is there a polynomial-time algorithm that checks if an input graph has three spanning trees whose edge sets have pairwise symmetric difference at least d?", "problemStatement": "Given a graph and a diversity threshold d, decide whether it contains three spanning trees whose edge sets are pairwise separated by symmetric difference at least d. Determine whether this fixed-cardinality problem is polynomial-time solvable or NP-hard. Existing results handle two solutions and parameterize jointly by solution count and diversity.", "statusEvidence": "No primary paper explicitly claiming a proof or counterexample to this precise formulation was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "diverse solutions", "spanning trees", "symmetric difference", "complexity" ], "literature": [], "source": { "title": "Diverse collections in matroids and graphs", "authors": [ "Fedor V. Fomin", "Petr A. Golovach", "Fahad Panolan", "Geevarghese Philip", "Saket Saurabh" ], "doi": "10.1007/s10107-023-01959-z", "url": "https://doi.org/10.1007/s10107-023-01959-z", "volume": "204", "issue": "1-2", "pages": "415-447" }, "number": 11 }, { "id": "mp-2023-013", "journal": "Mathematical Programming", "publicationYear": 2023, "title": "Weighted diverse bases FPT by solution count", "topic": "Combinatorial optimization and complexity", "publicationDate": "2023-04-15", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conclusion", "sourceQuote": "Is Weighted Diverse Bases FPT parameterized by k alone?", "problemStatement": "Weighted Diverse Bases asks for k high-weight bases that are pairwise sufficiently different. Determine whether the problem is fixed-parameter tractable when only the number k of requested bases is the parameter. The paper gives tractability for the combined parameter (k,d), while diversity d alone is unlikely to suffice.", "statusEvidence": "No primary paper explicitly claiming a proof or counterexample to this precise formulation was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "matroids", "diverse bases", "fixed-parameter tractability", "parameterized complexity" ], "literature": [], "source": { "title": "Diverse collections in matroids and graphs", "authors": [ "Fedor V. Fomin", "Petr A. Golovach", "Fahad Panolan", "Geevarghese Philip", "Saket Saurabh" ], "doi": "10.1007/s10107-023-01959-z", "url": "https://doi.org/10.1007/s10107-023-01959-z", "volume": "204", "issue": "1-2", "pages": "415-447" }, "number": 12 }, { "id": "mp-2023-014", "journal": "Mathematical Programming", "publicationYear": 2023, "title": "Weighted diverse common independent sets FPT by diversity", "topic": "Combinatorial optimization and complexity", "publicationDate": "2023-04-15", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conclusion", "sourceQuote": "Is Weighted Diverse Common Independent Sets FPT when parameterized by d?", "problemStatement": "The problem seeks multiple high-weight sets that are independent in two matroids and pairwise diverse. Determine whether it is fixed-parameter tractable when parameterized only by the diversity threshold d. Parameterization by the number of solutions is already para-NP-hard.", "statusEvidence": "No primary paper explicitly claiming a proof or counterexample to this precise formulation was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "matroid intersection", "diverse solutions", "fixed-parameter tractability", "parameterized complexity" ], "literature": [], "source": { "title": "Diverse collections in matroids and graphs", "authors": [ "Fedor V. Fomin", "Petr A. Golovach", "Fahad Panolan", "Geevarghese Philip", "Saket Saurabh" ], "doi": "10.1007/s10107-023-01959-z", "url": "https://doi.org/10.1007/s10107-023-01959-z", "volume": "204", "issue": "1-2", "pages": "415-447" }, "number": 13 }, { "id": "mp-2023-010", "journal": "Mathematical Programming", "publicationYear": 2023, "title": "Characterize three agent envy free tangles", "topic": "Fair division", "publicationDate": "2023-04-15", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conclusion, open question 2", "sourceQuote": "Can the methods used for the lips tangle be extended to characterize the class of tangles", "problemStatement": "The paper proves a positive three-agent result for the lips tangle. Characterize all tangles that guarantee connected envy-free allocations for three agents. The requested result is a structural classification, not merely additional isolated examples.", "statusEvidence": "No primary paper explicitly claiming a proof or counterexample to this precise formulation was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "envy-freeness", "tangles", "three agents", "connected allocation" ], "literature": [], "source": { "title": "Fair division of graphs and of tangled cakes", "authors": [ "Ayumi Igarashi", "William S. Zwicker" ], "doi": "10.1007/s10107-023-01945-5", "url": "https://doi.org/10.1007/s10107-023-01945-5", "volume": "203", "issue": "1-2", "pages": "931-975" }, "number": 14 }, { "id": "mp-2023-011", "journal": "Mathematical Programming", "publicationYear": 2023, "title": "Cutset characterization of tangle agent bound", "topic": "Fair division", "publicationDate": "2023-04-15", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conclusion, open question 3", "sourceQuote": "Does some generalization of the gap ≥ 2 cutset notion provide a plausible characterization of the agent bound for an arbitrary tangle?", "problemStatement": "A tangle is a connected space made by gluing finitely many unit intervals at endpoints, and its agent bound is the largest n for which every n-tuple of monotone continuous valuations admits a connected envy-free allocation. For a finite point set X of size t, the ordinary gap is the number of connected components of the complement minus t, so a gap-at-least-two cutset leaves at least t+2 components; the paper's generalized version replaces points by disjoint connected subtangles subject to specified contact-point conditions. Find a still more general cutset notion whose minimum cardinality equals the agent bound for every tangle, or disprove that such a characterization exists.", "statusEvidence": "No primary paper explicitly claiming a proof or counterexample to this precise formulation was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "tangles", "agent bound", "cutsets", "envy-freeness" ], "literature": [], "source": { "title": "Fair division of graphs and of tangled cakes", "authors": [ "Ayumi Igarashi", "William S. Zwicker" ], "doi": "10.1007/s10107-023-01945-5", "url": "https://doi.org/10.1007/s10107-023-01945-5", "volume": "203", "issue": "1-2", "pages": "931-975" }, "number": 15 }, { "id": "mp-2023-009", "journal": "Mathematical Programming", "publicationYear": 2023, "title": "Nontraceable graph EF1 for arbitrarily many agents", "topic": "Fair division", "publicationDate": "2023-04-15", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conclusion, open question 1", "sourceQuote": "Does there exist a single finite graph that is not traceable, yet guarantees contiguous EF1_outer allocations for arbitrarily many agents?", "problemStatement": "Find a finite nontraceable graph on which a contiguous outer-EF1 allocation is guaranteed for every number of agents under reasonable valuations. The graph itself must remain fixed as the number of agents grows. This would sharply separate traceability from universal approximate envy-freeness.", "statusEvidence": "No primary paper explicitly claiming a proof or counterexample to this precise formulation was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "EF1", "nontraceable graphs", "contiguous allocation", "arbitrarily many agents" ], "literature": [], "source": { "title": "Fair division of graphs and of tangled cakes", "authors": [ "Ayumi Igarashi", "William S. Zwicker" ], "doi": "10.1007/s10107-023-01945-5", "url": "https://doi.org/10.1007/s10107-023-01945-5", "volume": "203", "issue": "1-2", "pages": "931-975" }, "number": 16 }, { "id": "mp-2023-008", "journal": "Mathematical Programming", "publicationYear": 2023, "title": "Population monotonicity for tangle envy freeness", "topic": "Fair division", "publicationDate": "2023-04-15", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Notes on definitions", "sourceQuote": "We do not know if the following population monotonicity holds: if a tangle guarantees envy-freeness for n agents, then it guarantees it for n−1.", "problemStatement": "A tangle may guarantee a connected envy-free allocation for a specified number of agents. Determine whether that guarantee is downward monotone in the population size. In particular, does a guarantee for n agents always imply one for n−1 agents?", "statusEvidence": "No primary paper explicitly claiming a proof or counterexample to this precise formulation was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "envy-freeness", "tangles", "population monotonicity", "cake cutting" ], "literature": [], "source": { "title": "Fair division of graphs and of tangled cakes", "authors": [ "Ayumi Igarashi", "William S. Zwicker" ], "doi": "10.1007/s10107-023-01945-5", "url": "https://doi.org/10.1007/s10107-023-01945-5", "volume": "203", "issue": "1-2", "pages": "931-975" }, "number": 17 }, { "id": "mp-2023-007", "journal": "Mathematical Programming", "publicationYear": 2023, "title": "Subdivision preserves EF1 for complete bipartite graphs", "topic": "Fair division", "publicationDate": "2023-04-15", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Introduction", "sourceQuote": "We do not know whether adding degree 2 subdivision vertices along edges of Kₐ,ᵦ always preserves its EF1 guarantee.", "problemStatement": "Complete bipartite graphs Kₐ,ᵦ guarantee EF1 for n agents when both parts have size at least n. Determine whether every graph obtained by subdividing their edges retains that EF1 guarantee. A positive answer would lift the result from individual graphs to their entire topological classes.", "statusEvidence": "No primary paper explicitly claiming a proof or counterexample to this precise formulation was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "EF1", "graph subdivision", "complete bipartite graph", "connected allocation" ], "literature": [], "source": { "title": "Fair division of graphs and of tangled cakes", "authors": [ "Ayumi Igarashi", "William S. Zwicker" ], "doi": "10.1007/s10107-023-01945-5", "url": "https://doi.org/10.1007/s10107-023-01945-5", "volume": "203", "issue": "1-2", "pages": "931-975" }, "number": 18 }, { "id": "mor-2023-W3105682563_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2023, "title": "Characterize limiting competitive ratio of p-sample matroid secretary via i.i.d. analogues", "topic": "Prophet Inequalities", "publicationDate": "2023-04-17", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 5.3, pages 23–24 (discussion around the limit as p → 1 and matroid secretary problem consequences)", "sourceQuote": "It would be natural to ask whether there is an analog of the i.i.d. prophet inequality on matroids whose optimal competitive ratio equals L.", "problemStatement": "In the p-sample matroid secretary problem, each element is independently revealed as a training sample with probability p, and an online algorithm must select an independent set from the remaining random-order arrivals. Let β_M(p) be the best worst-case competitive ratio over all matroids; the source asks about its limit L as p approaches 1. Identify a natural full-information or IID prophet-inequality analogue for matroids whose optimal ratio is exactly L.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "matroid secretary problem", "sampling model", "competitive ratio", "prophet inequalities", "limit as p approaches one", "independence systems" ], "literature": [], "source": { "title": "Sample-Driven Optimal Stopping: From the Secretary Problem to the i.i.d. Prophet Inequality", "authors": [ "José Correa", "Andrés Cristi", "Boris Epstein", "José A. Soto" ], "doi": "10.1287/moor.2023.1363", "url": "https://doi.org/10.1287/moor.2023.1363", "volume": "49", "issue": "1", "pages": "441-475" }, "number": 19 }, { "id": "mp-2023-015", "journal": "Mathematical Programming", "publicationYear": 2023, "title": "Last iterate rates for stochastic monotone VI and saddle points", "topic": "Stochastic and privacy-preserving optimization", "publicationDate": "2023-04-19", "recordType": "reviewed", "status": "partial_progress", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Introduction", "sourceQuote": "Accuracy guarantees for the last iterate, which is currently an open problem for SVI and SSP", "problemStatement": "Single-pass differentially private methods need accuracy guarantees for the final iterate of a stochastic monotone variational inequality or saddle-point method. Establish such last-iterate rates without assuming strong monotonicity. Recent work proves rates for stochastic smooth monotone variational inequalities, but does not by itself complete the paper’s full private single-pass program.", "statusEvidence": "It establishes anytime and fixed-horizon last-iterate gap rates for stochastic smooth monotone variational inequalities, but does not settle every saddle-point and private single-pass formulation in the source question.", "verificationNote": "This status is tied to the cited primary-literature result and the precise scope stated above.", "keywords": [ "last iterate", "stochastic variational inequality", "saddle-point problem", "differential privacy" ], "literature": [ { "title": "Last-Iterate Convergence of Regularized Gradient Methods for Stochastic Monotone Variational Inequalities", "url": "https://openreview.net/pdf/0a48da15400a2b08307be7254bee0562990d0bd5.pdf", "relation": "It establishes anytime and fixed-horizon last-iterate gap rates for stochastic smooth monotone variational inequalities, but does not settle every saddle-point and private single-pass formulation in the source question." } ], "source": { "title": "Optimal algorithms for differentially private stochastic monotone variational inequalities and saddle-point problems", "authors": [ "Digvijay Boob", "Cristóbal Guzmán" ], "doi": "10.1007/s10107-023-01953-5", "url": "https://doi.org/10.1007/s10107-023-01953-5", "volume": "204", "issue": "1-2", "pages": "255-297" }, "number": 20 }, { "id": "focm-2023-invariant-feec-bases-exhaustive", "journal": "Foundations of Computational Mathematics", "publicationYear": 2023, "title": "Invariant FEEC Bases Exhaustive", "topic": "Finite element exterior calculus", "publicationDate": "2023-04-27", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Remark 8", "sourceQuote": "These conditions on R-invariance are sufficient and we conjecture that they are necessary as well.", "problemStatement": "Call a real basis R-invariant when every permutation of a simplex's vertex indices sends each basis vector to itself or its negative. Prove that the paper's sufficient degree lists are also necessary: for example, on triangles A P_r Lambda^1 exists exactly when r is not divisible by three and A P_r^- Lambda^1 exactly when r is not congruent to two modulo three, while on tetrahedra the stated untrimmed Lambda^1 and Lambda^2 degrees are 0,1,2,4,5,8. The necessity must hold for arbitrary real bases, not only the geometrically decomposed A-bases constructed in the paper, and likewise for the remaining trimmed and boundary-space degree lists in Theorems 8.2 and 8.5.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "invariant bases", "finite elements", "representation theory", "differential forms" ], "literature": [], "source": { "title": "Symmetry and Invariant Bases in Finite Element Exterior Calculus", "authors": [ "Martin W. Licht" ], "doi": "10.1007/s10208-023-09609-8", "url": "https://doi.org/10.1007/s10208-023-09609-8", "volume": "24", "issue": "4", "pages": "1185-1224" }, "number": 21 }, { "id": "mp-2023-016", "journal": "Mathematical Programming", "publicationYear": 2023, "title": "Strongly polynomial linear programming", "topic": "Linear and integer programming", "publicationDate": "2023-04-29", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Introduction", "sourceQuote": "An outstanding open question is the existence of a strongly polynomial algorithm for LP.", "problemStatement": "Find an algorithm for general linear programming whose number of arithmetic operations is bounded by a polynomial in the numbers of variables and constraints, independent of numerical bit length. This is the strongly polynomial linear-programming problem. Strongly polynomial algorithms are known only for important special classes.", "statusEvidence": "No primary paper explicitly claiming a proof or counterexample to this precise formulation was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "linear programming", "strongly polynomial algorithm", "complexity", "Smale problem" ], "literature": [], "source": { "title": "A scaling-invariant algorithm for linear programming whose running time depends only on the constraint matrix", "authors": [ "Daniel Dadush", "Sophie Huiberts", "Bento Natura", "László A. Végh" ], "doi": "10.1007/s10107-023-01956-2", "url": "https://doi.org/10.1007/s10107-023-01956-2", "volume": "204", "issue": "1-2", "pages": "135-206" }, "number": 22 }, { "id": "mp-2023-017", "journal": "Mathematical Programming", "publicationYear": 2023, "title": "Number of stieltjes ℓ1 solution path pieces", "topic": "Convex and nonlinear optimization", "publicationDate": "2023-05-04", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Problem setting", "sourceQuote": "It remains an open question whether the number of such pieces is an exponential or polynomial function of n when Q is a Stieltjes matrix.", "problemStatement": "For ℓ₁-regularized sparse quadratic minimization with a Stieltjes Hessian Q, the solution path has finitely many smooth pieces. Determine whether the maximum number of pieces grows polynomially or exponentially with the dimension n. Linear-size examples are known, while exponential examples for general Hessians do not use Stieltjes matrices.", "statusEvidence": "No primary paper explicitly claiming a proof or counterexample to this precise formulation was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "solution path", "Stieltjes matrix", "sparse optimization", "parametric complexity" ], "literature": [], "source": { "title": "Comparing solution paths of sparse quadratic minimization with a Stieltjes matrix", "authors": [ "Ziyu He", "Shaoning Han", "Andrés Gómez", "Ying Cui", "Jong-Shi Pang" ], "doi": "10.1007/s10107-023-01966-0", "url": "https://doi.org/10.1007/s10107-023-01966-0", "volume": "204", "issue": "1-2", "pages": "517-566" }, "number": 23 }, { "id": "mor-2023-W2953466159_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2023, "title": "Compute a competitive equilibrium utility profile for divisible chores with variable size", "topic": "Competitive Equilibrium Chores", "publicationDate": "2023-05-10", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 10, end of Related Work / start of Section 2 (statement just before Section 2), and reiterated in Section 3 (“What if both n and m are large?”), page 19.", "sourceQuote": "Computing an exact competitive equilibrium in polynomial time when both the number of agents and chores are variable remains an open question for additive utilities.", "problemStatement": "In a market of divisible chores, each agent has additive negative utility, a negative budget, and chooses a utility-maximizing affordable bundle at equilibrium prices. The paper can enumerate all competitive utility profiles in strongly polynomial time when either the number of agents or the number of chores is fixed. Design a polynomial-time algorithm that computes one exact competitive utility profile when both dimensions vary.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "competitive equilibrium", "chores", "fair division", "Fisher markets", "additive utilities", "equilibrium computation" ], "literature": [], "source": { "title": "Algorithms for Competitive Division of Chores", "authors": [ "Simina Brânzei", "Fedor Sandomirskiy" ], "doi": "10.1287/moor.2023.1361", "url": "https://doi.org/10.1287/moor.2023.1361", "preprintUrl": "https://arxiv.org/pdf/1907.01766", "volume": "49", "issue": "1", "pages": "398-429" }, "number": 24 }, { "id": "mor-2023-W4376149422_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2023, "title": "Determine minimax regret rate for contextual dynamic pricing with unknown noise distribution", "topic": "Contextual Dynamic Pricing", "publicationDate": "2023-05-11", "recordType": "reviewed", "status": "resolved", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 4 (Introduction, end of Abstract/Contributions discussion) and page 20 (Remark 2 following Theorem 1 in Section 3).", "sourceQuote": "we conjecture the obtained Õ(T^(2/3)) rate is close to the optimal rate.", "problemStatement": "In contextual dynamic pricing, each customer's valuation is a linear function of observed features plus noise with an unknown distribution, and the seller sees only whether a posted price is accepted. The known upper regret rate is nearly T to the two-thirds under Lipschitz regularity, and that exponent is conjectured to be minimax optimal. Determine matching upper and lower minimax regret rates without assuming a known parametric noise family.", "statusEvidence": "Matching bounds, up to logarithmic factors, establish the T^(2/3) minimax exponent under Lipschitz unknown noise.", "verificationNote": "This status is tied to the cited primary-literature result and the precise scope stated above.", "keywords": [ "contextual dynamic pricing", "linear valuation model", "unknown noise distribution", "regret minimax lower bound", "distribution-free learning", "bandit feedback" ], "literature": [ { "title": "Minimax Optimality in Contextual Dynamic Pricing with General Valuation Models", "url": "https://doi.org/10.1287/opre.2025.1779", "relation": "Matching bounds, up to logarithmic factors, establish the T^(2/3) minimax exponent under Lipschitz unknown noise." } ], "source": { "title": "Distribution-Free Contextual Dynamic Pricing", "authors": [ "Yiyun Luo", "Will Wei Sun", "Yufeng Liu" ], "doi": "10.1287/moor.2023.1369", "url": "https://doi.org/10.1287/moor.2023.1369", "volume": "49", "issue": "1", "pages": "599-618" }, "number": 25 }, { "id": "mp-2023-024", "journal": "Mathematical Programming", "publicationYear": 2023, "title": "Eventual heller formula threshold", "topic": "Discrete optimization and geometry of numbers", "publicationDate": "2023-05-13", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 6, Question 5", "sourceQuote": "Given m, is there a threshold Δ(m) such that h(Δ,m)=m²+m+1+2m(Δ−1) for every Δ≥Δ(m)?", "problemStatement": "For an integer m×n matrix A with pairwise distinct columns, h(Δ,m) is the largest n such that the maximum absolute m×m minor of A equals Δ. Determine whether, for each fixed m, there is a threshold Δ(m) beyond which h(Δ,m)=m²+m+1+2m(Δ−1) for every Δ≥Δ(m). Equivalently, decide whether only finitely many determinant bounds violate this affine formula in each dimension; the paper identifies h(4,3)=33 as one exception.", "statusEvidence": "No primary paper explicitly claiming a proof or counterexample to this precise formulation was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "Delta-modular matrices", "eventual formula", "threshold", "Heller constant" ], "literature": [], "source": { "title": "On the maximal number of columns of a Δ-modular integer matrix: bounds and computations", "authors": [ "Gennadiy Averkov", "Matthias Schymura" ], "doi": "10.1007/s10107-023-01964-2", "url": "https://doi.org/10.1007/s10107-023-01964-2", "volume": "206", "issue": "1-2", "pages": "61-89" }, "number": 26 }, { "id": "mp-2023-019", "journal": "Mathematical Programming", "publicationYear": 2023, "title": "Exact generalized heller constant", "topic": "Discrete optimization and geometry of numbers", "publicationDate": "2023-05-13", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 6, Open problems", "sourceQuote": "The determination of the exact value of h(Δ,m) remains the major open problem.", "problemStatement": "Determine the generalized Heller constant h(Δ,m) exactly for arbitrary positive integers Δ and m. The source computes several small cases and gives upper and lower bounds, but also constructs counterexamples to a previously proposed formula. An exact general theory remains unknown.", "statusEvidence": "No primary paper explicitly claiming a proof or counterexample to this precise formulation was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "Delta-modular matrices", "Heller constant", "extremal problem", "integer matrices" ], "literature": [], "source": { "title": "On the maximal number of columns of a Δ-modular integer matrix: bounds and computations", "authors": [ "Gennadiy Averkov", "Matthias Schymura" ], "doi": "10.1007/s10107-023-01964-2", "url": "https://doi.org/10.1007/s10107-023-01964-2", "volume": "206", "issue": "1-2", "pages": "61-89" }, "number": 27 }, { "id": "mp-2023-025", "journal": "Mathematical Programming", "publicationYear": 2023, "title": "Geometric representation of heller extremizers", "topic": "Discrete optimization and geometry of numbers", "publicationDate": "2023-05-13", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 6, Question 6", "sourceQuote": "Is there always an extremizer for h(Δ,m) that can be written as the set of integer points in the convex hull of a difference set?", "problemStatement": "Every small extremizer computed in the paper admits a geometric representation as the integer points in the convex hull of a difference set from a subset of ℤᵐ. Determine whether such a representation always exists for at least one extremizer of h(Δ,m). A positive answer could connect the matrix extremal problem to lattice polytope structure.", "statusEvidence": "No primary paper explicitly claiming a proof or counterexample to this precise formulation was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "Heller constant", "extremizer", "difference set", "lattice polytope" ], "literature": [], "source": { "title": "On the maximal number of columns of a Δ-modular integer matrix: bounds and computations", "authors": [ "Gennadiy Averkov", "Matthias Schymura" ], "doi": "10.1007/s10107-023-01964-2", "url": "https://doi.org/10.1007/s10107-023-01964-2", "volume": "206", "issue": "1-2", "pages": "61-89" }, "number": 28 }, { "id": "mp-2023-018", "journal": "Mathematical Programming", "publicationYear": 2023, "title": "Improve delta modular bound below fourth degree", "topic": "Discrete optimization and geometry of numbers", "publicationDate": "2023-05-13", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Introduction", "sourceQuote": "It remains an open question whether our bound can be improved, for all Δ, to O(mᵈ)Δ for some exponent d < 4.", "problemStatement": "For an integer m×n matrix A, let Δ_m(A) be the largest absolute determinant of an m×m submatrix; h(Δ,m) is the maximum n for which A has pairwise distinct columns and Δ_m(A)=Δ. The paper proves h(Δ,m)=O(m⁴Δ) for all Δ, with a cubic bound only when Δ is odd. Prove h(Δ,m)=O(mᵈΔ) for every Δ with one universal exponent d<4; the conjectured sharp order is O(m²Δ).", "statusEvidence": "No primary paper explicitly claiming a proof or counterexample to this precise formulation was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "Delta-modular matrices", "Heller constant", "upper bound", "integer matrices" ], "literature": [], "source": { "title": "On the maximal number of columns of a Δ-modular integer matrix: bounds and computations", "authors": [ "Gennadiy Averkov", "Matthias Schymura" ], "doi": "10.1007/s10107-023-01964-2", "url": "https://doi.org/10.1007/s10107-023-01964-2", "volume": "206", "issue": "1-2", "pages": "61-89" }, "number": 29 }, { "id": "mp-2023-023", "journal": "Mathematical Programming", "publicationYear": 2023, "title": "Prime delta heller formula", "topic": "Discrete optimization and geometry of numbers", "publicationDate": "2023-05-13", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 6, Question 4", "sourceQuote": "Does Conjecture 1 hold for every positive m and every prime Δ? In particular, does it hold for h(3,m)?", "problemStatement": "For an integer m×n matrix A with pairwise distinct columns, h(Δ,m) is the largest n such that the maximum absolute m×m minor of A equals Δ. Conjecture 1 predicts h(Δ,m)=m²+m+1+2m(Δ−1), although the paper finds counterexamples for some composite Δ. Prove or disprove this formula for every prime Δ and every m, in particular for Δ=3.", "statusEvidence": "No primary paper explicitly claiming a proof or counterexample to this precise formulation was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "Delta-modular matrices", "prime determinant bound", "Heller constant", "conjectured formula" ], "literature": [], "source": { "title": "On the maximal number of columns of a Δ-modular integer matrix: bounds and computations", "authors": [ "Gennadiy Averkov", "Matthias Schymura" ], "doi": "10.1007/s10107-023-01964-2", "url": "https://doi.org/10.1007/s10107-023-01964-2", "volume": "206", "issue": "1-2", "pages": "61-89" }, "number": 30 }, { "id": "mp-2023-022", "journal": "Mathematical Programming", "publicationYear": 2023, "title": "Quadratic linear generalized heller bound", "topic": "Discrete optimization and geometry of numbers", "publicationDate": "2023-05-13", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 6, Question 3", "sourceQuote": "Is it true that h(Δ,m) is O(m²)Δ?", "problemStatement": "Prove a generalized Heller upper bound that is quadratic in dimension m and linear in the subdeterminant bound Δ. This relaxes the paper’s more structural shifted-constant questions while preserving the conjectured asymptotic order. The paper itself proves only O(m⁴)Δ.", "statusEvidence": "No primary paper explicitly claiming a proof or counterexample to this precise formulation was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "Delta-modular matrices", "Heller constant", "asymptotic bound", "integer matrices" ], "literature": [], "source": { "title": "On the maximal number of columns of a Δ-modular integer matrix: bounds and computations", "authors": [ "Gennadiy Averkov", "Matthias Schymura" ], "doi": "10.1007/s10107-023-01964-2", "url": "https://doi.org/10.1007/s10107-023-01964-2", "volume": "206", "issue": "1-2", "pages": "61-89" }, "number": 31 }, { "id": "mp-2023-021", "journal": "Mathematical Programming", "publicationYear": 2023, "title": "Refined shifted heller constant bound", "topic": "Discrete optimization and geometry of numbers", "publicationDate": "2023-05-13", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 6, Question 2", "sourceQuote": "Do we have hₛ^Δ(m) ≤ h(1,m), for every positive m and Δ?", "problemStatement": "The refined shifted Heller constant hₛ^Δ(m) tracks the modular parameter Δ. Determine whether it is always bounded by h(1,m). This weaker route would still imply an O(m²)Δ bound for the generalized Heller constant. The source also leaves open whether the refined constant could grow sublinearly in m for fixed Δ.", "statusEvidence": "No primary paper explicitly claiming a proof or counterexample to this precise formulation was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "refined shifted Heller constant", "Delta-modular matrices", "extremal bound", "integer matrices" ], "literature": [], "source": { "title": "On the maximal number of columns of a Δ-modular integer matrix: bounds and computations", "authors": [ "Gennadiy Averkov", "Matthias Schymura" ], "doi": "10.1007/s10107-023-01964-2", "url": "https://doi.org/10.1007/s10107-023-01964-2", "volume": "206", "issue": "1-2", "pages": "61-89" }, "number": 32 }, { "id": "mp-2023-020", "journal": "Mathematical Programming", "publicationYear": 2023, "title": "Shifted heller constant bounded by unimodular constant", "topic": "Discrete optimization and geometry of numbers", "publicationDate": "2023-05-13", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 6, Question 1", "sourceQuote": "It is true that for every positive integer m, we have hₛ(m) ≤ h(1,m)?", "problemStatement": "The shifted Heller constant hₛ(m) is an auxiliary extremal quantity used to control h(Δ,m). Prove or disprove that it never exceeds the classical unimodular Heller constant h(1,m). A positive answer would yield the conjecturally best order O(m²)Δ through the paper’s method.", "statusEvidence": "No primary paper explicitly claiming a proof or counterexample to this precise formulation was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "shifted Heller constant", "unimodular matrices", "extremal bound", "integer matrices" ], "literature": [], "source": { "title": "On the maximal number of columns of a Δ-modular integer matrix: bounds and computations", "authors": [ "Gennadiy Averkov", "Matthias Schymura" ], "doi": "10.1007/s10107-023-01964-2", "url": "https://doi.org/10.1007/s10107-023-01964-2", "volume": "206", "issue": "1-2", "pages": "61-89" }, "number": 33 }, { "id": "mor-2023-W2922533999_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2023, "title": "Constant-factor welfare approximation without separability in multidimensional-signal combinatorial auctions", "topic": "Mechanism Design", "publicationDate": "2023-05-15", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 7 (problems), page 16 (in the provided PDF pagination)", "sourceQuote": "For combinatorial auctions with multidimensional signals, is separability a necessary condition for achieving constant approximation to welfare?", "problemStatement": "A combinatorial auction has bundle-specific signals and submodular-over-signals (SOS) valuations; ex post incentive compatibility and individual rationality must hold at every realized signal profile. The paper obtains a constant-factor welfare approximation only when each valuation is separable into own-signal and others'-signal terms. Determine whether a universally ex post IC-IR constant-factor mechanism exists without separability.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "interdependent valuations", "combinatorial auctions", "welfare approximation", "ex post incentive compatibility", "submodular over signals", "multidimensional signals", "separability" ], "literature": [], "source": { "title": "Combinatorial Auctions with Interdependent Valuations: SOS to the Rescue", "authors": [ "Alon Eden", "Michal Feldman", "Amos Fiat", "Kira Goldner", "Anna R. Karlin" ], "doi": "10.1287/moor.2023.1371", "url": "https://doi.org/10.1287/moor.2023.1371", "volume": "49", "issue": "2", "pages": "653-674" }, "number": 34 }, { "id": "mor-2023-W3217525437_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2023, "title": "Uniqueness in law criteria for weak solutions of multidimensional Ito SDEs", "topic": "Stochastic Differential Equations", "publicationDate": "2023-05-15", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 3, Section 1 (Introduction), footnote 1.", "sourceQuote": "It is still a problem to find conditions, sufficient and necessary at the same time, guaranteeing uniqueness in law of weak solutions of (2)", "problemStatement": "The paper studies weak solutions of uniformly elliptic Ito stochastic differential equations whose drift or diffusion coefficients may be merely measurable. In dimensions above two, no general conditions are known that are both necessary and sufficient for uniqueness in law. Characterize exactly when such multidimensional weak solutions are unique in law.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "weak solutions", "uniqueness in law", "uniform ellipticity", "ito sdes", "multidimensional diffusion", "measurable coefficients" ], "literature": [], "source": { "title": "Large Ranking Games with Diffusion Control", "authors": [ "Stefan Ankirchner", "Nabil Kazi-Tani", "Julian Wendt", "Chao Zhou" ], "doi": "10.1287/moor.2023.1373", "url": "https://doi.org/10.1287/moor.2023.1373", "preprintUrl": "https://hal.science/hal-03434678/document", "volume": "49", "issue": "2", "pages": "675-696" }, "number": 35 }, { "id": "siopt-2023-003", "journal": "SIAM Journal on Optimization", "publicationYear": 2023, "title": "Accelerated proximal bundle", "topic": "Nonsmooth optimization", "publicationDate": "2023-05-17", "recordType": "reviewed", "status": "resolved", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 2.3, smooth setting; cached preprint lines 407-412", "sourceQuote": "It is likely that a variant of the bundle method can achieve accelerated rates.", "problemStatement": "The paper establishes optimal convergence rates for the classical proximal bundle method in several convex regimes, but its smooth-case rate is slower than accelerated gradient descent. The unresolved task was to construct a bundle-method variant with accelerated smooth convex rates, possibly by strengthening the models or changing the update logic. Later work supplied accelerated proximal-bundle variants, so this record is marked resolved.", "statusEvidence": "Proves optimal O(1/sqrt(epsilon)) iteration complexity for a proximal bundle method on smooth convex objectives.", "verificationNote": "This status is tied to the cited primary-literature result and the precise scope stated above.", "keywords": [ "proximal bundle method", "acceleration", "smooth convex optimization" ], "literature": [ { "title": "An accelerated proximal bundle method for convex optimization", "url": "https://arxiv.org/abs/2512.04523", "relation": "Proves optimal O(1/sqrt(epsilon)) iteration complexity for a proximal bundle method on smooth convex objectives." } ], "source": { "title": "Optimal Convergence Rates for the Proximal Bundle Method", "authors": [ "Mateo Díaz", "Benjamin Grimmer" ], "doi": "10.1137/21m1428601", "url": "https://doi.org/10.1137/21m1428601", "preprintUrl": "https://arxiv.org/abs/2105.07874", "volume": "33", "issue": "2", "pages": "424-454" }, "number": 36 }, { "id": "mor-2023-W4377821221_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2023, "title": "Bound the integrality gap of time-indexed LP for bipartite AND/OR scheduling", "topic": "Scheduling With Precedence Constraints", "publicationDate": "2023-05-23", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 7 (Conclusion), page 22 (arXiv v1 pagination)", "sourceQuote": "it would be interesting to obtain stronger bounds on the integrality gap of the time-indexed formulation considered in Section 2.", "problemStatement": "Jobs on one machine obey AND precedences, which require every named predecessor, and OR precedences, which require at least one predecessor from a specified set. The paper rounds a time-indexed linear-programming relaxation but does not know whether its approximation analysis is tight. Determine the relaxation's worst-case integrality gap, as a universal constant or as a function of the maximum OR-predecessor set size.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "single-machine scheduling", "OR-precedence constraints", "time-indexed LP", "integrality gap", "alpha-point rounding", "min-sum objectives" ], "literature": [], "source": { "title": "Approximation Algorithms and Linear Programming Relaxations for Scheduling Problems Related to Min-Sum Set Cover", "authors": [ "Felix Happach", "Andreas S. Schulz" ], "doi": "10.1287/moor.2023.1368", "url": "https://doi.org/10.1287/moor.2023.1368", "volume": "49", "issue": "1", "pages": "578-598" }, "number": 37 }, { "id": "mor-2023-manual-10.1287-moor.2023.1374-1", "journal": "Mathematics of Operations Research", "publicationYear": 2023, "title": "General numeraire-invariant quadratic hedging", "topic": "Mathematical Finance", "publicationDate": "2023-05-24", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 13, Section 3.6", "sourceQuote": "As yet, we do not know how to obtain the solution of the hedging problem (1.2) for general X.", "problemStatement": "The quadratic hedging problem chooses a trading strategy for a price process S and payoff H while allowing a strictly positive numeraire X. The paper relates this problem to a discounted model with price process (1, Ŷ), payoff H/X, and a changed probability measure, but does not solve the general case. Derive the optimizer for arbitrary admissible X and prove the proposed numeraire transformation is valid.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "quadratic hedging", "numeraire invariance", "mean–variance portfolio", "change of measure" ], "literature": [], "source": { "title": "Numeraire-Invariant Quadratic Hedging and Mean–Variance Portfolio Allocation", "authors": [ "Aleš Černý", "Christoph Czichowsky", "Jan Kallsen" ], "doi": "10.1287/moor.2023.1374", "url": "https://doi.org/10.1287/moor.2023.1374", "preprintUrl": "https://openaccess.city.ac.uk/id/eprint/30220/1/2023-04-07-NumeraireMOR_final.pdf", "volume": "49", "issue": "2", "pages": "752-781" }, "number": 38 }, { "id": "siopt-2023-007", "journal": "SIAM Journal on Optimization", "publicationYear": 2023, "title": "Spectrahedral approximation rate", "topic": "Convex approximation", "publicationDate": "2023-06-01", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conclusion; extracted preprint lines 1007-1011", "sourceQuote": "An interesting open question is to obtain the approximation rate for (m, k)-spectrahedral functions", "problemStatement": "An (m,k)-spectrahedral function is represented through an m-by-m linear matrix pencil with block or rank parameter k; its sublevel geometry yields an (m,k)-spectratope. The paper obtains nearly matching approximation bounds for fixed k. Determine the approximation rates to Lipschitz convex functions and convex bodies when k grows with m, especially k=m.", "statusEvidence": "No forward-citing primary paper or exact-query result explicitly claiming a proof or counterexample was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "spectrahedra", "convex regression", "approximation rate", "spectratopes" ], "literature": [], "source": { "title": "Spectrahedral Regression", "authors": [ "Eliza O’Reilly", "Venkat Chandrasekaran" ], "doi": "10.1137/21m1455899", "url": "https://doi.org/10.1137/21m1455899", "preprintUrl": "https://arxiv.org/abs/2110.14779", "volume": "33", "issue": "2", "pages": "553-588" }, "number": 39 }, { "id": "mor-2023-W4379383737_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2023, "title": "Design accelerated proximal bundle methods for hybrid convex composite optimization problems", "topic": "Convex Composite Optimization", "publicationDate": "2023-06-05", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 6 (Concluding Remarks), page 21 (paragraph beginning 'First, under the assumption that the diameter D of dom h is finite, ...').", "sourceQuote": "It would be interesting to design an accelerated variant of GPB which is optimal for the aforementioned HCCO problem class.", "problemStatement": "Hybrid composite optimization minimizes f+h on a convex domain of diameter D, where oracle subgradients of f obey ‖f′(u)−f′(v)‖≤2M+L‖u−v‖. Design an accelerated proximal-bundle method that finds an ε-optimal point in O(√L D/√ε+M²D²/ε²) iterations, up to logarithmic factors. This would match known lower bounds in both smooth and nonsmooth regimes.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "proximal bundle methods", "hybrid convex composite optimization", "acceleration", "iteration complexity", "nonsmooth optimization", "first-order methods" ], "literature": [], "source": { "title": "A Unified Analysis of a Class of Proximal Bundle Methods for Solving Hybrid Convex Composite Optimization Problems", "authors": [ "Jiaming Liang", "Renato D. C. Monteiro" ], "doi": "10.1287/moor.2023.1372", "url": "https://doi.org/10.1287/moor.2023.1372", "volume": "49", "issue": "2", "pages": "832-855" }, "number": 40 }, { "id": "mor-2023-W4379646686_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2023, "title": "Compute the exact competitive ratio for minimization fractional prophet inequalities without in-house production", "topic": "Prophet Inequalities", "publicationDate": "2023-06-07", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 6, Section 1.3 (Our results) and reiterated Page 18, Section 4.2 (More than two sellers)", "sourceQuote": "We leave it as an open question to analytically resolve the competitive ratio.", "problemStatement": "In the minimization fractional prophet problem, independent costs supported on a known interval arrive sequentially and the algorithm must make fractional procurement decisions without in-house production. The paper gives bounds that depend on the power-cost exponent, number of arrivals, and support endpoints. Determine the exact worst-case competitive ratio as a closed-form function of those parameters.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "minimization prophet inequalities", "sequential procurement", "fractional allocations", "competitive ratio", "worst-case distributions", "monomial cost functions", "in-house production" ], "literature": [], "source": { "title": "Minimization Fractional Prophet Inequalities for Sequential Procurement", "authors": [ "Junjie Qin", "Shai Vardi", "Adam Wierman" ], "doi": "10.1287/moor.2021.173", "url": "https://doi.org/10.1287/moor.2021.173" }, "number": 41 }, { "id": "siopt-2023-009", "journal": "SIAM Journal on Optimization", "publicationYear": 2023, "title": "Coordinate descent saddle avoidance", "topic": "Nonconvex optimization", "publicationDate": "2023-06-09", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 1 discussion; extracted preprint lines 472-478", "sourceQuote": "It is an interesting open question whether it is possible to establish similar results without such assumptions.", "problemStatement": "The paper proves that randomized coordinate gradient descent almost surely avoids strict saddles under quantitative assumptions controlling the chosen coordinate gradient and random step. The authors conjecture that the bad initialization set still has Lebesgue measure zero without those assumptions. Prove the measure-zero saddle-avoidance claim under weaker hypotheses or construct a counterexample.", "statusEvidence": "No forward-citing primary paper or exact-query result explicitly claiming a proof or counterexample was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "coordinate descent", "strict saddles", "randomization", "measure zero" ], "literature": [], "source": { "title": "On the Global Convergence of Randomized Coordinate Gradient Descent for Nonconvex Optimization", "authors": [ "Ziang Chen", "Yingzhou Li", "Jianfeng Lu" ], "doi": "10.1137/21m1460375", "url": "https://doi.org/10.1137/21m1460375", "preprintUrl": "https://www.semanticscholar.org/paper/f5206d2a6d499fb1437efe165cb0b6af101e99c4", "volume": "33", "issue": "2", "pages": "713-738" }, "number": 42 }, { "id": "mp-2023-026", "journal": "Mathematical Programming", "publicationYear": 2023, "title": "Random subspace dimension for conic preconditioning", "topic": "Numerical optimization", "publicationDate": "2023-06-15", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Introduction", "sourceQuote": "It remains open how to choose the subspace dimension in order to obtain an efficient preconditioner.", "problemStatement": "The paper preconditions conic-bundle interior-point systems by projecting a low-rank factor onto a random subspace. Develop a principled rule for choosing the projected dimension so the resulting preconditioner is computationally efficient. The rule must balance construction cost against the condition number and iterative-solver work.", "statusEvidence": "No primary paper explicitly claiming a proof or counterexample to this precise formulation was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "preconditioning", "random projection", "conic bundle", "interior-point method" ], "literature": [], "source": { "title": "A preconditioned iterative interior point approach to the conic bundle subproblem", "authors": [ "Christoph Helmberg" ], "doi": "10.1007/s10107-023-01986-w", "url": "https://doi.org/10.1007/s10107-023-01986-w", "volume": "205", "issue": "1-2", "pages": "559-615" }, "number": 43 }, { "id": "siopt-2023-001", "journal": "SIAM Journal on Optimization", "publicationYear": 2023, "title": "Compact domain and complexity of I-AL", "topic": "Convex optimization", "publicationDate": "2023-06-23", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 7, concluding remarks; cached preprint lines 1455-1459", "sourceQuote": "the assumption that the domain of the function P is compact. One natural question is whether this assumption can be dropped.", "problemStatement": "The paper analyzes first-order inexact augmented-Lagrangian (I-AL) methods for convex conic programs assuming that the domain of the nonsmooth term is compact. Under that hypothesis it proves an O(epsilon^-1 log epsilon^-1) first-order complexity bound for finding an epsilon-KKT point. Determine whether compactness can be removed and whether an I-AL method can improve that complexity bound.", "statusEvidence": "No forward-citing primary paper or exact-query result explicitly claiming a proof or counterexample was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "augmented Lagrangian", "conic programming", "iteration complexity", "compactness" ], "literature": [], "source": { "title": "Iteration-Complexity of First-Order Augmented Lagrangian Methods for Convex Conic Programming", "authors": [ "Zhaosong Lu", "Zirui Zhou" ], "doi": "10.1137/21m1403837", "url": "https://doi.org/10.1137/21m1403837", "preprintUrl": "https://arxiv.org/abs/1803.09941", "volume": "33", "issue": "2", "pages": "1159-1190" }, "number": 44 }, { "id": "siopt-2023-023", "journal": "SIAM Journal on Optimization", "publicationYear": 2023, "title": "Pointwise variational convexity and sufficiency", "topic": "Variational analysis", "publicationDate": "2023-06-23", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 5 and conclusion; extracted preprint lines 994-1000 and 1633-1637", "sourceQuote": "Establishing results of this type for (nonstrong) variational convexity is an open question.", "problemStatement": "The paper develops neighborhood and second-order characterizations of variational convexity and strong variational convexity. It obtains a point-based characterization for the strong property, and strong variational sufficiency is comparatively well understood in constrained optimization. Find a point-based second-order characterization for nonstrong variational convexity and develop a parallel theory of merely variational sufficiency.", "statusEvidence": "No forward-citing primary paper or exact-query result explicitly claiming a proof or counterexample was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "variational convexity", "variational sufficiency", "second-order subdifferential", "pointwise criterion" ], "literature": [], "source": { "title": "Variational Convexity of Functions and Variational Sufficiency in Optimization", "authors": [ "Pham Duy Khanh", "Boris S. Mordukhovich", "Vo Thanh Phat" ], "doi": "10.1137/22m1519250", "url": "https://doi.org/10.1137/22m1519250", "preprintUrl": "https://arxiv.org/abs/2208.14399", "volume": "33", "issue": "2", "pages": "1121-1158" }, "number": 45 }, { "id": "mp-2023-027", "journal": "Mathematical Programming", "publicationYear": 2023, "title": "Redundancy of odd beta cycle inequalities", "topic": "Linear and integer programming", "publicationDate": "2023-07-07", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 5, Conjecture 12", "sourceQuote": "proper subsequence C′ of C is a β-cycle. Then the simple odd β-cycle inequality corresponding to (C,σ) is redundant for CR(G).", "problemStatement": "Let FR(G) be the factorable relaxation for binary polynomial optimization and let CR(G) be the polyhedron obtained by imposing every simple odd β-cycle inequality on FR(G). A β-cycle is a closed sequence of pairwise distinct nodes and edges in which each listed node belongs only to its two adjacent listed edges, and an odd signature σ on a closed walk C defines one such inequality. Prove or disprove that the inequality for (C,σ) is redundant for CR(G) whenever C contains a proper subsequence that is a β-cycle; only the case of a repeated edge carrying two positive signs is proved.", "statusEvidence": "No primary paper explicitly claiming a proof or counterexample to this precise formulation was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "binary polynomial optimization", "beta-cycle inequality", "redundancy", "polyhedral combinatorics" ], "literature": [], "source": { "title": "Simple odd β-cycle inequalities for binary polynomial optimization", "authors": [ "Alberto Del Pia", "Matthias Walter" ], "doi": "10.1007/s10107-023-01992-y", "url": "https://doi.org/10.1007/s10107-023-01992-y", "volume": "206", "issue": "1-2", "pages": "203-238" }, "number": 46 }, { "id": "mp-2023-028", "journal": "Mathematical Programming", "publicationYear": 2023, "title": "Ideal sparsity for PSD and CPSD ranks", "topic": "Polynomial and semidefinite optimization", "publicationDate": "2023-07-08", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Concluding remarks, application to other matrix factorization ranks", "sourceQuote": "(or X_iX_j=0). Thus the techniques in the present paper may extend to this general setting. We leave this extension to future work.", "problemStatement": "The positive-semidefinite rank of M∈ℝ_+^(m×n) is the least r admitting positive-semidefinite r×r matrices X_i,Y_j with M_(ij)=⟨X_i,Y_j⟩, while the completely positive-semidefinite rank of A∈S^n uses one family X_i with A_(ij)=⟨X_i,X_j⟩. Zero entries impose noncommutative ideal constraints X_iY_j=0 or X_iX_j=0, analogous to the scalar monomial ideals exploited by the paper's sparse moment hierarchy for nonnegative and completely positive ranks. Construct and validate an ideal-sparse noncommutative moment hierarchy for these two ranks, and compare its bounds and computational size with the dense hierarchy.", "statusEvidence": "No primary paper explicitly claiming a proof or counterexample to this precise formulation was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "ideal sparsity", "positive semidefinite rank", "cpsd rank", "moment hierarchy" ], "literature": [], "source": { "title": "Exploiting ideal-sparsity in the generalized moment problem with application to matrix factorization ranks", "authors": [ "Milan Korda", "Monique Laurent", "Victor Magron", "Andries Steenkamp" ], "doi": "10.1007/s10107-023-01993-x", "url": "https://doi.org/10.1007/s10107-023-01993-x", "volume": "205", "issue": "1-2", "pages": "703-744" }, "number": 47 }, { "id": "focm-2023-shallow-network-approximation-space-hardness", "journal": "Foundations of Computational Mathematics", "publicationYear": 2023, "title": "Shallow Network Approximation Space Hardness", "topic": "Computational learning theory", "publicationDate": "2023-07-12", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 5, after Theorem 5.1", "sourceQuote": "whether a similar result even holds for approximation spaces associated to shallow networks", "problemStatement": "For depth and weight-growth rules ell(n) and c(n), let the approximation space consist of functions on [0,1]^d that admit uniform error O(n^-alpha) from ReLU networks with at most n nonzero weights, depth at most ell(n), and weights bounded by c(n). The paper proves severe deterministic and randomized point-sampling lower bounds when the limiting depth is at least three, meaning at least two hidden layers. Determine whether an analogous hardness theorem holds for the shallow limiting-depth-two space, with one hidden layer and the same complexity and weight constraints.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "shallow neural networks", "approximation spaces", "complexity lower bounds" ], "literature": [], "source": { "title": "Proof of the Theory-to-Practice Gap in Deep Learning via Sampling Complexity bounds for Neural Network Approximation Spaces", "authors": [ "Philipp Grohs", "Felix Voigtlaender" ], "doi": "10.1007/s10208-023-09607-w", "url": "https://doi.org/10.1007/s10208-023-09607-w", "volume": "24", "issue": "4", "pages": "1085-1143" }, "number": 48 }, { "id": "mp-2023-029", "journal": "Mathematical Programming", "publicationYear": 2023, "title": "FC GCG fast convergence for dynamic inverse transport", "topic": "Convex and nonlinear optimization", "publicationDate": "2023-07-13", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 4.4, dynamic inverse problems", "sourceQuote": "The verification of hypotheses (B1)–(B5), necessary for ensuring fast convergence of Algorithm 1, is non-trivial and is left to future work.", "problemStatement": "Consider min_u F(Ku)+G(u), where G is convex and positively one-homogeneous, B is its unit ball, and FC-GCG adds extremal points of B before fully reoptimizing their coefficients. Conditions (B1)–(B4) require local strong convexity of F, finitely many dual-maximizing extremal points, linear independence of their K-images, and strictly positive optimal coefficients. Condition (B5) requires a distance g for which K is locally Lipschitz and the dual certificate has quadratic growth near each active extremal point. Verify these conditions for time-dependent measure recovery regularized by the coercive Benamou–Brenier transport energy, thereby upgrading the known O(1/k) objective residual to an asymptotically geometric bound O(ζ^k) with ζ<1.", "statusEvidence": "No primary paper explicitly claiming a proof or counterexample to this precise formulation was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "conditional gradient", "dynamic inverse problem", "optimal transport", "linear convergence" ], "literature": [], "source": { "title": "Asymptotic linear convergence of fully-corrective generalized conditional gradient methods", "authors": [ "Kristian Bredies", "Marcello Carioni", "Silvio Fanzon", "Daniel Walter" ], "doi": "10.1007/s10107-023-01975-z", "url": "https://doi.org/10.1007/s10107-023-01975-z", "volume": "205", "issue": "1-2", "pages": "135-202" }, "number": 49 }, { "id": "mp-2023-035", "journal": "Mathematical Programming", "publicationYear": 2023, "title": "Algebraic number relaxation complexity", "topic": "Linear and integer programming", "publicationDate": "2023-07-13", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 5, Problem P6", "sourceQuote": "When F is the field of real algebraic numbers, determine if rc_F(X) is computable for a given finite X⊂ℤᵈ.", "problemStatement": "Restrict relaxation coefficients to the field of real algebraic numbers. Determine whether the resulting field-restricted complexity is computable for every finite integer set X and whether it always equals unrestricted real relaxation complexity. This asks whether transcendental coefficients ever provide a genuine advantage.", "statusEvidence": "No primary paper explicitly claiming a proof or counterexample to this precise formulation was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "relaxation complexity", "algebraic numbers", "computability", "coefficient fields" ], "literature": [], "source": { "title": "The role of rationality in integer-programming relaxations", "authors": [ "Manuel Aprile", "Gennadiy Averkov", "Marco Di Summa", "Christopher Hojny" ], "doi": "10.1007/s10107-023-01994-w", "url": "https://doi.org/10.1007/s10107-023-01994-w", "volume": "205", "issue": "1-2", "pages": "745-771" }, "number": 50 }, { "id": "mp-2023-031", "journal": "Mathematical Programming", "publicationYear": 2023, "title": "Asymptotic relaxation complexity of simplex", "topic": "Linear and integer programming", "publicationDate": "2023-07-13", "recordType": "reviewed", "status": "resolved", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 5, Problem P2", "sourceQuote": "Improve the asymptotic upper bound on rc(Δ_d).", "problemStatement": "Determine the asymptotic order of the relaxation complexity of the discrete standard simplex Δ_d. The source proved an upper bound O(d/√log d), while an Ω(log d) lower bound was known. A 2026 construction closes the gap by proving an O(log d) upper bound.", "statusEvidence": "It gives an explicit O(log d) construction matching the previously known Ω(log d) lower bound, thereby determining the asymptotic order.", "verificationNote": "This status is tied to the cited primary-literature result and the precise scope stated above.", "keywords": [ "relaxation complexity", "standard simplex", "asymptotic bound", "integer programming" ], "literature": [ { "title": "The relaxation complexity of the standard simplex is logarithmic", "url": "https://arxiv.org/abs/2606.11852", "relation": "It gives an explicit O(log d) construction matching the previously known Ω(log d) lower bound, thereby determining the asymptotic order." } ], "source": { "title": "The role of rationality in integer-programming relaxations", "authors": [ "Manuel Aprile", "Gennadiy Averkov", "Marco Di Summa", "Christopher Hojny" ], "doi": "10.1007/s10107-023-01994-w", "url": "https://doi.org/10.1007/s10107-023-01994-w", "volume": "205", "issue": "1-2", "pages": "745-771" }, "number": 51 }, { "id": "mp-2023-030", "journal": "Mathematical Programming", "publicationYear": 2023, "title": "Characterize irrational relaxation advantage", "topic": "Linear and integer programming", "publicationDate": "2023-07-13", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 5, Problem P1", "sourceQuote": "For given d, characterize all finite subsets X of ℤᵈ satisfying rc(X) < rc_Q(X).", "problemStatement": "Relaxation complexity counts the fewest facets of a polyhedron whose integer points equal a prescribed finite set X. Characterize the sets for which allowing irrational facet coefficients strictly reduces this number compared with rational relaxations. The first open ambient dimension is five.", "statusEvidence": "No primary paper explicitly claiming a proof or counterexample to this precise formulation was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "relaxation complexity", "irrational polyhedra", "rational polyhedra", "integer points" ], "literature": [], "source": { "title": "The role of rationality in integer-programming relaxations", "authors": [ "Manuel Aprile", "Gennadiy Averkov", "Marco Di Summa", "Christopher Hojny" ], "doi": "10.1007/s10107-023-01994-w", "url": "https://doi.org/10.1007/s10107-023-01994-w", "volume": "205", "issue": "1-2", "pages": "745-771" }, "number": 52 }, { "id": "mp-2023-032", "journal": "Mathematical Programming", "publicationYear": 2023, "title": "Computability of relaxation complexities", "topic": "Linear and integer programming", "publicationDate": "2023-07-13", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 5, Problem P3", "sourceQuote": "Determine whether rc(X) and rc_Q(X) are algorithmically computable.", "problemStatement": "Given a finite integer set X, decide whether its unrestricted relaxation complexity rc(X) and rational relaxation complexity rc_Q(X) can be computed algorithmically. The question applies both in every fixed dimension and when dimension is part of the input. Existing computable upper-bound sequences lack a general stopping certificate.", "statusEvidence": "No primary paper explicitly claiming a proof or counterexample to this precise formulation was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "relaxation complexity", "computability", "rational relaxation", "algorithmic decidability" ], "literature": [], "source": { "title": "The role of rationality in integer-programming relaxations", "authors": [ "Manuel Aprile", "Gennadiy Averkov", "Marco Di Summa", "Christopher Hojny" ], "doi": "10.1007/s10107-023-01994-w", "url": "https://doi.org/10.1007/s10107-023-01994-w", "volume": "205", "issue": "1-2", "pages": "745-771" }, "number": 53 }, { "id": "mp-2023-034", "journal": "Mathematical Programming", "publicationYear": 2023, "title": "Field dependence of simplex relaxation complexity", "topic": "Linear and integer programming", "publicationDate": "2023-07-13", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 5, Problem P5", "sourceQuote": "It is likely that rc_F(Δ₅)=5 holds for every field F⊂ℝ of dimension two over ℚ.", "problemStatement": "Study how relaxation complexity changes when facet coefficients are restricted to an arbitrary subfield F of the reals. A concrete conjecture is that rc over ℚ[r] of Δ₅ equals five for every irrational real r. More generally, determine the dependence of field-restricted relaxation complexity on algebraic properties of F.", "statusEvidence": "No primary paper explicitly claiming a proof or counterexample to this precise formulation was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "relaxation complexity", "coefficient fields", "standard simplex", "irrationality" ], "literature": [], "source": { "title": "The role of rationality in integer-programming relaxations", "authors": [ "Manuel Aprile", "Gennadiy Averkov", "Marco Di Summa", "Christopher Hojny" ], "doi": "10.1007/s10107-023-01994-w", "url": "https://doi.org/10.1007/s10107-023-01994-w", "volume": "205", "issue": "1-2", "pages": "745-771" }, "number": 54 }, { "id": "mp-2023-033", "journal": "Mathematical Programming", "publicationYear": 2023, "title": "Minimum relaxation complexity in fixed dimension", "topic": "Linear and integer programming", "publicationDate": "2023-07-13", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 5, Problem P4", "sourceQuote": "Determine the minimum of rc(X) over all sets X ⊂ ℤᵈ with dim(X)=d.", "problemStatement": "Among all full-dimensional finite integer sets in dimension d, determine the smallest possible relaxation complexity. Study both exact values for fixed d and asymptotic growth. The source asks in particular whether the discrete standard simplex attains the minimum.", "statusEvidence": "No primary paper explicitly claiming a proof or counterexample to this precise formulation was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "relaxation complexity", "extremal problem", "integer sets", "dimension" ], "literature": [], "source": { "title": "The role of rationality in integer-programming relaxations", "authors": [ "Manuel Aprile", "Gennadiy Averkov", "Marco Di Summa", "Christopher Hojny" ], "doi": "10.1007/s10107-023-01994-w", "url": "https://doi.org/10.1007/s10107-023-01994-w", "volume": "205", "issue": "1-2", "pages": "745-771" }, "number": 55 }, { "id": "siopt-2023-012", "journal": "SIAM Journal on Optimization", "publicationYear": 2023, "title": "Bounded gradient trajectories overparameterized networks", "topic": "Optimization for machine learning", "publicationDate": "2023-07-13", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 4 discussion; extracted preprint lines 852-857", "sourceQuote": "it is still an open question whether the gradient trajectories will be bounded in the over-parameterized case", "problemStatement": "The paper certifies absence of spurious local minima at infinity by analyzing boundedness and limiting behavior of gradient trajectories. A one-hidden-layer counterexample rules out a general boundedness claim with multiple data points. Determine whether trajectories are bounded in the overparameterized regime where a global minimum is attainable.", "statusEvidence": "No forward-citing primary paper or exact-query result explicitly claiming a proof or counterexample was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "gradient flow", "overparameterization", "neural networks", "minima at infinity" ], "literature": [], "source": { "title": "Certifying the Absence of Spurious Local Minima at Infinity", "authors": [ "Cédric Josz", "Xiaopeng Li" ], "doi": "10.1137/22m1479531", "url": "https://doi.org/10.1137/22m1479531", "preprintUrl": "https://arxiv.org/abs/2303.03536", "volume": "33", "issue": "3", "pages": "1416-1439" }, "number": 56 }, { "id": "mor-2023-W4384342505_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2023, "title": "Prove convergence of iterative KKT-point exclusion for nonconvex polynomial Nash equilibria", "topic": "Polynomial Nash Equilibrium", "publicationDate": "2023-07-14", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 6 (Conclusions and Discussions), page 26 (arXiv v2 pdf pagination), near the end of the section beginning 'There is much interesting future work to do.'", "sourceQuote": "However, the convergence property of Algorithm 3.1 is not known when there are infinitely many KKT points.", "problemStatement": "For a polynomial Nash equilibrium problem, the paper enumerates Karush–Kuhn–Tucker points and excludes a candidate whenever a player's polynomial best-response problem finds a profitable deviation. The method terminates when the set of non-equilibrium KKT points is finite. Prove convergence or give a counterexample when infinitely many non-equilibrium KKT points can occur.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "nash equilibrium", "nonconvex games", "KKT points", "moment-sos hierarchy", "semidefinite relaxation", "algorithmic convergence" ], "literature": [], "source": { "title": "Nash Equilibrium Problems of Polynomials", "authors": [ "Jiawang Nie", "Xindong Tang" ], "doi": "10.1287/moor.2022.0334", "url": "https://doi.org/10.1287/moor.2022.0334", "volume": "49", "issue": "2", "pages": "1065-1090" }, "number": 57 }, { "id": "mor-2023-W2965273013_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2023, "title": "Deciding existence of equilibria in Fisher markets with earning and utility limits", "topic": "Fisher Market Equilibrium", "publicationDate": "2023-07-18", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 5 (Future Directions), page 32.", "sourceQuote": "In addition to deciding existence, can we efficiently find exact equilibria in general (if they exist) or in money-clearing markets?", "problemStatement": "A Fisher market with earning caps limits the revenue obtainable from each good, while a utility cap limits how much utility a buyer seeks. A thrifty bundle uses only maximum-bang-per-buck goods, and a modest bundle does not exceed the buyer's utility cap. Decide whether a thrifty and modest market equilibrium always exists under both kinds of caps, and compute one when it does.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "Fisher markets", "market equilibrium", "earning caps", "utility caps", "existence complexity", "NP-hardness", "polynomial characterization" ], "literature": [], "source": { "title": "Satiation in Fisher Markets and Approximation of Nash Social Welfare", "authors": [ "Jugal Garg", "Martin Hoefer", "Kurt Mehlhorn" ], "doi": "10.1287/moor.2019.0129", "url": "https://doi.org/10.1287/moor.2019.0129", "volume": "49", "issue": "2", "pages": "1109-1139" }, "number": 58 }, { "id": "mor-2023-W4385350470_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2023, "title": "Determine the asymptotically tight constant for traveling repairman latency under i.i.d. sampling", "topic": "Traveling Repairman Problem", "publicationDate": "2023-07-28", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 29, Section 6 (Conclusion).", "sourceQuote": "A natural question is whether this constant beta_(TSP) is tight.", "problemStatement": "The traveling repairman problem orders IID Euclidean sample points to minimize their total waiting time, which grows on the order of n√n. The paper obtains constant-factor asymptotics using the universal Beardwood–Halton–Hammersley constant β_TSP from Euclidean traveling-salesman tours. Determine the exact leading repairman constant and whether it equals β_TSP.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "traveling repairman problem", "minimum latency", "Euclidean random optimization", "asymptotic constant", "Beardwood-Halton-Hammersley constant", "probabilistic bounds" ], "literature": [], "source": { "title": "Probabilistic Bounds on the k-Traveling Salesman Problem and the Traveling Repairman Problem", "authors": [ "Moïse Blanchard", "Alexandre Jacquillat", "Patrick Jaillet" ], "doi": "10.1287/moor.2021.0286", "url": "https://doi.org/10.1287/moor.2021.0286", "volume": "49", "issue": "2", "pages": "1169-1191" }, "number": 59 }, { "id": "mor-2023-W3186360956_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2023, "title": "Prove existence of complete EFX allocations for general valuations", "topic": "Fair Division", "publicationDate": "2023-08-01", "recordType": "reviewed", "status": "resolved", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 2, Introduction (discussion of prior work and open status of EFX existence for general valuations)", "sourceQuote": "It is not known whether EFX allocations always exist even when n = 3 for general valuations.", "problemStatement": "A complete allocation assigns every indivisible good to an agent. It is envy-free up to any item (EFX) if, after removing any one good from another agent's bundle, no agent prefers the remainder to her own bundle. Determine whether every instance with normalized monotone valuations admits a complete EFX allocation; a later counterexample resolves the general question negatively.", "statusEvidence": "An explicit three-agent, eight-good monotone-valuation instance has no complete EFX allocation, refuting universal existence.", "verificationNote": "This status is tied to the cited primary-literature result and the precise scope stated above.", "keywords": [ "fair division", "indivisible goods", "EFX", "general valuations", "existence theorem", "envy-freeness" ], "literature": [ { "title": "A Counterexample to EFX; n >= 3 Agents, m >= n + 5 Items, Monotone Valuations; via SAT-Solving", "url": "https://arxiv.org/abs/2604.18216", "relation": "An explicit three-agent, eight-good monotone-valuation instance has no complete EFX allocation, refuting universal existence." } ], "source": { "title": "Extension of Additive Valuations to General Valuations on the Existence of EFX", "authors": [ "Ryoga Mahara" ], "doi": "10.1287/moor.2022.0044", "url": "https://doi.org/10.1287/moor.2022.0044", "volume": "49", "issue": "2", "pages": "1263-1277" }, "number": 60 }, { "id": "siopt-2023-013", "journal": "SIAM Journal on Optimization", "publicationYear": 2023, "title": "Adaptive second order realization of third order method", "topic": "Higher-order methods", "publicationDate": "2023-08-02", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 6 discussion; extracted preprint lines 1667-1671", "sourceQuote": "the development of adaptive strategies in this context remains a very interesting open question.", "problemStatement": "Let f be three-times continuously differentiable with L_f-Lipschitz third derivative. The second-order surrogate T_τ(h)=[∇f(x+τh)+∇f(x−τh)−2∇f(x)]/τ² approximates D³f(x)[h]² with error at most (L_f/3)τ‖h‖³, but the known choice of τ needs L_f. Choose τ adaptively without knowing L_f while preserving the basic O(|log ε|ε^(−1/3)) and accelerated O(|log ε|ε^(−1/4)) inner-iteration bounds of the third-order methods.", "statusEvidence": "No forward-citing primary paper or exact-query result explicitly claiming a proof or counterexample was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "third-order methods", "adaptivity", "second-order oracle", "Lipschitz Hessian" ], "literature": [], "source": { "title": "Adaptive Third-Order Methods for Composite Convex Optimization", "authors": [ "G. N. Grapiglia", "Yu. Nesterov" ], "doi": "10.1137/22m1480872", "url": "https://doi.org/10.1137/22m1480872", "preprintUrl": "https://arxiv.org/abs/2202.12730", "volume": "33", "issue": "3", "pages": "1855-1883" }, "number": 61 }, { "id": "siopt-2023-008", "journal": "SIAM Journal on Optimization", "publicationYear": 2023, "title": "Exact penalty under uncertainty", "topic": "PDE-constrained optimization", "publicationDate": "2023-08-02", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Performance-bound discussion; cached preprint lines 1145-1150", "sourceQuote": "Under additional assumptions, we conjecture that ‘exactness’ is possible and then θν can remain bounded", "problemStatement": "The paper studies penalty and augmented-Lagrangian bounds for PDE-constrained optimization under uncertainty, using a penalty parameter theta_nu that may diverge as approximation error vanishes. It conjectures that additional regularity can make the penalization exact with theta_nu uniformly bounded. Identify sufficient assumptions and prove or disprove that exactness statement.", "statusEvidence": "No forward-citing primary paper or exact-query result explicitly claiming a proof or counterexample was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "exact penalty", "uncertainty", "PDE constraints", "augmented Lagrangian" ], "literature": [], "source": { "title": "Performance Bounds for PDE-Constrained Optimization under Uncertainty", "authors": [ "Peng Chen", "Johannes O. Royset" ], "doi": "10.1137/21m1457916", "url": "https://doi.org/10.1137/21m1457916", "preprintUrl": "https://www.semanticscholar.org/paper/a615377cda3dc25f52e62289583254943271f4d7", "volume": "33", "issue": "3", "pages": "1828-1854" }, "number": 62 }, { "id": "siopt-2023-022", "journal": "SIAM Journal on Optimization", "publicationYear": 2023, "title": "Low rank SOS landscape beyond univariate", "topic": "Polynomial optimization", "publicationDate": "2023-08-08", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conclusion; extracted preprint lines 946-954", "sourceQuote": "we conjecture that a version of Theorem 1.1 is true for ternary quartics and matrix polynomials", "problemStatement": "Theorem 1.1 gives a rank threshold above which the factored sum-of-squares objective for a nonnegative univariate polynomial has no spurious second-order critical points. Ternary quartics and matrix polynomials are further settings where nonnegativity is equivalent to an SOS representation. Establish an analogous rank threshold and benign-landscape theorem for those classes, or find a counterexample.", "statusEvidence": "No forward-citing primary paper or exact-query result explicitly claiming a proof or counterexample was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "sum of squares", "low-rank factorization", "spurious local minima", "ternary quartics" ], "literature": [], "source": { "title": "Low-Rank Univariate Sum of Squares Has No Spurious Local Minima", "authors": [ "Benoît Legat", "Chenyang Yuan", "Pablo Parrilo" ], "doi": "10.1137/22m1516208", "url": "https://doi.org/10.1137/22m1516208", "preprintUrl": "https://arxiv.org/abs/2205.11466", "volume": "33", "issue": "3", "pages": "2041-2061" }, "number": 63 }, { "id": "siopt-2023-032", "journal": "SIAM Journal on Optimization", "publicationYear": 2023, "title": "Optimal assumption free trigonometric SOS rate", "topic": "Polynomial optimization", "publicationDate": "2023-08-11", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 3 discussion; extracted preprint lines 470-475", "sourceQuote": "A natural open question is the optimality of the ‘assumption-free’ bound in O(1/s2)", "problemStatement": "The paper analyzes a sum-of-squares hierarchy for minimizing trigonometric polynomials and proves an assumption-free O(1/s^2) convergence bound at relaxation order s. Faster rates follow under extra nondegeneracy assumptions. Determine whether O(1/s^2) is the sharp worst-case rate without additional assumptions by proving a matching lower bound or improving the upper bound.", "statusEvidence": "No forward-citing primary paper or exact-query result explicitly claiming a proof or counterexample was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "trigonometric polynomial", "sum-of-squares hierarchy", "convergence rate", "lower bound" ], "literature": [], "source": { "title": "Exponential Convergence of Sum-of-Squares Hierarchies for Trigonometric Polynomials", "authors": [ "Francis Bach", "Alessandro Rudi" ], "doi": "10.1137/22m1540818", "url": "https://doi.org/10.1137/22m1540818", "preprintUrl": "https://arxiv.org/abs/2211.04889", "volume": "33", "issue": "3", "pages": "2137-2159" }, "number": 64 }, { "id": "mor-2023-W4385878155_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2023, "title": "Verify sharp growth constant for max-linear regression with absolute loss", "topic": "Max Linear Regression", "publicationDate": "2023-08-16", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 5.2.2 (Max-linear regression and PolyakSGM), page 38", "sourceQuote": "if the solution set is isolated, semialgebraicity of f implies that (A2) holds. On the other hand, verifying Assumption (A1) remains an intriguing open problem.", "problemStatement": "Max-linear regression minimizes a nonsmooth objective formed from the maximum of several linear predictors. Existing local convergence guarantees require sharp growth: objective error must dominate distance to the minimizer set in a neighborhood. Give verifiable measurement or statistical conditions that guarantee this sharp error bound.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "nonsmooth optimization", "sharp growth", "error bounds", "max-linear regression", "semialgebraic functions", "subgradient methods" ], "literature": [], "source": { "title": "A Superlinearly Convergent Subgradient Method for Sharp Semismooth Problems", "authors": [ "Vasileios Charisopoulos", "Damek Davis" ], "doi": "10.1287/moor.2023.1390", "url": "https://doi.org/10.1287/moor.2023.1390", "volume": "49", "issue": "3", "pages": "1678-1709" }, "number": 65 }, { "id": "mor-2023-W4385873460_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2023, "title": "Existence of epsilon-equilibria in multiplayer stochastic games with Borel payoffs", "topic": "Stochastic Games", "publicationDate": "2023-08-16", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 2, Introduction (end of item [4] discussion)", "sourceQuote": "every multiplayer stochastic game with finite action spaces, finite or countably infinite state space, and bounded and Borel-measurable payoffs admits an ε-equilibrium for every ε>0.", "problemStatement": "A multiplayer stochastic game has finitely many actions, finitely or countably many states, and an arbitrary bounded Borel payoff on the infinite play. An ε-equilibrium lets no player gain more than ε by deviating, uniformly from every initial state. Determine whether every such game with random state transitions admits an ε-equilibrium for every ε > 0.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "stochastic games", "borel measurable payoffs", "epsilon equilibrium", "nash equilibrium", "countable state space", "existence theorem" ], "literature": [], "source": { "title": "Stochastic Games with General Payoff Functions", "authors": [ "János Flesch", "Eilon Solan" ], "doi": "10.1287/moor.2023.1385", "url": "https://doi.org/10.1287/moor.2023.1385", "preprintUrl": "https://cris.maastrichtuniversity.nl/ws/files/189808346/Flesch-2023-stochastic-games-with-general-payoff-functions.pdf", "volume": "49", "issue": "3", "pages": "1349-1371" }, "number": 66 }, { "id": "mor-2023-W4385980778_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2023, "title": "Improve contention resolution for points in the exact matching polytope", "topic": "Online Matching", "publicationDate": "2023-08-18", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 5 (Discussion), page 20", "sourceQuote": "A natural question is whether better (RO-)OCRS can be achieved for points in the matching polytope, i.e., points which also satisfy Edmonds’ odd-set constraints [Edm65].", "problemStatement": "A random-order online contention-resolution scheme sees active edges in random order and must accept a matching; c-balance requires each edge to be accepted with probability at least c times its fractional value. Existing analysis reaches 0.456 in a restricted setting. Determine whether one can exceed 0.456 for general matchings when the fractional input lies in the exact matching polytope, including all Edmonds odd-set constraints rather than degree constraints alone.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "contention resolution schemes", "random-order online algorithms", "matching polytope", "Edmonds odd-set constraints", "correlation gap" ], "literature": [], "source": { "title": "Improved Online Contention Resolution for Matchings and Applications to the Gig Economy", "authors": [ "Tristan Pollner", "Mohammad Roghani", "Amin Saberi", "David Wajc" ], "doi": "10.1287/moor.2023.1388", "url": "https://doi.org/10.1287/moor.2023.1388", "volume": "49", "issue": "3", "pages": "1582-1606" }, "number": 67 }, { "id": "mor-2023-W3170420696_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2023, "title": "Optimal rank among simplex-isomorphic binarizations for bounded integer variables", "topic": "Integer Programming Formulations", "publicationDate": "2023-08-24", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 5 (Conclusion), page 17.", "sourceQuote": "We do not know whether B_U(d) is optimal with respect to all binarizations that are isomorphic to a d-simplex.", "problemStatement": "A natural binarization represents x∈0,…,d by a polytope in one continuous and d binary variables whose integer slices encode x; restrict attention to polytopes combinatorially equivalent to a d-simplex. For a split between x≤α and x≥α+1, its rank is the minimum number of sequential binary-variable convexifications needed to remove all vertices with α <= f-infinity(u) for every asymptotic direction u. Decide whether dom(f*)=B(f-infinity) always holds, rather than only the proved equality after closing dom(f*) or under the paper's additional geometric hypotheses.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "Hadamard spaces", "Fenchel conjugate", "recession function", "domain closure" ], "literature": [], "source": { "title": "Convex Analysis on Hadamard Spaces and Scaling Problems", "authors": [ "Hiroshi Hirai" ], "doi": "10.1007/s10208-023-09628-5", "url": "https://doi.org/10.1007/s10208-023-09628-5", "preprintUrl": "https://arxiv.org/abs/2203.03193", "volume": "24", "issue": "6", "pages": "1979-2016" }, "number": 82 }, { "id": "focm-2023-gaussian-beams-bounded-domain-semidiscretization", "journal": "Foundations of Computational Mathematics", "publicationYear": 2023, "title": "Gaussian Beams Bounded Domain Semidiscretization", "topic": "Numerical wave propagation", "publicationDate": "2023-10-17", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 7, open problem 1", "sourceQuote": "the case of finite difference approximations is, to the best of our knowledge, still missing", "problemStatement": "Extend the Gaussian-beam construction for semi-discrete wave equations to bounded domains with Dirichlet boundary conditions. The construction must incorporate reflected beams and a discrete analogue of Snell's law. A Gaussian beam is a high-frequency wave packet localized around a bicharacteristic ray; the paper proves approximation estimates only on the whole Euclidean space.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "Gaussian beams", "finite differences", "bounded domains", "wave reflection" ], "literature": [], "source": { "title": "Gaussian Beam Ansatz for Finite Difference Wave Equations", "authors": [ "Umberto Biccari", "Enrique Zuazua" ], "doi": "10.1007/s10208-023-09632-9", "url": "https://doi.org/10.1007/s10208-023-09632-9", "preprintUrl": "https://arxiv.org/abs/2209.13976", "volume": "25", "issue": "1", "pages": "1-54" }, "number": 83 }, { "id": "focm-2023-gaussian-beams-nonuniform-meshes", "journal": "Foundations of Computational Mathematics", "publicationYear": 2023, "title": "Gaussian Beams Nonuniform Meshes", "topic": "Numerical wave propagation", "publicationDate": "2023-10-17", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 7, open problem 2", "sourceQuote": "A natural extension of our work would then be to address the case of non-uniform meshes", "problemStatement": "Develop and analyze Gaussian-beam approximations for semi-discrete wave equations on nonuniform meshes. The theory must account for high-frequency internal reflection and rodeo effects created by mesh grading. The established construction assumes a uniform finite-difference grid, where the discrete Hamiltonian rays and their group velocities can be controlled explicitly.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "Gaussian beams", "nonuniform meshes", "high-frequency waves", "internal reflection" ], "literature": [], "source": { "title": "Gaussian Beam Ansatz for Finite Difference Wave Equations", "authors": [ "Umberto Biccari", "Enrique Zuazua" ], "doi": "10.1007/s10208-023-09632-9", "url": "https://doi.org/10.1007/s10208-023-09632-9", "preprintUrl": "https://arxiv.org/abs/2209.13976", "volume": "25", "issue": "1", "pages": "1-54" }, "number": 84 }, { "id": "mor-2023-W4387738269_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2023, "title": "Find a Hall blocker with operator Sinkhorn in the nonscalable case", "topic": "Operator Scaling", "publicationDate": "2023-10-18", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 3, Introduction (discussion of operator scaling extension and problem raised in prior work)", "sourceQuote": "It is a natural question whether the operator Sinkhorn algorithm can find a Hall blocker for nonscalable case.", "problemStatement": "A completely positive operator Φ maps positive semidefinite matrices to positive semidefinite matrices, and operator Sinkhorn alternately normalizes its two marginals. When Φ is nonscalable, a Hall blocker—equivalently, an appropriate shrunk-subspace certificate—proves that no invertible scaling can make it approximately doubly stochastic. Prove that the standard operator Sinkhorn iterates identify such a blocker and give an extraction procedure, or construct a nonscalable operator for which they do not.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "operator scaling", "operator Sinkhorn algorithm", "completely positive operators", "nonscalable case", "limits of iterates", "shrunk subspaces", "certificates of nonscalability" ], "literature": [], "source": { "title": "Finding Hall Blockers by Matrix Scaling", "authors": [ "Koyo Hayashi", "Hiroshi Hirai", "Keiya Sakabe" ], "doi": "10.1287/moor.2022.0198", "url": "https://doi.org/10.1287/moor.2022.0198", "volume": "49", "issue": "4", "pages": "2166-2179" }, "number": 85 }, { "id": "siopt-2023-017", "journal": "SIAM Journal on Optimization", "publicationYear": 2023, "title": "Riemannian minimax acceleration and lower bound", "topic": "Riemannian optimization", "publicationDate": "2023-10-18", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Open questions near the conclusion; extracted preprint lines 1180-1200", "sourceQuote": "a matching lower bound analysis like [49] is still lacking for minimax problems in Riemannian geometry.", "problemStatement": "The paper establishes minimax theory and an extragradient algorithm on geodesic metric spaces and Riemannian manifolds. Its rates are not accelerated, while curvature prevents a direct transfer of full Euclidean acceleration, and no matching Riemannian minimax lower bound is known. Determine attainable accelerated rates under geometric restrictions such as constant curvature and prove matching oracle lower bounds.", "statusEvidence": "No forward-citing primary paper or exact-query result explicitly claiming a proof or counterexample was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "minimax", "Riemannian geometry", "acceleration", "oracle lower bounds" ], "literature": [], "source": { "title": "Sion’s Minimax Theorem in Geodesic Metric Spaces and a Riemannian Extragradient Algorithm", "authors": [ "Peiyuan Zhang", "Jingzhao Zhang", "Suvrit Sra" ], "doi": "10.1137/22m1505475", "url": "https://doi.org/10.1137/22m1505475", "preprintUrl": "https://arxiv.org/abs/2202.06950", "volume": "33", "issue": "4", "pages": "2885-2908" }, "number": 86 }, { "id": "siopt-2023-011", "journal": "SIAM Journal on Optimization", "publicationYear": 2023, "title": "Coder assumption relaxation and lower bounds", "topic": "Coordinate methods", "publicationDate": "2023-10-25", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conclusion; extracted preprint lines 1231-1236", "sourceQuote": "one such question is understanding the complexity of standard optimization problem classes under our new Lipschitz condition by obtaining new oracle lower bounds.", "problemStatement": "Partition a monotone operator F:ℝ^d→ℝ^d into coordinate blocks F^j. The Lipschitz condition requires positive-semidefinite matrices Q_j with ‖F^j(x)−F^j(y)‖≤√((x−y)ᵀQ_j(x−y)); if Q̂_j is Q_j with its first j−1 block rows and columns zeroed, then L̂=√‖Σ_jQ̂_j‖. Derive oracle lower bounds for standard convex-composite and convex-concave minimax classes under this condition, thereby deciding whether CODER's dependence on L̂ and the number of blocks is optimal.", "statusEvidence": "No forward-citing primary paper or exact-query result explicitly claiming a proof or counterexample was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "cyclic coordinates", "dual averaging", "oracle lower bounds", "Lipschitz operators" ], "literature": [], "source": { "title": "Cyclic Coordinate Dual Averaging with Extrapolation", "authors": [ "Chaobing Song", "Jelena Diakonikolas" ], "doi": "10.1137/22m1470104", "url": "https://doi.org/10.1137/22m1470104", "preprintUrl": "https://arxiv.org/abs/2102.13244", "volume": "33", "issue": "4", "pages": "2935-2961" }, "number": 87 }, { "id": "focm-2023-stochastic-navier-stokes-boundary-second-moment", "journal": "Foundations of Computational Mathematics", "publicationYear": 2023, "title": "Stochastic Navier Stokes Boundary Second Moment", "topic": "Stochastic partial differential equations", "publicationDate": "2023-10-26", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Introduction, equation following no-slip boundary condition (1.2)", "sourceQuote": "it is still an open problem if the solution satisfies", "problemStatement": "Consider the two-dimensional incompressible stochastic Navier-Stokes equation on a smooth bounded domain, driven by cylindrical Wiener noise with the paper's solenoidal Hilbert-Schmidt diffusion coefficient, and impose u=0 on the boundary. Under the initial-data and diffusion growth and regularity assumptions that give the global strong pathwise solution, prove for every deterministic finite T that E[||grad u(T)||^2_L2] is finite. Existing estimates in the no-slip case reach only a possibly large stopping time, unlike the periodic-domain estimates.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "stochastic Navier-Stokes", "no-slip boundary", "moment bounds", "regularity" ], "literature": [], "source": { "title": "Error Analysis for 2D Stochastic Navier–Stokes Equations in Bounded Domains with Dirichlet Data", "authors": [ "Dominic Breit", "Andreas Prohl" ], "doi": "10.1007/s10208-023-09621-y", "url": "https://doi.org/10.1007/s10208-023-09621-y", "volume": "24", "issue": "5", "pages": "1643-1672" }, "number": 88 }, { "id": "mor-2023-W3135365769_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2023, "title": "Determine optimal approximation constant for AnyPrice share allocations with additive valuations", "topic": "Fair Division", "publicationDate": "2023-10-26", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 5, Introduction (paragraph after Theorem 1.3); reiterated page 5 (after Theorem 1.4) and page 23, Section 6.3 (Open Questions, items 2–3).", "sourceQuote": "Finding the best approximation is open even for the case of equal entitlements.", "problemStatement": "The AnyPrice share (APS) is a fairness benchmark for allocating indivisible goods to agents with arbitrary entitlements. For additive valuations, the paper guarantees each agent at least 3/5 of her APS and shows that the exact share is not always attainable. Determine the optimal universal approximation factor and, if possible, a polynomial-time allocation algorithm achieving it.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "fair division", "indivisible goods", "anyprice share", "approximation ratio", "additive valuations", "unequal entitlements" ], "literature": [], "source": { "title": "Fair-Share Allocations for Agents with Arbitrary Entitlements", "authors": [ "Moshe Babaioff", "Tomer Ezra", "Uriel Feige" ], "doi": "10.1287/moor.2021.0199", "url": "https://doi.org/10.1287/moor.2021.0199", "volume": "49", "issue": "4", "pages": "2180-2211" }, "number": 89 }, { "id": "mor-2023-W3176856692_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2023, "title": "Convergence guarantees when stochastic gradient estimates diverge in vanishing-merit regime", "topic": "Stochastic Constrained Optimization", "publicationDate": "2023-10-30", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 4.4 (Complementary Events), page 25.", "sourceQuote": "However, it remains an open question whether it is possible to obtain a similar guarantee if/when a subsequence of stochastic gradient estimates diverges.", "problemStatement": "A stochastic sequential quadratic programming method uses noisy objective gradients and an exact-penalty merit parameter to solve equality-constrained problems even when the constraint Jacobian is rank deficient. The source proves convergence in several regimes but leaves open the event where the merit parameter tends to zero while gradient errors are unbounded. Establish convergence guarantees in that vanishing-merit, unbounded-noise regime or construct a failure example.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "stochastic SQP", "equality constraints", "rank-deficient Jacobians", "merit parameter", "constraint violation stationarity", "unbounded gradient noise" ], "literature": [], "source": { "title": "A Stochastic Sequential Quadratic Optimization Algorithm for Nonlinear-Equality-Constrained Optimization with Rank-Deficient Jacobians", "authors": [ "Albert S. Berahas", "Frank E. Curtis", "Michael J. O’Neill", "Daniel P. Robinson" ], "doi": "10.1287/moor.2021.0154", "url": "https://doi.org/10.1287/moor.2021.0154", "volume": "49", "issue": "4", "pages": "2212-2248" }, "number": 90 }, { "id": "mor-2023-W4388047886_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2023, "title": "Determine diameter of acyclic and preorder preference spaces under top-difference semimetric", "topic": "Social Choice Theory", "publicationDate": "2023-10-31", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 4, page 25 (end of Section 4, after Table 1)", "sourceQuote": "diam_D(A(X))gediam_D(P(X))≥ n2^(n-1)+2-2^(⌊ n/2⌋)-2^(⌈ n/2⌉). We do not presently know whether or not either of these inequalities hold as equalities.", "problemStatement": "The top-difference semimetric compares two preference relations by summing, over every subset of alternatives, the symmetric difference between their maximal-element sets. The paper studies this metric on acyclic orders and on preorders. Determine the exact diameter of either class as a function of the number n of alternatives.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "preference relations", "acyclic orders", "preorders", "metric diameter", "finite metric spaces", "choice correspondences" ], "literature": [], "source": { "title": "A Class of Dissimilarity Semimetrics for Preference Relations", "authors": [ "Hiroki Nishimura", "Efe A. Ok" ], "doi": "10.1287/moor.2022.0351", "url": "https://doi.org/10.1287/moor.2022.0351", "volume": "49", "issue": "4", "pages": "2249-2270" }, "number": 91 }, { "id": "mor-2023-W2947637889_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2023, "title": "Extend vector-contraction inequalities for Rademacher complexity to general norms", "topic": "Rademacher Complexity", "publicationDate": "2023-11-01", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 2, Introduction (discussion following margin-based bounds and contraction lemmas); reiterated on page 8, start of Section 4.2 (margin-based generalization bounds).", "sourceQuote": "extending the vector-contraction inequality of [17] to an arbitrary norm setting (or even the case of general ell_q-norms) remains an open question", "problemStatement": "Rademacher complexity measures how well a hypothesis class correlates with random signs and is a central tool for learning-theory generalization bounds. Vector-contraction inequalities transfer Lipschitz losses to vector-valued hypothesis classes, but the cited theorem covers the Euclidean norm and only limited non-Euclidean cases. Prove an analogous contraction inequality for arbitrary norms, including general ℓ_q norms, with sharp dimension dependence.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "rademacher complexity", "vector contraction", "lipschitz functions", "general norms", "ell_q norms", "generalization bounds" ], "literature": [], "source": { "title": "Generalization Bounds in the Predict-Then-Optimize Framework", "authors": [ "Othman El Balghiti", "Adam N. Elmachtoub", "Paul Grigas", "Ambuj Tewari" ], "doi": "10.1287/moor.2022.1330", "url": "https://doi.org/10.1287/moor.2022.1330", "volume": "48", "issue": "4", "pages": "2043-2065" }, "number": 92 }, { "id": "siopt-2023-002", "journal": "SIAM Journal on Optimization", "publicationYear": 2023, "title": "Universal nonsmooth conditional gradient", "topic": "First-order methods", "publicationDate": "2023-11-03", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Concluding discussion; extracted preprint lines 1541-1546", "sourceQuote": "is unknown whether it can be generalized to a universal method", "problemStatement": "Let X have diameter D_X and suppose f(y)≤f(x)+⟨g_x,y−x⟩+[M_ν/(1+ν)]‖x−y‖^(1+ν) for exponent ν∈[0,1], where g_x is a gradient or subgradient, ν=0 is the Lipschitz nonsmooth endpoint, and ν=1 is smooth. A known nonsmooth conditional-gradient method uses O((M_0D_X/ε)^2) subgradient calls and the same number of linear-optimization calls, whereas UCGS needs O((M_0D_X/ε)^4) linear calls at ν=0. Develop one method valid for every ν∈[0,1] that retains the better ν=0 linear-oracle bound and improves UCGS for ν∈(0,1].", "statusEvidence": "No forward-citing primary paper or exact-query result explicitly claiming a proof or counterexample was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "conditional gradient", "universality", "Holder smoothness", "linear minimization oracle" ], "literature": [], "source": { "title": "Universal Conditional Gradient Sliding for Convex Optimization", "authors": [ "Yuyuan Ouyang", "Trevor Squires" ], "doi": "10.1137/21m1406234", "url": "https://doi.org/10.1137/21m1406234", "preprintUrl": "https://arxiv.org/abs/2103.11026", "volume": "33", "issue": "4", "pages": "2962-2987" }, "number": 93 }, { "id": "mp-2023-037", "journal": "Mathematical Programming", "publicationYear": 2023, "title": "Asymptotic beta hierarchy limit", "topic": "Combinatorial optimization and graph theory", "publicationDate": "2023-11-20", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Derived lower bounds", "sourceQuote": "Figure 2 gives rise to the question whether 8β_k/(k(k−1)) tends to 1 as k tends to infinity.", "problemStatement": "The paper computes semidefinite-programming lower bounds β_k used in a hierarchy for crossing numbers. Determine whether the normalized sequence 8β_k divided by k(k−1) converges to one. A positive answer would support an asymptotically exact lower-bound route toward Zarankiewicz’s conjecture.", "statusEvidence": "No primary paper explicitly claiming a proof or counterexample to this precise formulation was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "crossing number", "semidefinite hierarchy", "asymptotic limit", "Zarankiewicz conjecture" ], "literature": [], "source": { "title": "New lower bounds on crossing numbers of Kₘ,ₙ from semidefinite programming", "authors": [ "Daniel Brosch", "Sven C. Polak" ], "doi": "10.1007/s10107-023-02028-1", "url": "https://doi.org/10.1007/s10107-023-02028-1", "volume": "207", "issue": "1-2", "pages": "693-715" }, "number": 94 }, { "id": "mp-2023-036", "journal": "Mathematical Programming", "publicationYear": 2023, "title": "Zarankiewicz crossing number conjecture", "topic": "Combinatorial optimization and graph theory", "publicationDate": "2023-11-20", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Introduction", "sourceQuote": "Zarankiewicz conjectured that cr(Kₘ,ₙ)=Z(m,n), where Z(m,n) is the Zarankiewicz number.", "problemStatement": "Determine the crossing number of every complete bipartite graph Kₘ,ₙ. Zarankiewicz’s conjecture asserts that the standard drawing with Z(m,n) crossings is optimal for all m and n. The conjecture is proved for limited parameter ranges but remains open in general.", "statusEvidence": "No primary paper explicitly claiming a proof or counterexample to this precise formulation was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "crossing number", "complete bipartite graph", "Zarankiewicz conjecture", "graph drawing" ], "literature": [], "source": { "title": "New lower bounds on crossing numbers of Kₘ,ₙ from semidefinite programming", "authors": [ "Daniel Brosch", "Sven C. Polak" ], "doi": "10.1007/s10107-023-02028-1", "url": "https://doi.org/10.1007/s10107-023-02028-1", "volume": "207", "issue": "1-2", "pages": "693-715" }, "number": 95 }, { "id": "mor-2023-W4388912937_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2023, "title": "Determine the asymptotically tight rainbow cycle number in bounded-partite digraphs", "topic": "Extremal Digraph Theory", "publicationDate": "2023-11-22", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 5, end of Section 1.2 (immediately after the discussion of \\(R(2)\\le 2\\)); reiterated on page 16 right after Theorem 6 in Section 5.", "sourceQuote": "better upper bounds on R(d) is a natural combinatorial question, and better upper bounds to R(d) imply the existence of better relaxations of EFX allocations.", "problemStatement": "The rainbow cycle number R(d) is the largest number of parts in a directed multipartite graph whose parts have size at most d, every pair of parts supplies incoming neighbors to one another, and no directed cycle visits a part more than once. The source gives nonmatching bounds. Determine the asymptotic growth of R(d), in particular whether R(d) = O(d).", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "fair division", "envy freeness up to any good", "extremal digraph theory", "multipartite digraphs", "rainbow cycles", "Ramsey-type bounds" ], "literature": [], "source": { "title": "Improving Envy Freeness up to Any Good Guarantees Through Rainbow Cycle Number", "authors": [ "Bhaskar Ray Chaudhury", "Jugal Garg", "Kurt Mehlhorn", "Ruta Mehta", "Pranabendu Misra" ], "doi": "10.1287/moor.2021.0252", "url": "https://doi.org/10.1287/moor.2021.0252", "volume": "49", "issue": "4", "pages": "2323-2340" }, "number": 96 }, { "id": "mor-2023-W4389193896_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2023, "title": "Determine the MMS guarantee of EFX pure Nash equilibria for two agents", "topic": "Fair Division", "publicationDate": "2023-11-30", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 22, Section 5 (Discussion)", "sourceQuote": "Theorems 4.5 and 4.7 leave an open question on the MMS guarantee that the equilibria of mechanisms that always have PNE and these are EFX.", "problemStatement": "For two agents with additive values, a maximin share is the value an agent can guarantee by partitioning the goods into two bundles and receiving the worse one. The mechanisms studied always have a pure Nash equilibrium and every equilibrium allocation is EFX. Determine the tight fraction of each agent's maximin share guaranteed at every such equilibrium.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "fair division", "indivisible goods", "pure nash equilibrium", "EFX", "maximin share", "mechanism design" ], "literature": [], "source": { "title": "Allocating Indivisible Goods to Strategic Agents: Pure Nash Equilibria and Fairness", "authors": [ "Georgios Amanatidis", "Georgios Birmpas", "Federico Fusco", "Philip Lazos", "Stefano Leonardi", "Rebecca Reiffenhäuser" ], "doi": "10.1287/moor.2022.0058", "url": "https://doi.org/10.1287/moor.2022.0058", "volume": "49", "issue": "4", "pages": "2425-2445" }, "number": 97 }, { "id": "mor-2023-W4389224581_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2023, "title": "Prove square-root lower bound for LP-compatible policies in degenerate restless bandits", "topic": "Restless Bandits", "publicationDate": "2023-12-01", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 4.3.2 (page 16–18, around the end of the section containing 'We actually conjecture that ...')", "sourceQuote": "This implies that as long as the RB is degenerate, we can not expect a faster than square root convergence to the LP relaxation bound", "problemStatement": "A finite-horizon restless bandit is degenerate when every optimal fluid linear-program solution has a time at which no state is split between active and passive actions. The paper exhibits a degenerate instance where every LP-compatible policy has an optimality gap of order 1/√N. Prove or refute that an Ω(1/√N) gap, with a matching upper bound, holds for every degenerate instance and every LP-compatible policy.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "restless bandits", "LP relaxation", "asymptotic optimality", "degeneracy", "convergence rate", "finite horizon" ], "literature": [], "source": { "title": "Linear Program-Based Policies for Restless Bandits: Necessary and Sufficient Conditions for (Exponentially Fast) Asymptotic Optimality", "authors": [ "Nicolas Gast", "Bruno Gaujal", "Chen Yan" ], "doi": "10.1287/moor.2022.0101", "url": "https://doi.org/10.1287/moor.2022.0101", "preprintUrl": "https://inria.hal.science/hal-03262307v4/file/LP_finite_infinite.pdf", "volume": "49", "issue": "4", "pages": "2468-2491" }, "number": 98 }, { "id": "mor-2023-W4389224571_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2023, "title": "Bound lattice width of lattice-free polytopes using only determinant minor parameter", "topic": "Geometry Of Numbers", "publicationDate": "2023-12-01", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 3, Introduction (discussion following Theorem 4)", "sourceQuote": "It is open whether the lattice width of lattice-free polytopes can be bounded solely by Delta_n(A).", "problemStatement": "A full-dimensional polyhedron is lattice-free if its interior contains no integer point, and its lattice width is the smallest span measured in an integer direction. Let Δ_n(A) be the largest absolute full-size subdeterminant of its integer constraint matrix A. Determine whether every such polyhedron has width bounded by a function of Δ_n(A) alone, independent of the ambient dimension.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "integer optimization", "lattice-free polyhedra", "lattice width", "subdeterminants", "flatness", "polyhedral theory" ], "literature": [], "source": { "title": "Proximity and Flatness Bounds for Linear Integer Optimization", "authors": [ "Marcel Celaya", "Stefan Kuhlmann", "Joseph Paat", "Robert Weismantel" ], "doi": "10.1287/moor.2022.0335", "url": "https://doi.org/10.1287/moor.2022.0335", "volume": "49", "issue": "4", "pages": "2446-2467" }, "number": 99 }, { "id": "mor-2023-W4389389242_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2023, "title": "Lower bounds on acceleration thresholds for stochastic momentum methods under growth condition", "topic": "Stochastic Convex Optimization", "publicationDate": "2023-12-06", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 27, Section 7 (Conclusion)", "sourceQuote": "it is unclear whether the condition of having a small δ for different methods is necessary for acceleration; and how sharp such a condition is.", "problemStatement": "Nesterov's accelerated method, robust momentum, and implicit accelerated dual averaging minimize an L-smooth, m-strongly convex objective with condition number κ=L/m. Their gradient oracle has mean-zero error satisfying E‖error(x)‖²≤δ‖∇f(x)‖²+σ². Determine sharp necessary thresholds on δ for these methods to achieve accelerated bias contraction 1−Θ(1/√κ), rather than the 1−Θ(1/κ) scale of stochastic gradient descent.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "stochastic optimization", "accelerated methods", "growth condition", "multiplicative noise", "lower bounds", "momentum methods" ], "literature": [], "source": { "title": "Convergence Analysis of Accelerated Stochastic Gradient Descent Under the Growth Condition", "authors": [ "You-Lin Chen", "Sen Na", "Mladen Kolar" ], "doi": "10.1287/moor.2021.0293", "url": "https://doi.org/10.1287/moor.2021.0293", "volume": "49", "issue": "4", "pages": "2492-2526" }, "number": 100 }, { "id": "mor-2023-W4389673940_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2023, "title": "Determine complexity of minimizing sum of boundary cut sizes in hypergraph k-partitions", "topic": "Hypergraph K Partitioning", "publicationDate": "2023-12-13", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 24, Section 7 (Conclusion)", "sourceQuote": "Resolving the complexity of this variant of the hypergraph k-partitioning problem for k ≥ 5 remains open.", "problemStatement": "For a hypergraph, the boundary of a vertex part is the set of hyperedges meeting both that part and its complement. For fixed k≥5, minimize the sum of the k boundary cardinalities over all k-partitions of the vertices. Decide whether this unweighted problem is polynomial-time solvable for each fixed k or is NP-hard.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "hypergraph partitioning", "k-partition", "submodular partitioning", "computational complexity", "cut function", "polynomial-time algorithm" ], "literature": [], "source": { "title": "Counting and Enumerating Optimum Cut Sets for Hypergraph k-Partitioning Problems for Fixed k", "authors": [ "Calvin Beideman", "Karthekeyan Chandrasekaran", "Weihang Wang" ], "doi": "10.1287/moor.2022.0259", "url": "https://doi.org/10.1287/moor.2022.0259", "volume": "49", "issue": "4", "pages": "2579-2601" }, "number": 101 }, { "id": "mp-2023-041", "journal": "Mathematical Programming", "publicationYear": 2023, "title": "Better approximation with trivial principal partition", "topic": "Combinatorial optimization and approximation", "publicationDate": "2023-12-14", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 9, Open question 4", "sourceQuote": "Do there exist better polynomial time approximation algorithms for monotone submodular MLOP", "problemStatement": "For monotone submodular MLOP whose principal partition is trivial, design an approximation algorithm that improves on the arbitrary ordering returned by the principal-partition method. The condition requires f(S)|E|≥f(E)|S| for every subset S and f(∅)=0. A general 2026 submodular-MLOP algorithm does not improve the source’s near-two guarantee on this special class.", "statusEvidence": "No primary paper explicitly claiming a proof or counterexample to this precise formulation was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "submodular MLOP", "principal partition", "approximation algorithm", "monotone submodular function" ], "literature": [], "source": { "title": "Hardness and approximation of submodular minimum linear ordering problems", "authors": [ "Majid Farhadi", "Swati Gupta", "Shengding Sun", "Prasad Tetali", "Michael C. Wigal" ], "doi": "10.1007/s10107-023-02038-z", "url": "https://doi.org/10.1007/s10107-023-02038-z", "volume": "208", "issue": "1-2", "pages": "277-318" }, "number": 102 }, { "id": "mp-2023-038", "journal": "Mathematical Programming", "publicationYear": 2023, "title": "Fixed basis graphic MLOP hardness", "topic": "Combinatorial optimization and complexity", "publicationDate": "2023-12-14", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 9, Open question 1", "sourceQuote": "For what families of trees is this problem NP-hard?", "problemStatement": "Given a graph G, a fixed spanning tree T, and a threshold, order the tree edges to minimize the aggregate latest position on each fundamental cycle. Classify tree families for which this fixed-basis graphic MLOP problem is NP-hard. Hardness is known when T is a star, while even the path case remains open.", "statusEvidence": "No primary paper explicitly claiming a proof or counterexample to this precise formulation was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "minimum linear ordering", "spanning tree", "fixed basis", "NP-hardness" ], "literature": [], "source": { "title": "Hardness and approximation of submodular minimum linear ordering problems", "authors": [ "Majid Farhadi", "Swati Gupta", "Shengding Sun", "Prasad Tetali", "Michael C. Wigal" ], "doi": "10.1007/s10107-023-02038-z", "url": "https://doi.org/10.1007/s10107-023-02038-z", "volume": "208", "issue": "1-2", "pages": "277-318" }, "number": 103 }, { "id": "mp-2023-040", "journal": "Mathematical Programming", "publicationYear": 2023, "title": "Hardness of ordering problems on regular graphs", "topic": "Combinatorial optimization and complexity", "publicationDate": "2023-12-14", "recordType": "reviewed", "status": "partial_progress", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 9, Open question 3", "sourceQuote": "Are MLA, MLVC, and MSVC NP-hard for the family of simple regular graphs?", "problemStatement": "Determine unconditional NP-hardness of minimum linear arrangement, minimum latency vertex cover, and minimum sum vertex cover on simple regular graphs. The source proves that their decision versions are equivalent on regular graphs. Later work establishes Unique-Games-conditional hardness of approximation for regular-graph MSVC, but not the requested unconditional theorem.", "statusEvidence": "It proves Unique-Games-conditional hardness of approximation for MSVC on regular graphs; via the source equivalence this is strong conditional evidence, not an unconditional NP-hardness proof.", "verificationNote": "This status is tied to the cited primary-literature result and the precise scope stated above.", "keywords": [ "regular graphs", "minimum linear arrangement", "minimum sum vertex cover", "NP-hardness" ], "literature": [ { "title": "Some Results on Approximability of Minimum Sum Vertex Cover", "url": "https://doi.org/10.1145/3716551", "relation": "It proves Unique-Games-conditional hardness of approximation for MSVC on regular graphs; via the source equivalence this is strong conditional evidence, not an unconditional NP-hardness proof." } ], "source": { "title": "Hardness and approximation of submodular minimum linear ordering problems", "authors": [ "Majid Farhadi", "Swati Gupta", "Shengding Sun", "Prasad Tetali", "Michael C. Wigal" ], "doi": "10.1007/s10107-023-02038-z", "url": "https://doi.org/10.1007/s10107-023-02038-z", "volume": "208", "issue": "1-2", "pages": "277-318" }, "number": 104 }, { "id": "mp-2023-039", "journal": "Mathematical Programming", "publicationYear": 2023, "title": "LP approximation for uniform MLSC", "topic": "Combinatorial optimization and approximation", "publicationDate": "2023-12-14", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 9, Open question 2", "sourceQuote": "Does solving the LP relaxation provide an approximation guarantee for MLSC on ℓ-uniform hypergraphs by a factor of 2−2/(1+ℓ)?", "problemStatement": "For minimum latency set cover on an ℓ-uniform hypergraph, determine whether the natural LP relaxation can always be rounded within factor 2−2/(1+ℓ). The factor is proved for regular ℓ-uniform hypergraphs and matches a general integrality-gap lower bound. The nonregular case remains unsettled.", "statusEvidence": "No primary paper explicitly claiming a proof or counterexample to this precise formulation was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "minimum latency set cover", "uniform hypergraph", "LP relaxation", "approximation" ], "literature": [], "source": { "title": "Hardness and approximation of submodular minimum linear ordering problems", "authors": [ "Majid Farhadi", "Swati Gupta", "Shengding Sun", "Prasad Tetali", "Michael C. Wigal" ], "doi": "10.1007/s10107-023-02038-z", "url": "https://doi.org/10.1007/s10107-023-02038-z", "volume": "208", "issue": "1-2", "pages": "277-318" }, "number": 105 }, { "id": "mp-2023-042", "journal": "Mathematical Programming", "publicationYear": 2023, "title": "Sublinear approximation for symmetric submodular MLOP", "topic": "Combinatorial optimization and approximation", "publicationDate": "2023-12-14", "recordType": "reviewed", "status": "resolved", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 9, Open question 5", "sourceQuote": "Can symmetric submodular MLOP over a ground set E be approximated to a factor better than O(|E|)?", "problemStatement": "Find a polynomial-time approximation for symmetric submodular MLOP with ratio asymptotically better than linear in the ground-set size. A 2026 result gives an O(√(n/log n)) algorithm for the more general class of all nonnegative submodular functions. It also proves a matching value-oracle lower bound.", "statusEvidence": "It gives an O(√(n/log n)) approximation for general nonnegative submodular MLOP, which directly improves the linear factor for the symmetric subclass.", "verificationNote": "This status is tied to the cited primary-literature result and the precise scope stated above.", "keywords": [ "submodular MLOP", "symmetric submodular function", "approximation algorithm", "value oracle" ], "literature": [ { "title": "Tight Algorithm and Hardness for Submodular Linear Ordering", "url": "https://doi.org/10.4230/LIPIcs.ICALP.2026.4", "relation": "It gives an O(√(n/log n)) approximation for general nonnegative submodular MLOP, which directly improves the linear factor for the symmetric subclass." } ], "source": { "title": "Hardness and approximation of submodular minimum linear ordering problems", "authors": [ "Majid Farhadi", "Swati Gupta", "Shengding Sun", "Prasad Tetali", "Michael C. Wigal" ], "doi": "10.1007/s10107-023-02038-z", "url": "https://doi.org/10.1007/s10107-023-02038-z", "volume": "208", "issue": "1-2", "pages": "277-318" }, "number": 106 }, { "id": "mp-2023-044", "journal": "Mathematical Programming", "publicationYear": 2023, "title": "Ideal clutters without intersecting minor are MFMC", "topic": "Combinatorial optimization and polyhedral theory", "publicationDate": "2023-12-18", "recordType": "reviewed", "status": "partial_progress", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section on k-wise intersecting clutters", "sourceQuote": "It is conjectured that ideal clutters with no intersecting minor also have the max-flow min-cut property.", "problemStatement": "Prove that an ideal clutter with no intersecting minor must have the max-flow min-cut property. The statement is equivalent to a central τ=2 conjecture in clutter theory. Later work verifies the related conjecture for coordinate-subspace multipartite clutters, leaving the general case open.", "statusEvidence": "It characterizes MFMC clutters in this coordinate-subspace class by excluding Δ₃ and the intersecting clutter Q₆; hence an ideal member with no intersecting minor is MFMC in this class.", "verificationNote": "This status is tied to the cited primary-literature result and the precise scope stated above.", "keywords": [ "clutters", "max-flow min-cut property", "intersecting minor", "idealness" ], "literature": [ { "title": "From coordinate subspaces over finite fields to ideal multipartite uniform clutters", "url": "https://doi.org/10.1007/s10107-024-02155-3", "relation": "It characterizes MFMC clutters in this coordinate-subspace class by excluding Δ₃ and the intersecting clutter Q₆; hence an ideal member with no intersecting minor is MFMC in this class." } ], "source": { "title": "Intersecting and dense restrictions of clutters in polynomial time", "authors": [ "Martin Drees" ], "doi": "10.1007/s10107-023-02034-3", "url": "https://doi.org/10.1007/s10107-023-02034-3", "volume": "206", "issue": "1-2", "pages": "461-477" }, "number": 107 }, { "id": "mp-2023-043", "journal": "Mathematical Programming", "publicationYear": 2023, "title": "K wise intersecting clutters nonideal", "topic": "Combinatorial optimization and polyhedral theory", "publicationDate": "2023-12-18", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Introduction", "sourceQuote": "Abdi, Cornuéjols, Huynh and Lee conjectured that for k≥4 these clutters are non-ideal.", "problemStatement": "A nontrivial clutter is k-wise intersecting when every k members share an element but no single element covers all members. Prove that every such clutter is nonideal for k=4, or at least for some universal k≥4. The conjecture is known for binary clutters but remains open in general.", "statusEvidence": "No primary paper explicitly claiming a proof or counterexample to this precise formulation was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "clutters", "k-wise intersection", "ideal matrix", "set covering" ], "literature": [], "source": { "title": "Intersecting and dense restrictions of clutters in polynomial time", "authors": [ "Martin Drees" ], "doi": "10.1007/s10107-023-02034-3", "url": "https://doi.org/10.1007/s10107-023-02034-3", "volume": "206", "issue": "1-2", "pages": "461-477" }, "number": 108 }, { "id": "mp-2023-046", "journal": "Mathematical Programming", "publicationYear": 2023, "title": "ADMM dual residual rate with two strongly convex terms", "topic": "Convex and nonlinear optimization", "publicationDate": "2023-12-26", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 3, after Theorem 5", "sourceQuote": "We could not manage to guess a closed-form formula for the residual dual in this case.", "problemStatement": "For ADMM with both objective terms strongly convex, determine a closed-form worst-case convergence rate for the dual residual. The paper obtains exact formulas for related dual-gap and primal-residual quantities in a weaker setting but reports no formula for this measure. The question calls for both the rate and its parameter dependence.", "statusEvidence": "No primary paper explicitly claiming a proof or counterexample to this precise formulation was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "ADMM", "dual residual", "worst-case convergence", "strong convexity" ], "literature": [], "source": { "title": "The exact worst-case convergence rate of the alternating direction method of multipliers", "authors": [ "Moslem Zamani", "Hadi Abbaszadehpeivasti", "Etienne de Klerk" ], "doi": "10.1007/s10107-023-02037-0", "url": "https://doi.org/10.1007/s10107-023-02037-0", "volume": "208", "issue": "1-2", "pages": "243-276" }, "number": 109 }, { "id": "mp-2023-045", "journal": "Mathematical Programming", "publicationYear": 2023, "title": "ADMM two strongly convex gap and primal residual rates", "topic": "Convex and nonlinear optimization", "publicationDate": "2023-12-26", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 3, after Theorem 5", "sourceQuote": "we conjecture, under the assumptions of Theorem 3, if g is c₂-strongly convex relative to ‖·‖_B, Algorithm 1 enjoys the following convergence rates: D(λ*)−D(λᴺ)≤[‖λ⁰−λ*‖²+t²‖z⁰−z*‖²_B]/[4Nt+2c₁c₂/(c₁+c₂)], ‖Axᴺ+Bzᴺ−b‖≤√[‖λ⁰−λ*‖²+t²‖z⁰−z*‖²_B]/[Nt+c₁c₂/(c₁+c₂)].", "problemStatement": "Consider ADMM with penalty t for min f(x)+g(z) subject to Ax+Bz=b, where f is c₁-strongly convex in ‖·‖_A and g is c₂-strongly convex in ‖·‖_B. Let R₀²=‖λ⁰−λ*‖²+t²‖z⁰−z*‖_B². Prove the PEP-inferred bounds D(λ*)−D(λᴺ)≤R₀²/[4Nt+2c₁c₂/(c₁+c₂)] and ‖Axᴺ+Bzᴺ−b‖≤R₀/[Nt+c₁c₂/(c₁+c₂)] for N≥4 and t≤c₁.", "statusEvidence": "No primary paper explicitly claiming a proof or counterexample to this precise formulation was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "ADMM", "worst-case convergence", "strong convexity", "primal residual" ], "literature": [], "source": { "title": "The exact worst-case convergence rate of the alternating direction method of multipliers", "authors": [ "Moslem Zamani", "Hadi Abbaszadehpeivasti", "Etienne de Klerk" ], "doi": "10.1007/s10107-023-02037-0", "url": "https://doi.org/10.1007/s10107-023-02037-0", "volume": "208", "issue": "1-2", "pages": "243-276" }, "number": 110 }, { "id": "mor-2024-W4390588799_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2024, "title": "Certificate for unboundedness of polynomial optimization when moment-ray condition fails", "topic": "Polynomial Optimization", "publicationDate": "2024-01-04", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 7 (Conclusions and discussions), page 24 (Question 7.1); see also Appendix A discussion around pages 28–29.", "sourceQuote": "When (A.8) is infeasible, what is a computationally convenient certificate for unboundedness of (A.1)?", "problemStatement": "Let g be a polynomial minimized over a basic semialgebraic set K. The existing moment-ray certificate searches for a truncated moment sequence supported on recession directions of K whose pairing with the leading homogeneous part of g is negative, but this system can be infeasible even when the infimum is −∞. Develop a computationally checkable certificate of unboundedness that also covers those cases.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "polynomial optimization", "unboundedness detection", "moment sos relaxations", "homogenization", "Positivstellensatz certificates", "semialgebraic sets" ], "literature": [], "source": { "title": "The Multi-Objective Polynomial Optimization", "authors": [ "Jiawang Nie", "Zi Yang" ], "doi": "10.1287/moor.2023.0200", "url": "https://doi.org/10.1287/moor.2023.0200", "volume": "49", "issue": "4", "pages": "2723-2748" }, "number": 111 }, { "id": "siopt-2024-021", "journal": "SIAM Journal on Optimization", "publicationYear": 2024, "title": "CS LME without archimedean condition", "topic": "Polynomial optimization", "publicationDate": "2024-01-05", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conclusion; extracted preprint lines 2188-2194", "sourceQuote": "whether the CS-LME approach has guaranteed asymptotic or finite convergence without the archimedean condition", "problemStatement": "Correlatively sparse Lagrange multiplier expressions (CS-LMEs) yield sparse moment-SOS relaxations for polynomial optimization. The paper proves convergence under Archimedean assumptions on the local quadratic modules. Determine whether asymptotic or finite convergence holds for every sparsity block without that compactness condition, and identify any necessary replacement assumptions.", "statusEvidence": "No forward-citing primary paper or exact-query result explicitly claiming a proof or counterexample was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "Lagrange multiplier expressions", "correlative sparsity", "moment-SOS", "Archimedean condition" ], "literature": [], "source": { "title": "A Correlatively Sparse Lagrange Multiplier Expression Relaxation for Polynomial Optimization", "authors": [ "Zheng Qu", "Xindong Tang" ], "doi": "10.1137/22m1515689", "url": "https://doi.org/10.1137/22m1515689", "preprintUrl": "https://arxiv.org/abs/2208.03979", "volume": "34", "issue": "1", "pages": "127-162" }, "number": 112 }, { "id": "siopt-2024-026", "journal": "SIAM Journal on Optimization", "publicationYear": 2024, "title": "Aggregation certificate full convex hull", "topic": "Quadratic optimization", "publicationDate": "2024-01-05", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conjecture 3.3 introduction; extracted preprint lines 668-674", "sourceQuote": "We conjecture that under hidden hyperplane convexity assumption, in this case conv(S) is also Rn.", "problemStatement": "The paper characterizes when aggregations certify that a strict quadratic system has empty convex complement, except when every semidefinite nonnegative aggregation collapses to a negative constant. Under hidden hyperplane convexity the authors conjecture that this exceptional case still implies conv(S)=R^n. Prove the aggregation certificate is complete or give a counterexample.", "statusEvidence": "No forward-citing primary paper or exact-query result explicitly claiming a proof or counterexample was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "aggregation certificates", "hidden hyperplane convexity", "full convex hull", "quadratic systems" ], "literature": [], "source": { "title": "Aggregations of Quadratic Inequalities and Hidden Hyperplane Convexity", "authors": [ "Grigoriy Blekherman", "Santanu S. Dey", "Shengding Sun" ], "doi": "10.1137/22m1528215", "url": "https://doi.org/10.1137/22m1528215", "preprintUrl": "https://arxiv.org/abs/2210.01722", "volume": "34", "issue": "1", "pages": "98-126" }, "number": 113 }, { "id": "siopt-2024-024", "journal": "SIAM Journal on Optimization", "publicationYear": 2024, "title": "Hidden hyperplane infinite aggregations", "topic": "Quadratic optimization", "publicationDate": "2024-01-05", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conjecture 3.1; cached preprint lines 643-646", "sourceQuote": "There exists a set S described by quadratic inequalities satisfying hidden hyperplane convexity, such that conv(S) cannot be described by finitely many good aggregations.", "problemStatement": "For a set S defined by strict quadratic inequalities, a good aggregation is a nonnegative combination of the inequalities whose convex component contains S. Hidden hyperplane convexity guarantees that all good aggregations describe conv(S), and additional triple-PDLC assumptions give finiteness. Construct an HHC example for which no finite family of good aggregations describes conv(S), or prove that none exists.", "statusEvidence": "No forward-citing primary paper or exact-query result explicitly claiming a proof or counterexample was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "quadratic inequalities", "hidden hyperplane convexity", "convex hull", "aggregation" ], "literature": [], "source": { "title": "Aggregations of Quadratic Inequalities and Hidden Hyperplane Convexity", "authors": [ "Grigoriy Blekherman", "Santanu S. Dey", "Shengding Sun" ], "doi": "10.1137/22m1528215", "url": "https://doi.org/10.1137/22m1528215", "preprintUrl": "https://arxiv.org/abs/2210.01722", "volume": "34", "issue": "1", "pages": "98-126" }, "number": 114 }, { "id": "siopt-2024-025", "journal": "SIAM Journal on Optimization", "publicationYear": 2024, "title": "Six aggregations necessary", "topic": "Quadratic optimization", "publicationDate": "2024-01-05", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conjecture 3.2; extracted preprint lines 662-665", "sourceQuote": "There exists a set S described by three quadratic inequalities satisfying PDLC", "problemStatement": "For three quadratic inequalities satisfying the positive-definite linear-combination condition, the paper proves that at most six good aggregations describe the convex hull and exhibits an example needing four. Determine whether the upper bound is sharp. Specifically, construct a PDLC instance whose convex hull requires exactly six aggregations, or lower the universal bound.", "statusEvidence": "No forward-citing primary paper or exact-query result explicitly claiming a proof or counterexample was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "PDLC", "convex hull", "aggregation", "sharp bound" ], "literature": [], "source": { "title": "Aggregations of Quadratic Inequalities and Hidden Hyperplane Convexity", "authors": [ "Grigoriy Blekherman", "Santanu S. Dey", "Shengding Sun" ], "doi": "10.1137/22m1528215", "url": "https://doi.org/10.1137/22m1528215", "preprintUrl": "https://arxiv.org/abs/2210.01722", "volume": "34", "issue": "1", "pages": "98-126" }, "number": 115 }, { "id": "mor-2024-W4390763638_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2024, "title": "Design mechanisms to mitigate information leakage and privacy paradox in data marketplaces", "topic": "Mechanism Design Privacy", "publicationDate": "2024-01-09", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 15, Section 5 (Concluding Remarks)", "sourceQuote": "what approaches can be brought to bear to mitigate the impact of information leakage and the privacy paradox in data marketplaces?", "problemStatement": "In a data marketplace, purchasing one person's data may reveal information about other agents because their records are correlated. This creates a privacy paradox: an agent can suffer leakage even after declining to sell, undermining ordinary consent and compensation rules. Design and analyze a mechanism that limits such external leakage while still enabling efficient data trade.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "data marketplaces", "privacy externalities", "information leakage", "correlated data", "mechanism design", "regulation" ], "literature": [], "source": { "title": "The Privacy Paradox and Optimal Bias–Variance Trade-offs in Data Acquisition", "authors": [ "Guocheng Liao", "Yu Su", "Juba Ziani", "Adam Wierman", "Jianwei Huang" ], "doi": "10.1287/moor.2023.0022", "url": "https://doi.org/10.1287/moor.2023.0022", "volume": "49", "issue": "4", "pages": "2749-2767" }, "number": 116 }, { "id": "siopt-2024-038", "journal": "SIAM Journal on Optimization", "publicationYear": 2024, "title": "Integer caratheodory optimal bounds", "topic": "Integer optimization", "publicationDate": "2024-01-09", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Introduction, open questions; extracted preprint lines 104-113", "sourceQuote": "What are the optimal upper bounds for CR(C) and CRa(C) in terms of n?", "problemStatement": "CR(C) and CR_a(C) are the integer and asymptotic integer Caratheodory ranks of an n-dimensional rational cone. The paper improves the asymptotic upper bound to floor(3n/2) and gives lower examples of order 7n/6, while the best general bound for CR(C) remains much older. Determine the optimal dimension-dependent upper bounds for both ranks and close the remaining linear-factor gaps.", "statusEvidence": "No forward-citing primary paper or exact-query result explicitly claiming a proof or counterexample was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "integer Caratheodory rank", "rational cones", "upper bounds", "normal polytopes" ], "literature": [], "source": { "title": "New Bounds for the Integer Carathéodory Rank", "authors": [ "Iskander Aliev", "Martin Henk", "Mark Hogan", "Stefan Kuhlmann", "Timm Oertel" ], "doi": "10.1137/23m1561312", "url": "https://doi.org/10.1137/23m1561312", "preprintUrl": "https://arxiv.org/abs/2211.03150", "volume": "34", "issue": "1", "pages": "190-200" }, "number": 117 }, { "id": "mp-2024-first-002", "journal": "Mathematical Programming", "publicationYear": 2024, "title": "Nonmigratory throughput maximization", "topic": "Scheduling and online algorithms", "publicationDate": "2024-01-10", "recordType": "reviewed", "status": "partial_progress", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conclusion", "sourceQuote": "We leave it as an open question to develop a non-migratory constant-competitive algorithm for throughput maximization on multiple machines.", "problemStatement": "The source algorithm moves some low-laxity jobs between machines, so it does not establish that migration can be avoided. Design a constant-competitive online algorithm for preemptive throughput maximization on multiple identical machines that never migrates a job and uses neither a slack assumption nor speed augmentation. A later result achieves this under uniform slack, leaving the unrestricted model open.", "statusEvidence": "Under a uniform ε-slack assumption, the paper gives an O(1/ε)-competitive non-migratory algorithm on unrelated machines and, on identical machines, compares it with a migratory optimum. It does not remove the source question's slack-free requirement.", "verificationNote": "This status is tied to the cited primary-literature result and the precise scope stated above.", "keywords": [ "throughput maximization", "non-migratory scheduling", "online scheduling", "identical machines" ], "literature": [ { "title": "O(1/ε) Is the Answer in Online Weighted Throughput Maximization", "url": "https://doi.org/10.4230/LIPIcs.STACS.2024.32", "relation": "Under a uniform ε-slack assumption, the paper gives an O(1/ε)-competitive non-migratory algorithm on unrelated machines and, on identical machines, compares it with a migratory optimum. It does not remove the source question's slack-free requirement." } ], "source": { "title": "A competitive algorithm for throughput maximization on identical machines", "authors": [ "Benjamin Moseley", "Kirk Pruhs", "Clifford Stein", "Rudy Zhou" ], "doi": "10.1007/s10107-023-02045-0", "url": "https://doi.org/10.1007/s10107-023-02045-0", "volume": "206", "issue": "1-2", "pages": "497-514" }, "number": 118 }, { "id": "mp-2024-first-001", "journal": "Mathematical Programming", "publicationYear": 2024, "title": "Optimal throughput competitive ratio", "topic": "Scheduling and online algorithms", "publicationDate": "2024-01-10", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conclusion", "sourceQuote": "What is the right competitive ratio for throughput maximization? Can we give an improved upper bound, or any lower bound?", "problemStatement": "The paper obtains a deterministic constant-competitive online algorithm for preemptive throughput maximization on multiple identical machines, but its constant is large and unspecified. Determine the optimal competitive ratio in the unrestricted model without high-laxity or speed augmentation, either by sharpening the upper bound or by proving a nontrivial lower bound. The target should apply for any number of machines greater than one and clarify the remaining quantitative gap.", "statusEvidence": "No primary paper explicitly claiming a proof or counterexample to this precise formulation was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "throughput maximization", "competitive ratio", "online scheduling", "identical machines" ], "literature": [], "source": { "title": "A competitive algorithm for throughput maximization on identical machines", "authors": [ "Benjamin Moseley", "Kirk Pruhs", "Clifford Stein", "Rudy Zhou" ], "doi": "10.1007/s10107-023-02045-0", "url": "https://doi.org/10.1007/s10107-023-02045-0", "volume": "206", "issue": "1-2", "pages": "497-514" }, "number": 119 }, { "id": "siopt-2024-034", "journal": "SIAM Journal on Optimization", "publicationYear": 2024, "title": "Variable mass newton asymptotics", "topic": "Continuous-time optimization", "publicationDate": "2024-01-18", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 4; cached preprint lines 998-1003", "sourceQuote": "derive precise-enough asymptotic estimates for the solution of (VM-DIN-AVD) using this strategy. Whether this is possible or not remains an open question.", "problemStatement": "VM-DIN-AVD is the flow ε(t)ẍ(t)+α(t)ẋ(t)+β∇²f(x(t))ẋ(t)+∇f(x(t))=0, with variable mass ε, damping α, and β>0. It reduces to continuous Newton when ε=α=0 and to Levenberg–Marquardt when ε=0. For strongly convex f with Lipschitz gradient, determine sharp asymptotic comparisons by extending the quadratic-sandwich analysis, which currently gives no precise estimates. Also determine whether the Newton and Levenberg–Marquardt proximity and acceleration properties persist when ε is chosen as state feedback depending on x(t), rather than as a prescribed function of time.", "statusEvidence": "No forward-citing primary paper or exact-query result explicitly claiming a proof or counterexample was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "Newton flow", "inertia", "variable mass", "asymptotic analysis" ], "literature": [], "source": { "title": "Continuous Newton-like Methods Featuring Inertia and Variable Mass", "authors": [ "Camille Castera", "Hedy Attouch", "Jalal Fadili", "Peter Ochs" ], "doi": "10.1137/23m1549675", "url": "https://doi.org/10.1137/23m1549675", "preprintUrl": "https://arxiv.org/abs/2301.08726", "volume": "34", "issue": "1", "pages": "251-277" }, "number": 120 }, { "id": "mor-2024-W4391111008_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2024, "title": "Classify the complexity of Nash equilibrium existence in edge-ranking flow games", "topic": "Algorithmic Game Theory", "publicationDate": "2024-01-22", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 5.1, Remark 21 (page 38 in the provided text pagination)", "sourceQuote": "whether the problem of deciding existence of a pure Nash equilibrium in an edge-ranking game is in NP or not", "problemStatement": "In an edge-ranking flow game, each network node orders its outgoing edges and greedily routes all available supply through them subject to capacities; utility is total outgoing flow in the maximal clearing state. A pure Nash equilibrium is a profile of rankings with no profitable unilateral reorder. Determine the exact complexity of deciding whether an equilibrium exists, including whether the problem is in NP or complete for the second level of the polynomial hierarchy.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "flow allocation games", "edge-ranking strategies", "pure Nash equilibrium", "complexity theory", "polynomial hierarchy", "Sigma2p completeness" ], "literature": [], "source": { "title": "Flow Allocation Games", "authors": [ "Nils Bertschinger", "Martin Hoefer", "Daniel Schmand" ], "doi": "10.1287/moor.2022.0355", "url": "https://doi.org/10.1287/moor.2022.0355", "volume": "50", "issue": "1", "pages": "68-89" }, "number": 121 }, { "id": "mor-2024-W4391315199_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2024, "title": "Extend exchangeable joint-mix optimality for uncertain subsets under general convex costs", "topic": "Robust Optimal Transport", "publicationDate": "2024-01-29", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 6 (Conclusion), page 19 (in the provided parsed text; labeled page 19/32).", "sourceQuote": "the possible optimality of some exchangeable joint mix for general convex cost under uncertainty, but we do not have a proof.", "problemStatement": "A joint mix couples random variables with a prescribed common marginal distribution, and it is exchangeable when its law is invariant under coordinate permutations. The paper optimizes convex functions of subset sums over such couplings under a robust weighting of subsets. Determine whether an exchangeable coupling always attains the robust optimum for every convex integrand, or characterize the marginals and objectives for which it does.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "joint mixability", "negative dependence", "multi-marginal optimal transport", "robust optimization", "convex cost", "exchangeability" ], "literature": [], "source": { "title": "Joint Mixability and Notions of Negative Dependence", "authors": [ "Takaaki Koike", "Liyuan Lin", "Ruodu Wang" ], "doi": "10.1287/moor.2022.0121", "url": "https://doi.org/10.1287/moor.2022.0121", "volume": "49", "issue": "4", "pages": "2786-2802" }, "number": 122 }, { "id": "mor-2024-W4391601457_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2024, "title": "Large-deviation bounds for stake-volume Markov chain in intermediate deviation regime", "topic": "Large Deviations", "publicationDate": "2024-02-07", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Appendix, Section 5.1 (page 20).", "sourceQuote": "It remains open to prove such bounds in the range (lambda_-(α), lambda_+(α)).", "problemStatement": "The Markov chain starts at N and, from state n, increases to n+1 with probability n to the power −α and otherwise stays put. Its typical state at time t scales as t to the power 1/(1+α), but exponential tail bounds are known only for sufficiently small or sufficiently large normalized deviations. Extend matching large-deviation bounds to the intermediate deviation range.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "large deviations", "Markov chains", "proof of stake", "urn models", "fluid limits", "concentration inequalities" ], "literature": [], "source": { "title": "Polynomial Voting Rules", "authors": [ "Wenpin Tang", "David D. Yao" ], "doi": "10.1287/moor.2023.0080", "url": "https://doi.org/10.1287/moor.2023.0080", "volume": "50", "issue": "1", "pages": "90-106" }, "number": 123 }, { "id": "focm-2024-point-cloud-invariant-optimal-cardinality", "journal": "Foundations of Computational Mathematics", "publicationYear": 2024, "title": "Point Cloud Invariant Optimal Cardinality", "topic": "Invariant learning", "publicationDate": "2024-02-08", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 7, conclusion", "sourceQuote": "One example is studying the optimal cardinality necessary for separation.", "problemStatement": "For a point cloud X in R^(d x n) and one of the paper's permutation, orthogonal, special-orthogonal, Lorentz, general-linear, special-linear, scaling, or translation actions, a separating invariant map is constant on group orbits and injective on the quotient. Determine the minimum number of efficiently computable scalar invariant coordinates needed for separation for each action, with the stated full-rank restrictions where required. The construction gives a generic count near twice dim(R^(d x n)/G) plus one, but optimal cardinality is conjectured to be slightly smaller.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "point clouds", "separating invariants", "embedding dimension", "group actions" ], "literature": [], "source": { "title": "Low-Dimensional Invariant Embeddings for Universal Geometric Learning", "authors": [ "Nadav Dym", "Steven J. Gortler" ], "doi": "10.1007/s10208-024-09641-2", "url": "https://doi.org/10.1007/s10208-024-09641-2", "volume": "25", "issue": "2", "pages": "375-415" }, "number": 124 }, { "id": "siopt-2024-040", "journal": "SIAM Journal on Optimization", "publicationYear": 2024, "title": "Efficient riemannian proximal mapping", "topic": "Riemannian optimization", "publicationDate": "2024-02-09", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Introduction; extracted preprint lines 105-109", "sourceQuote": "it is still unknown whether Subproblem (1.4) can be solved efficiently in general.", "problemStatement": "Subproblem (1.4) is the exact Riemannian proximal mapping for a composite objective on an embedded manifold. It gives clean global and local theory, but solving it may be as hard as the original constrained nonsmooth problem; the paper instead uses an inexact Newton direction. Determine whether the exact mapping has a generally efficient algorithm, or identify a complexity obstruction and the largest tractable manifold-function classes.", "statusEvidence": "No forward-citing primary paper or exact-query result explicitly claiming a proof or counterexample was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "Riemannian proximal map", "composite optimization", "subproblem complexity" ], "literature": [], "source": { "title": "A Riemannian Proximal Newton Method", "authors": [ "Wutao Si", "P.-A. Absil", "Wen Huang", "Rujun Jiang", "Simon Vary" ], "doi": "10.1137/23m1565097", "url": "https://doi.org/10.1137/23m1565097", "preprintUrl": "https://arxiv.org/abs/2304.04032", "volume": "34", "issue": "1", "pages": "654-681" }, "number": 125 }, { "id": "focm-2024-gw-squared-distance-no-monge-map", "journal": "Foundations of Computational Mathematics", "publicationYear": 2024, "title": "GW Squared Distance No Monge Map", "topic": "Optimal transport", "publicationDate": "2024-02-15", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conjecture 3.11", "sourceQuote": "We conjecture that an optimal coupling is never induced by a map in this setting.", "problemStatement": "For compactly supported measures mu on R^n and nu on R^d with n at least d and mu absolutely continuous, the squared-distance GW problem minimizes the double integral of (||x-x'||^2-||y-y'||^2)^2 over couplings pi. Prove the tightness conjecture that there exist such mu and nu for which no optimal coupling is induced by one map, while an optimum is always supported on two graphs, either two maps or one map and one anti-map. This is an existential claim, including examples with a source density, rather than a claim about every unequal-dimensional pair.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "Gromov-Wasserstein", "Monge map", "optimal coupling", "squared distance" ], "literature": [], "source": { "title": "On the Existence of Monge Maps for the Gromov–Wasserstein Problem", "authors": [ "Théo Dumont", "Théo Lacombe", "François-Xavier Vialard" ], "doi": "10.1007/s10208-024-09643-0", "url": "https://doi.org/10.1007/s10208-024-09643-0", "preprintUrl": "https://arxiv.org/abs/2210.11945", "volume": "25", "issue": "2", "pages": "463-510" }, "number": 126 }, { "id": "siopt-2024-037", "journal": "SIAM Journal on Optimization", "publicationYear": 2024, "title": "Stiefel feasible set regularity", "topic": "Manifold optimization", "publicationDate": "2024-02-15", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 6 conclusion; extracted preprint lines 1062-1068", "sourceQuote": "When and only when LICQ holds at all feasible points of (LPS) is unknown.", "problemStatement": "For LPS, first- and second-order local conditions become globally sufficient when LICQ holds and the dimensions satisfy p+1<=n-k. The paper does not characterize when the linear constraints and Stiefel equations satisfy LICQ everywhere. Give necessary and sufficient data conditions for global LICQ, including a characterization of when a hyperplane intersection with St(n,p) remains a manifold.", "statusEvidence": "No forward-citing primary paper or exact-query result explicitly claiming a proof or counterexample was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "Stiefel manifold", "LICQ", "manifold intersection", "constraint qualification" ], "literature": [], "source": { "title": "Linear Programming on the Stiefel Manifold", "authors": [ "Mengmeng Song", "Yong Xia" ], "doi": "10.1137/23m1552243", "url": "https://doi.org/10.1137/23m1552243", "preprintUrl": "https://www.semanticscholar.org/paper/d621f16ef325c6e24b52ae5a26a023badeb4da27", "volume": "34", "issue": "1", "pages": "718-741" }, "number": 127 }, { "id": "siopt-2024-036", "journal": "SIAM Journal on Optimization", "publicationYear": 2024, "title": "Stiefel linear program hidden convexity", "topic": "Manifold optimization", "publicationDate": "2024-02-15", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 6 conclusion; extracted preprint lines 1050-1062", "sourceQuote": "The hidden convexity, particularly in the case of p-1 = n-k, remains unknown.", "problemStatement": "Linear programming on the Stiefel manifold (LPS) optimizes a linear objective over St(n,p) with k additional linear constraints. The paper proves exact SDP relaxation under broad dimension inequalities but leaves the boundary case p-1=n-k unresolved. Characterize hidden convexity and exact relaxation at that boundary.", "statusEvidence": "No forward-citing primary paper or exact-query result explicitly claiming a proof or counterexample was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "Stiefel manifold", "hidden convexity", "SDP relaxation", "linear constraints" ], "literature": [], "source": { "title": "Linear Programming on the Stiefel Manifold", "authors": [ "Mengmeng Song", "Yong Xia" ], "doi": "10.1137/23m1552243", "url": "https://doi.org/10.1137/23m1552243", "preprintUrl": "https://www.semanticscholar.org/paper/d621f16ef325c6e24b52ae5a26a023badeb4da27", "volume": "34", "issue": "1", "pages": "718-741" }, "number": 128 }, { "id": "mp-2024-first-003", "journal": "Mathematical Programming", "publicationYear": 2024, "title": "Construct orthonormal or orthogonal lattice basis", "topic": "Lattices and computational complexity", "publicationDate": "2024-02-19", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conclusion", "sourceQuote": "Therefore, being able to find orthonormal, or even orthogonal bases remains an interesting open problem.", "problemStatement": "Given a basis of a Euclidean lattice known to admit an orthonormal or orthogonal basis, construct such a basis substantially faster than the current route through shortest-vector or closest-vector oracles. The best cited reduction takes exponential time in the dimension, while the rotated-standard-lattice decision problem lies in both NP and co-NP. A polynomial-time construction would answer the strongest natural form, but any provably faster complexity bound would advance the question.", "statusEvidence": "No primary paper explicitly claiming a proof or counterexample to this precise formulation was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "lattice basis", "orthonormal basis", "orthogonal basis", "rotated standard lattice" ], "literature": [], "source": { "title": "Deciding whether a lattice has an orthonormal basis is in co-NP", "authors": [ "Christoph Hunkenschröder" ], "doi": "10.1007/s10107-023-02052-1", "url": "https://doi.org/10.1007/s10107-023-02052-1", "volume": "208", "issue": "1-2", "pages": "763-775" }, "number": 129 }, { "id": "mor-2024-W4391954556_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2024, "title": "Characterize least (a,b)-majorized elements for fixed scalings and shifts in feasible sets", "topic": "Majorization And Convex Order", "publicationDate": "2024-02-20", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 25, Section 7 (Conclusions and directions for future research), item 1 in the numbered list of future directions.", "sourceQuote": "whether the existence of least (a, b)-majorized elements for specific (sets of) values of a and b can be characterized.", "problemStatement": "An element of a feasible set is least (a,b)-majorized if every symmetric separable convex objective after the fixed scaling a and shift b is minimized there. The paper characterizes sets admitting such an element simultaneously for all scalings and shifts. Characterize the weaker property for one fixed pair (a,b), especially the unshifted case b = 0.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "majorization", "separable convex optimization", "submodular polyhedra", "base polyhedra", "resource allocation", "structural characterization" ], "literature": [], "source": { "title": "A Characterization of Simultaneous Optimization, Majorization, and (Bi-)Submodular Polyhedra", "authors": [ "Martijn H. H. Schoot Uiterkamp" ], "doi": "10.1287/moor.2023.0054", "url": "https://doi.org/10.1287/moor.2023.0054", "volume": "50", "issue": "1", "pages": "252-276" }, "number": 130 }, { "id": "focm-2024-henselian-polynomial-factorization-limit-augmentations", "journal": "Foundations of Computational Mathematics", "publicationYear": 2024, "title": "Henselian Polynomial Factorization Limit Augmentations", "topic": "Computational algebra", "publicationDate": "2024-02-21", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Introduction, discussion of limit augmentations", "sourceQuote": "The extension of the algorithm to arbitrary henselian fields is still an open problem.", "problemStatement": "Extend the terminating polynomial-factorization algorithm to arbitrary henselian valued fields. The missing case requires controlling limit augmentations rather than only the defectless situations handled in the paper. A henselian valued field supports unique lifting of simple residual factors, while limit augmentations are transfinite valuation steps that can prevent the present OM procedure from terminating.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "henselian fields", "polynomial factorization", "valuations", "limit augmentations" ], "literature": [], "source": { "title": "Polynomial Factorization Over Henselian Fields", "authors": [ "Maria Alberich-Carramiñana", "Jordi Guàrdia", "Enric Nart", "Adrien Poteaux", "Joaquim Roé", "Martin Weimann" ], "doi": "10.1007/s10208-024-09646-x", "url": "https://doi.org/10.1007/s10208-024-09646-x", "preprintUrl": "https://arxiv.org/abs/2207.02139", "volume": "25", "issue": "2", "pages": "631-681" }, "number": 131 }, { "id": "mor-2024-W4391994328_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2024, "title": "Characterize inf-convolution of multiple Lambda value-at-risk plus risk measures", "topic": "Risk Sharing", "publicationDate": "2024-02-21", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 1 Introduction (page 3) and Section 6 Conclusion (page 17).", "sourceQuote": "There are still some unsolved problems such as the inf-convolution of multiple LambdaVaR^+, and the inf-convolution of multiple LambdaVaR under heterogenous beliefs.", "problemStatement": "Lambda value-at-risk plus (ΛVaR+) is a monetary risk measure whose acceptance threshold varies with the probability level. Inf-convolution represents optimal risk sharing by splitting a loss among several agents. Characterize the inf-convolution of multiple ΛVaR+ measures, a case not covered by existing formulas for ΛVaR or mixed pairs.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "lambda value at risk", "inf-convolution", "risk sharing", "optimal allocation", "robust risk aggregation", "dependence uncertainty" ], "literature": [], "source": { "title": "Risk Sharing with Lambda Value at Risk", "authors": [ "Peng Liu" ], "doi": "10.1287/moor.2023.0246", "url": "https://doi.org/10.1287/moor.2023.0246", "volume": "50", "issue": "1", "pages": "313-333" }, "number": 132 }, { "id": "mor-2024-W4221167288_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2024, "title": "Uniqueness of Sobolev solutions to min-max variational inequality with obstacle and gradient constraints", "topic": "Variational Inequalities", "publicationDate": "2024-02-21", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 7, Section 3 (Remark 3.5).", "sourceQuote": "Uniqueness of the solution to Problem A remains an open question. Methods used in, e.g., [35], do not apply", "problemStatement": "A finite-horizon singular-controller-versus-stopper game leads to a min–max variational inequality combining an obstacle with a gradient constraint. The paper proves existence of a Sobolev solution on an unbounded domain under local ellipticity and quadratic growth. Prove uniqueness in the same strong Sobolev class, or identify additional assumptions needed for uniqueness.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "variational inequality", "gradient constraint", "obstacle problem", "uniqueness", "controller-stopper games", "Sobolev solution" ], "literature": [], "source": { "title": "Variational Inequalities on Unbounded Domains for Zero-Sum Singular Controller vs. Stopper Games", "authors": [ "Andrea Bovo", "Tiziano De Angelis", "Elena Issoglio" ], "doi": "10.1287/moor.2023.0029", "url": "https://doi.org/10.1287/moor.2023.0029", "preprintUrl": "https://arxiv.org/pdf/2203.06247", "volume": "50", "issue": "1", "pages": "277-312" }, "number": 133 }, { "id": "focm-2024-morley-cubic-3d-generalization", "journal": "Foundations of Computational Mathematics", "publicationYear": 2024, "title": "Morley Cubic 3d Generalization", "topic": "Finite element methods", "publicationDate": "2024-03-01", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Remark 7.6", "sourceQuote": "A generalization of the Morley finite element space to cubic polynomials in 3D is not known so far.", "problemStatement": "Construct a three-dimensional cubic-polynomial analogue of the Morley nonconforming finite element. Use it to characterize the relevant piecewise-linear symmetric tensor fields as second derivatives of nonconforming functions.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "Morley element", "three dimensions", "nonconforming FEM", "discrete Hessian" ], "literature": [], "source": { "title": "Discrete Helmholtz Decompositions of Piecewise Constant and Piecewise Affine Vector and Tensor Fields", "authors": [ "Philipp Bringmann", "Jonas W. Ketteler", "Mira Schedensack" ], "doi": "10.1007/s10208-024-09642-1", "url": "https://doi.org/10.1007/s10208-024-09642-1", "volume": "25", "issue": "2", "pages": "417-461" }, "number": 134 }, { "id": "mp-2024-first-005", "journal": "Mathematical Programming", "publicationYear": 2024, "title": "Full trajectory convergence of hessian barrier methods", "topic": "Conic and nonconvex optimization", "publicationDate": "2024-03-04", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conclusion", "sourceQuote": "Additionally, the question of convergence of the trajectory (xᵏ) generated by either scheme is open.", "problemStatement": "The paper gives first- and second-order Hessian-barrier algorithms, FAHBA and SAHBA, with finite worst-case iteration bounds for reaching approximate optimality conditions. Prove convergence of the entire sequence of iterates generated by either scheme, rather than only a bound on reaching an approximate point. The result should state assumptions under which the trajectory has a limit and identify the stationarity properties of that limit.", "statusEvidence": "No primary paper explicitly claiming a proof or counterexample to this precise formulation was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "Hessian barrier", "trajectory convergence", "FAHBA", "SAHBA" ], "literature": [], "source": { "title": "Hessian barrier algorithms for non-convex conic optimization", "authors": [ "Pavel Dvurechensky", "Mathias Staudigl" ], "doi": "10.1007/s10107-024-02062-7", "url": "https://doi.org/10.1007/s10107-024-02062-7", "volume": "209", "issue": "1-2", "pages": "171-229" }, "number": 135 }, { "id": "mp-2024-first-004", "journal": "Mathematical Programming", "publicationYear": 2024, "title": "Reverse inclusion for smooth lifts", "topic": "Nonconvex optimization and smooth parametrizations", "publicationDate": "2024-03-04", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Characterizations of lifts, after Theorem 3.16", "sourceQuote": "We conjecture that the reverse inclusion in Theorem 3.16 always holds", "problemStatement": "Fix a smooth parametrization from a lifted manifold onto a constraint set, a lifted point, and its image. Form the cone of all objective gradients at the image for which the lifted point is second-order critical after composition with the parametrization; the dual tangent cone of the constraint set is always contained in this gradient cone. Prove the reverse inclusion for every smooth parametrization, which is equivalent to every second-order critical lifted point mapping to a stationary point of the original problem.", "statusEvidence": "No primary paper explicitly claiming a proof or counterexample to this precise formulation was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "smooth parametrization", "nonconvex landscape", "second-order criticality", "tangent cone" ], "literature": [], "source": { "title": "The effect of smooth parametrizations on nonconvex optimization landscapes", "authors": [ "Eitan Levin", "Joe Kileel", "Nicolas Boumal" ], "doi": "10.1007/s10107-024-02058-3", "url": "https://doi.org/10.1007/s10107-024-02058-3", "volume": "209", "issue": "1-2", "pages": "63-111" }, "number": 136 }, { "id": "mp-2024-first-006", "journal": "Mathematical Programming", "publicationYear": 2024, "title": "Analytic center complexity in level constrained subproblems", "topic": "Constrained nonlinear optimization", "publicationDate": "2024-03-06", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Inexact LCPG", "sourceQuote": "It remains to show how to control the complexity of approximating the analytic center.", "problemStatement": "In the inexact level-constrained proximal-gradient framework, each convex subproblem starts from a point known to be strictly feasible but not known to lie near its analytic center. Bound the cost of moving from that point to an approximate analytic center and incorporate this phase into the subproblem's computational-complexity analysis. This would close the gap between the paper's gradient-complexity guarantees and a full interior-point cost bound.", "statusEvidence": "No primary paper explicitly claiming a proof or counterexample to this precise formulation was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "analytic center", "level-constrained method", "interior-point complexity", "inexact subproblem" ], "literature": [], "source": { "title": "Level constrained first order methods for function constrained optimization", "authors": [ "Digvijay Boob", "Qi Deng", "Guanghui Lan" ], "doi": "10.1007/s10107-024-02057-4", "url": "https://doi.org/10.1007/s10107-024-02057-4", "volume": "209", "issue": "1-2", "pages": "1-61" }, "number": 137 }, { "id": "mp-2024-first-008", "journal": "Mathematical Programming", "publicationYear": 2024, "title": "Half integral worst case TSP conjecture", "topic": "Combinatorial optimization and approximation", "publicationDate": "2024-03-06", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Introduction", "sourceQuote": "the Schalekamp, Williamson, and van Zuylen conjecture, which says that the integrality gap is achieved on instances where the LP has optimal half-integral solutions", "problemStatement": "Prove or disprove that the worst integrality gap of the subtour relaxation is attained on a traveling-salesman instance with an optimal half-integral relaxation solution. This is the Schalekamp-Williamson-van Zuylen conjecture and is stronger than merely improving the approximation ratio within selected half-integral subclasses. Restricted four-thirds results for cycle-cut and sparse-support instances do not establish the claimed global extremal reduction.", "statusEvidence": "No primary paper explicitly claiming a proof or counterexample to this precise formulation was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "traveling salesman problem", "half-integral solution", "integrality gap", "Schalekamp-Williamson-van Zuylen conjecture" ], "literature": [], "source": { "title": "Matroid-based TSP rounding for half-integral solutions", "authors": [ "Anupam Gupta", "Euiwoong Lee", "Jason Li", "Marcin Mucha", "Heather Newman", "Sherry Sarkar" ], "doi": "10.1007/s10107-024-02065-4", "url": "https://doi.org/10.1007/s10107-024-02065-4", "volume": "206", "issue": "1-2", "pages": "541-576" }, "number": 138 }, { "id": "mp-2024-first-007", "journal": "Mathematical Programming", "publicationYear": 2024, "title": "TSP subtour integrality gap four thirds", "topic": "Combinatorial optimization and approximation", "publicationDate": "2024-03-06", "recordType": "reviewed", "status": "partial_progress", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Introduction", "sourceQuote": "thereby making the first progress towards the conjectured optimal value of 4/3 in nearly half a century.", "problemStatement": "Determine the exact integrality gap of the subtour-elimination linear relaxation for the symmetric metric traveling-salesman problem. Prove the conjectured upper bound of four thirds, which would match the known lower-bound constructions. A 2025 result verifies the conjecture when an optimal relaxation solution has at most the number of vertices plus six nonzero components, but the general case remains open.", "statusEvidence": "The paper proves the four-thirds bound when an optimal subtour-relaxation solution on n vertices has at most n+6 nonzero components. This is a strict structural subclass of the general conjecture.", "verificationNote": "This status is tied to the cited primary-literature result and the precise scope stated above.", "keywords": [ "traveling salesman problem", "subtour relaxation", "integrality gap", "four-thirds conjecture" ], "literature": [ { "title": "The Integrality Gap of the Traveling Salesman Problem is 4/3 if the LP Solution Has at Most n+6 Non-zero Components", "url": "https://arxiv.org/abs/2507.07003", "relation": "The paper proves the four-thirds bound when an optimal subtour-relaxation solution on n vertices has at most n+6 nonzero components. This is a strict structural subclass of the general conjecture." } ], "source": { "title": "Matroid-based TSP rounding for half-integral solutions", "authors": [ "Anupam Gupta", "Euiwoong Lee", "Jason Li", "Marcin Mucha", "Heather Newman", "Sherry Sarkar" ], "doi": "10.1007/s10107-024-02065-4", "url": "https://doi.org/10.1007/s10107-024-02065-4", "volume": "206", "issue": "1-2", "pages": "541-576" }, "number": 139 }, { "id": "mp-2024-first-012", "journal": "Mathematical Programming", "publicationYear": 2024, "title": "Conjugacy on extended modular complexes", "topic": "Discrete convex analysis and duality", "publicationDate": "2024-03-07", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 7, Open questions, conjugacy question", "sourceQuote": "Is this also true for extended modular complexes?", "problemStatement": "For ordinary modular complexes, discrete conjugation carries an L-convex function on a complex to an L-convex conjugate on the dual complex. Prove or disprove the same conjugacy closure for extended modular complexes. A positive result would extend this central duality property of discrete convex analysis to the broader class of complexes.", "statusEvidence": "No primary paper explicitly claiming a proof or counterexample to this precise formulation was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "discrete conjugacy", "L-convexity", "extended modular complex", "duality" ], "literature": [], "source": { "title": "Generalized minimum 0-extension problem and discrete convexity", "authors": [ "Martin Dvorak", "Vladimir Kolmogorov" ], "doi": "10.1007/s10107-024-02064-5", "url": "https://doi.org/10.1007/s10107-024-02064-5", "volume": "209", "issue": "1-2", "pages": "279-322" }, "number": 140 }, { "id": "mp-2024-first-014", "journal": "Mathematical Programming", "publicationYear": 2024, "title": "Explicit symmetric fractional polymorphism", "topic": "Valued constraint satisfaction and discrete convexity", "publicationDate": "2024-03-07", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 7, Open questions, fractional polymorphism", "sourceQuote": "Is there a more explicit construction of this polymorphism?", "problemStatement": "L-convex functions on extended modular complexes are known nonconstructively to admit a binary symmetric fractional polymorphism. Give an explicit construction of that polymorphism from the complex and the function language. A constructive proof could expose how the polymorphism yields an algorithm and clarify the role of orientable modular graphs.", "statusEvidence": "No primary paper explicitly claiming a proof or counterexample to this precise formulation was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "fractional polymorphism", "L-convexity", "extended modular complex", "constructive proof" ], "literature": [], "source": { "title": "Generalized minimum 0-extension problem and discrete convexity", "authors": [ "Martin Dvorak", "Vladimir Kolmogorov" ], "doi": "10.1007/s10107-024-02064-5", "url": "https://doi.org/10.1007/s10107-024-02064-5", "volume": "209", "issue": "1-2", "pages": "279-322" }, "number": 141 }, { "id": "mp-2024-first-011", "journal": "Mathematical Programming", "publicationYear": 2024, "title": "Polynomial iteration bound for normal SDA", "topic": "Discrete convex optimization", "publicationDate": "2024-03-07", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 7, Open questions", "sourceQuote": "Is it possible to show that the normal SDA algorithm terminates after a polynomial number of iterations?", "problemStatement": "For minimizing an L-convex function on an extended modular complex, prove that the normal steepest-descent algorithm uses only polynomially many iterations. Each local minimization subproblem is polynomial-time solvable, yet no polynomial bound is known on the number of normal descent steps. Such a bound would make the normal algorithm itself polynomial time; tractability is currently obtained with a different descent variant.", "statusEvidence": "No primary paper explicitly claiming a proof or counterexample to this precise formulation was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "steepest descent", "L-convexity", "extended modular complex", "iteration complexity" ], "literature": [], "source": { "title": "Generalized minimum 0-extension problem and discrete convexity", "authors": [ "Martin Dvorak", "Vladimir Kolmogorov" ], "doi": "10.1007/s10107-024-02064-5", "url": "https://doi.org/10.1007/s10107-024-02064-5", "volume": "209", "issue": "1-2", "pages": "279-322" }, "number": 142 }, { "id": "mp-2024-first-013", "journal": "Mathematical Programming", "publicationYear": 2024, "title": "Refinement of admissible relations preserves l convexity", "topic": "Discrete convex analysis", "publicationDate": "2024-03-07", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 7, Open questions, admissible relations", "sourceQuote": "Is it true that if function f is L-convex on (Γ,⊑) then it is also L-convex on (Γ,⊑′)?", "problemStatement": "Let two admissible orientations be placed on the same modular complex, with the first relation coarser than the second. Determine whether every function that is L-convex for the coarser relation remains L-convex after the refinement. The property holds for the oriented-path case connecting discrete midpoint convexity to submodularity on an integer lattice, but the general case is neither proved nor disproved.", "statusEvidence": "No primary paper explicitly claiming a proof or counterexample to this precise formulation was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "admissible relation", "L-convexity", "modular complex", "orientation refinement" ], "literature": [], "source": { "title": "Generalized minimum 0-extension problem and discrete convexity", "authors": [ "Martin Dvorak", "Vladimir Kolmogorov" ], "doi": "10.1007/s10107-024-02064-5", "url": "https://doi.org/10.1007/s10107-024-02064-5", "volume": "209", "issue": "1-2", "pages": "279-322" }, "number": 143 }, { "id": "mp-2024-first-009", "journal": "Mathematical Programming", "publicationYear": 2024, "title": "Tractable languages as L-convex families", "topic": "Discrete convex analysis and valued constraints", "publicationDate": "2024-03-07", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 7, Open questions", "sourceQuote": "Can it be case that for every tractable finite-valued language Φ there exists an extended modular complex Γ", "problemStatement": "Characterize every tractable finite-valued constraint language as a family of functions that is L-convex on one common extended modular complex. Prove that such a complex always exists or exhibit a tractable language outside this representation. A negative answer would reveal tractable structure not captured by the discrete-convex framework developed in the paper.", "statusEvidence": "No primary paper explicitly claiming a proof or counterexample to this precise formulation was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "finite-valued language", "L-convexity", "extended modular complex", "tractability" ], "literature": [], "source": { "title": "Generalized minimum 0-extension problem and discrete convexity", "authors": [ "Martin Dvorak", "Vladimir Kolmogorov" ], "doi": "10.1007/s10107-024-02064-5", "url": "https://doi.org/10.1007/s10107-024-02064-5", "volume": "209", "issue": "1-2", "pages": "279-322" }, "number": 144 }, { "id": "mp-2024-first-010", "journal": "Mathematical Programming", "publicationYear": 2024, "title": "Zero extension and directed metric representation", "topic": "Discrete convex analysis and metric labeling", "publicationDate": "2024-03-07", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 7, Open questions", "sourceQuote": "Resolve the question above for (a) languages of the form 0-Ext[μ,F] where μ is a metric on V", "problemStatement": "For generalized zero-extension languages specified by a metric and allowed terminal sets, and for valued languages arising from directed metrics, determine whether every tractable member is L-convex on a common extended modular complex. These two families are structured test cases for the broader claim that all tractable finite-valued languages admit such a representation. Even showing that known tractable directed-star metrics admit such a complex would settle a meaningful special case.", "statusEvidence": "No primary paper explicitly claiming a proof or counterexample to this precise formulation was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "zero-extension", "directed metric", "L-convexity", "extended modular complex" ], "literature": [], "source": { "title": "Generalized minimum 0-extension problem and discrete convexity", "authors": [ "Martin Dvorak", "Vladimir Kolmogorov" ], "doi": "10.1007/s10107-024-02064-5", "url": "https://doi.org/10.1007/s10107-024-02064-5", "volume": "209", "issue": "1-2", "pages": "279-322" }, "number": 145 }, { "id": "siopt-2024-005", "journal": "SIAM Journal on Optimization", "publicationYear": 2024, "title": "Large solutions without strict feasibility", "topic": "Semidefinite optimization", "publicationDate": "2024-03-08", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Concluding section; extracted preprint lines 1629-1634", "sourceQuote": "what can we say about large solutions in semidefinite programs that are not strictly feasible?", "problemStatement": "The paper gives geometric mechanisms and certificates for exponentially large solutions in strictly feasible semidefinite programs. Its arguments do not characterize programs lying on proper faces of the positive-semidefinite cone. Develop a corresponding theory for large solutions when strict feasibility fails.", "statusEvidence": "No forward-citing primary paper or exact-query result explicitly claiming a proof or counterexample was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "large solutions", "strict feasibility", "facial structure", "bit complexity" ], "literature": [], "source": { "title": "How Do Exponential Size Solutions Arise in Semidefinite Programming?", "authors": [ "Gábor Pataki", "Aleksandr Touzov" ], "doi": "10.1137/21m1434945", "url": "https://doi.org/10.1137/21m1434945", "preprintUrl": "https://arxiv.org/abs/2103.00041", "volume": "34", "issue": "1", "pages": "977-1005" }, "number": 146 }, { "id": "siopt-2024-004", "journal": "SIAM Journal on Optimization", "publicationYear": 2024, "title": "Polynomial time SDP feasibility", "topic": "Semidefinite optimization", "publicationDate": "2024-03-08", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 1, displayed open problem; extracted preprint lines 119-125", "sourceQuote": "Can we decide feasibility of (P) in polynomial time?", "problemStatement": "Here (P) is a semidefinite feasibility system with rational input data. Feasible solutions can require exponentially many bits, obstructing standard polynomial-time arguments based on short certificates. Decide whether feasibility of arbitrary semidefinite programs can nevertheless be determined in polynomial time.", "statusEvidence": "No forward-citing primary paper or exact-query result explicitly claiming a proof or counterexample was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "semidefinite feasibility", "polynomial time", "bit complexity" ], "literature": [], "source": { "title": "How Do Exponential Size Solutions Arise in Semidefinite Programming?", "authors": [ "Gábor Pataki", "Aleksandr Touzov" ], "doi": "10.1137/21m1434945", "url": "https://doi.org/10.1137/21m1434945", "preprintUrl": "https://arxiv.org/abs/2103.00041", "volume": "34", "issue": "1", "pages": "977-1005" }, "number": 147 }, { "id": "mor-2024-W4392646050_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2024, "title": "Verify gamma^{-beta} truncation schedule meets summability condition in policy gradient algorithm", "topic": "Risk Sensitive Mdp", "publicationDate": "2024-03-11", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 12, Section 7 (Conclusion).", "sourceQuote": "gamma_m^(-β) always works. An interesting open question is to show that this sequence satisfies the conditions provided in our main theorem.", "problemStatement": "Let γₘ be positive stochastic-approximation step sizes decreasing to zero, and let Mₘ=γₘ^(−β), with 0<β<1/2, be the truncation level used to approximate the exponential risk-sensitive average cost. The varying-truncation convergence theorem requires the cumulative worst-case increase between successive truncated objectives to be summable. Prove that this explicit Mₘ sequence satisfies that summability condition under the paper's hypotheses.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "risk-sensitive MDP", "policy gradient", "stochastic approximation", "truncation schedule", "regenerative simulation", "convergence conditions" ], "literature": [], "source": { "title": "A Policy Gradient Algorithm for the Risk-Sensitive Exponential Cost MDP", "authors": [ "Mehrdad Moharrami", "Yashaswini Murthy", "Arghyadip Roy", "R. Srikant" ], "doi": "10.1287/moor.2022.0139", "url": "https://doi.org/10.1287/moor.2022.0139", "volume": "50", "issue": "1", "pages": "431-458" }, "number": 148 }, { "id": "mor-2024-W4392748341_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2024, "title": "Find stronger closed-form eigenvalue bounds for balanced independent sets in regular bipartite graphs", "topic": "Spectral Graph Theory", "publicationDate": "2024-03-13", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 7 (Concluding remarks), page 34.", "sourceQuote": "Hence, finding a stronger closed-form bound for alpha_(bal)(G) that is able to take advantage of the restriction to balanced independent sets remains a problem.", "problemStatement": "For an r-regular bipartite graph with n vertices per side, αbal is the maximum total size of edge-free subsets A and B on opposite sides with |A|=|B|. Writing λ₂ for the second adjacency eigenvalue, current symmetric semidefinite relaxations yield αbal(G)≤4bh(G), where bh(G)=nλ₂/[2(r+λ₂)]. Find a valid closed-form spectral upper bound that exploits balance more sharply and is strictly smaller for some graphs.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "balanced independent set", "bipartite regular graphs", "spectral bounds", "semidefinite programming relaxation", "lovasz theta number", "eigenvalues" ], "literature": [], "source": { "title": "Semidefinite Approximations for Bicliques and Bi-Independent Pairs", "authors": [ "Monique Laurent", "Sven Polak", "Luis Felipe Vargas" ], "doi": "10.1287/moor.2023.0046", "url": "https://doi.org/10.1287/moor.2023.0046", "volume": "50", "issue": "1", "pages": "537-572" }, "number": 149 }, { "id": "mor-2024-W4393233856_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2024, "title": "Characterize when irreducible Steiner-cut facets remain valid for global cut dominants", "topic": "Cut Polyhedra", "publicationDate": "2024-03-27", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 9 (Conclusion), paragraph titled 'Upwards validity', page 22 (of 24 in the provided text)", "sourceQuote": "whether every irreducible facet defining Steiner cut inequality for CUT^+(G, T) in fact is valid (and thus facet defining) for CUT^+(G).", "problemStatement": "A T-Steiner cut separates the terminal set T, and its cut dominant is the upper hull of all such cut incidence vectors. The source identifies irreducible facet inequalities that cannot be obtained by subdivision or gluing, but enlarging T can destroy validity in general. Decide whether every irreducible Steiner-cut facet remains valid for the global cut dominant obtained by making every vertex a terminal.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "cut dominants", "Steiner cuts", "polyhedral combinatorics", "facet validity", "terminal sets", "graph minors" ], "literature": [], "source": { "title": "Steiner Cut Dominants", "authors": [ "Michele Conforti", "Volker Kaibel" ], "doi": "10.1287/moor.2022.0280", "url": "https://doi.org/10.1287/moor.2022.0280", "volume": "50", "issue": "1", "pages": "764-781" }, "number": 150 }, { "id": "mor-2024-W4393423947_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2024, "title": "Prove covariance inequality between number in service and work in M GI n queues", "topic": "Queueing Theory", "publicationDate": "2024-04-02", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 22, Section 3 (\"Towards bounds with no large prefactors\"), immediately after Theorem 9 (Conjecture 1).", "sourceQuote": "In any M/G/n queue in which E[S^2] < ∞ and all relevant steady-state distributions exist, it holds that E[Num_(service)(∞)]times E[Work_(service)(∞)] ≤ E[Num_(service)(∞)times Work_(service)(∞)].", "problemStatement": "In a stationary first-come-first-served M/GI/n queue, let N be the number of busy servers and W their total residual workload. The conjectured inequality is equivalent to Cov(N,W)≥0. Prove this nonnegative covariance under a finite second service-time moment, or give conditions and counterexamples delimiting when it holds.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "many-server queues", "M/GI/n queue", "steady state", "covariance inequality", "residual workload", "association" ], "literature": [], "source": { "title": "Simple and Explicit Bounds for Multiserver Queues with 1/1−ρ Scaling", "authors": [ "Yuan Li", "David A. Goldberg" ], "doi": "10.1287/moor.2022.0131", "url": "https://doi.org/10.1287/moor.2022.0131", "volume": "50", "issue": "2", "pages": "813-837" }, "number": 151 }, { "id": "mp-2024-first-015", "journal": "Mathematical Programming", "publicationYear": 2024, "title": "Scenario beta bound without nonempty interior", "topic": "Scenario and chance-constrained optimization", "publicationDate": "2024-04-08", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Remark 8, digression into the convex setup", "sourceQuote": "Whether this claim preserves its validity without the assumption on the existence of a nonempty interior is at present an open problem.", "problemStatement": "For an optimizer obtained from N scenarios in a convex decision space of dimension bar d, an earlier theorem bounds the distribution of its violation probability by a Beta distribution with parameters bar d and N minus bar d plus one, even for degenerate problems whose feasibility domains have nonempty interior. Determine whether this stochastic-dominance bound remains valid after removing the nonempty-interior assumption. Neither a proof nor a counterexample is known for the resulting dimension-based statement.", "statusEvidence": "No primary paper explicitly claiming a proof or counterexample to this precise formulation was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "scenario optimization", "Beta distribution", "nonempty interior", "degeneracy" ], "literature": [], "source": { "title": "Non-convex scenario optimization", "authors": [ "Simone Garatti", "Marco C. Campi" ], "doi": "10.1007/s10107-024-02074-3", "url": "https://doi.org/10.1007/s10107-024-02074-3", "volume": "209", "issue": "1-2", "pages": "557-608" }, "number": 152 }, { "id": "mp-2024-first-016", "journal": "Mathematical Programming", "publicationYear": 2024, "title": "Support complexity beta failure with nonempty interior", "topic": "Scenario and chance-constrained optimization", "publicationDate": "2024-04-08", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Remark 8, footnote 26", "sourceQuote": "Whether this conclusion maintains its validity in the presence of a nonempty interior of the feasibility domain is at present another open problem.", "problemStatement": "The paper gives a degenerate empty-interior example in which a Beta bound based on support complexity d smaller than ambient dimension bar d fails. Determine whether the same failure can occur when the scenario-feasibility domain has nonempty interior. A counterexample would separate dimension-based and support-complexity bounds even under the regularity assumption used in earlier convex scenario theory.", "statusEvidence": "No primary paper explicitly claiming a proof or counterexample to this precise formulation was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "scenario optimization", "support complexity", "Beta distribution", "nonempty interior" ], "literature": [], "source": { "title": "Non-convex scenario optimization", "authors": [ "Simone Garatti", "Marco C. Campi" ], "doi": "10.1007/s10107-024-02074-3", "url": "https://doi.org/10.1007/s10107-024-02074-3", "volume": "209", "issue": "1-2", "pages": "557-608" }, "number": 153 }, { "id": "siopt-2024-016", "journal": "SIAM Journal on Optimization", "publicationYear": 2024, "title": "Gradient complexity gap varying condition numbers", "topic": "First-order complexity", "publicationDate": "2024-04-11", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conclusion; extracted preprint lines 1966-1972", "sourceQuote": "How to narrow down the theoretical gap between the lower and upper bounds for the general model remains an interesting question", "problemStatement": "The model minimizes a sum of strongly convex functions whose individual condition numbers may differ, and counts gradients of each component separately. The paper derives distinct lower and upper component-gradient bounds; generalized Katyusha improves on vanilla acceleration but does not close the gap. Determine the optimal gradient-complexity vector for the general model.", "statusEvidence": "No forward-citing primary paper or exact-query result explicitly claiming a proof or counterexample was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "finite sums", "condition numbers", "gradient complexity", "lower bounds" ], "literature": [], "source": { "title": "A Gradient Complexity Analysis for Minimizing the Sum of Strongly Convex Functions with Varying Condition Numbers", "authors": [ "Nuozhou Wang", "Shuzhong Zhang" ], "doi": "10.1137/22m1503646", "url": "https://doi.org/10.1137/22m1503646", "preprintUrl": "https://arxiv.org/abs/2208.06524", "volume": "34", "issue": "2", "pages": "1374-1401" }, "number": 154 }, { "id": "siopt-2024-045", "journal": "SIAM Journal on Optimization", "publicationYear": 2024, "title": "Complex orthogonal synchronization threshold", "topic": "Nonconvex relaxations", "publicationDate": "2024-04-11", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 1.3 discussion; extracted preprint lines 411-417", "sourceQuote": "We conjecture that this is the correct condition and that therefore Theorem 4 is suboptimal.", "problemStatement": "In noiseless synchronization of unknown r×r unitary matrices from exact relative measurements on a connected graph, the rank-p Burer–Monteiro relaxation replaces each unitary variable by a complex r×p Stiefel matrix Y_i satisfying Y_iY_i*=I_r. Thus r is the matrix/group dimension and p≥r is the relaxation rank, or equivalently the number of columns of each Stiefel factor. Theorem 4 proves that every second-order critical point is globally optimal when 2p≥3r, but complex St(r,p) is simply connected already when p≥r+1; prove the conjectured benign landscape at this sharper threshold for every connected graph, or find a counterexample.", "statusEvidence": "No forward-citing primary paper or exact-query result explicitly claiming a proof or counterexample was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "orthogonal synchronization", "Stiefel manifold", "benign landscape", "simple connectivity" ], "literature": [], "source": { "title": "Benign Landscapes of Low-Dimensional Relaxations for Orthogonal Synchronization on General Graphs", "authors": [ "Andrew D. McRae", "Nicolas Boumal" ], "doi": "10.1137/23m1584642", "url": "https://doi.org/10.1137/23m1584642", "preprintUrl": "https://arxiv.org/abs/2307.02941", "volume": "34", "issue": "2", "pages": "1427-1454" }, "number": 155 }, { "id": "siopt-2024-020", "journal": "SIAM Journal on Optimization", "publicationYear": 2024, "title": "Ising failure certificate complexity", "topic": "Mixed-integer and quantum optimization", "publicationDate": "2024-04-15", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conclusion; extracted preprint lines 1191-1198", "sourceQuote": "Could one design an efficient algorithm that circumvents the ambiguity due to failure to find a certificate of non-copositivity", "problemStatement": "The paper combines a classical cutting-plane method with an Ising solver that searches for certificates of non-copositivity. Failure of the heuristic Ising subroutine is ambiguous: it neither certifies copositivity nor supplies a cut. Determine whether an efficient fallback, perhaps using SOS inner approximations, removes this ambiguity, or prove a complexity barrier.", "statusEvidence": "No forward-citing primary paper or exact-query result explicitly claiming a proof or counterexample was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "Ising solver", "copositive optimization", "certificates", "complexity barrier" ], "literature": [], "source": { "title": "A Copositive Framework for Analysis of Hybrid Ising-Classical Algorithms", "authors": [ "Robin Brown", "David E. Bernal Neira", "Davide Venturelli", "Marco Pavone" ], "doi": "10.1137/22m1514581", "url": "https://doi.org/10.1137/22m1514581", "preprintUrl": "https://arxiv.org/abs/2207.13630", "volume": "34", "issue": "2", "pages": "1455-1489" }, "number": 156 }, { "id": "mor-2024-W2800787305_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2024, "title": "Decidability of feasible-set membership for robust-to-linear-dynamics linear programs", "topic": "Invariant Set Verification", "publicationDate": "2024-04-16", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 4 (Future directions and a broader agenda), page 34", "sourceQuote": "In higher dimensions, however, decidability is unknown even in the special case where the input polyhedron is a single halfspace.", "problemStatement": "For the rational linear dynamics xₖ₊₁=Gxₖ and rational polyhedron P=x:Ax≤b, the invariant set contains exactly the initial states whose entire forward orbit remains in P. This set is an infinite intersection of linear preimages and may lack a finite description. Decide algorithmically whether a rational point z belongs to it, equivalently whether every iterate Gᵏz remains in P.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "linear dynamical systems", "polyhedral invariance", "decidability", "orbit problems", "Skolem-Pisot problem", "semi-infinite constraints" ], "literature": [], "source": { "title": "Robust-to-Dynamics Optimization", "authors": [ "Amir Ali Ahmadi", "Oktay Günlük" ], "doi": "10.1287/moor.2023.0116", "url": "https://doi.org/10.1287/moor.2023.0116", "volume": "50", "issue": "2", "pages": "965-992" }, "number": 157 }, { "id": "siopt-2024-029", "journal": "SIAM Journal on Optimization", "publicationYear": 2024, "title": "Optimal SOC representation complexity", "topic": "Conic optimization", "publicationDate": "2024-04-16", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conclusion, open problem 1; extracted preprint lines 1672-1676", "sourceQuote": "Is there a polynomial time algorithm for computing an optimal simple second-order cone representation", "problemStatement": "A simple second-order-cone representation of a weighted geometric-mean inequality is encoded by a mediated set, and optimality means using the fewest auxiliary SOC constraints. The paper gives exact combinatorial formulations and heuristics. Decide whether an optimal representation is computable in polynomial time or whether the problem is NP-hard.", "statusEvidence": "No forward-citing primary paper or exact-query result explicitly claiming a proof or counterexample was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "second-order cone", "geometric mean", "extended formulation", "computational complexity" ], "literature": [], "source": { "title": "Weighted Geometric Mean, Minimum Mediated Set, and Optimal Simple Second-Order Cone Representation", "authors": [ "Jie Wang" ], "doi": "10.1137/22m1531257", "url": "https://doi.org/10.1137/22m1531257", "preprintUrl": "https://arxiv.org/abs/2206.05924", "volume": "34", "issue": "2", "pages": "1490-1514" }, "number": 158 }, { "id": "siopt-2024-030", "journal": "SIAM Journal on Optimization", "publicationYear": 2024, "title": "SOC representation lower bounds", "topic": "Conic optimization", "publicationDate": "2024-04-16", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conclusion, open problems 2-3; extracted preprint lines 1676-1684", "sourceQuote": "Can we prove L(s1,..., sm) >= m given that s-hat is odd?", "problemStatement": "L(s1,...,sm) is the minimum size of a simple SOC representation for a rational weighted geometric mean, while s-hat is the common denominator of the weights. The paper proves separate lower bounds in the dimension m and in s-hat and observes stronger behavior when s-hat is odd. Prove the odd-denominator bound L>=m and seek a sharper lower bound that combines m and s-hat.", "statusEvidence": "No forward-citing primary paper or exact-query result explicitly claiming a proof or counterexample was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "SOC representation", "geometric mean", "lower bounds", "mediated sets" ], "literature": [], "source": { "title": "Weighted Geometric Mean, Minimum Mediated Set, and Optimal Simple Second-Order Cone Representation", "authors": [ "Jie Wang" ], "doi": "10.1137/22m1531257", "url": "https://doi.org/10.1137/22m1531257", "preprintUrl": "https://arxiv.org/abs/2206.05924", "volume": "34", "issue": "2", "pages": "1490-1514" }, "number": 159 }, { "id": "mp-2024-first-017", "journal": "Mathematical Programming", "publicationYear": 2024, "title": "Singly exponential exact integer programming", "topic": "Integer programming and algorithms", "publicationDate": "2024-04-24", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 4, From approximate to exact integer programming", "sourceQuote": "The question whether there exists a singly exponential time, i.e., a 2^O(n)-algorithm for integer programming is one of the most prominent open problems", "problemStatement": "Design an algorithm for exact general integer programming whose running time is singly exponential in the dimension, apart from polynomial dependence on the input encoding. A twofold-dilation approximation is available in singly exponential time, and exact feasibility can be recovered at that speed when the residue class of a solution is supplied. Enumerating all possible residue classes recovers the larger classical bound, and no general singly exponential reduction from approximate to exact feasibility is known.", "statusEvidence": "No primary paper explicitly claiming a proof or counterexample to this precise formulation was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "integer programming", "singly exponential algorithm", "exact feasibility", "dimension" ], "literature": [], "source": { "title": "From approximate to exact integer programming", "authors": [ "Daniel Dadush", "Friedrich Eisenbrand", "Thomas Rothvoss" ], "doi": "10.1007/s10107-024-02084-1", "url": "https://doi.org/10.1007/s10107-024-02084-1", "volume": "210", "issue": "1-2", "pages": "223-241" }, "number": 160 }, { "id": "mor-2024-W4396581165_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2024, "title": "Prove consistency and rates for smoothness-minimizing estimator under variance target", "topic": "Smoothing Splines", "publicationDate": "2024-05-02", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 7.3 (page 15 of 27 in the provided text) and Remark 4 (page 10 of 27)", "sourceQuote": "We conjecture that the conclusions of Corollary 2 hold under a more natural condition that S_n → σ^2 as n → ∞.", "problemStatement": "Given regression data Yᵢ=f*(Xᵢ)+εᵢ with conditional noise variance σ², let ĝₙ minimize a Sobolev roughness functional subject to its empirical mean squared residual being at most the tolerance Sₙ. Existing consistency results require a stronger condition on Sₙ than mere convergence to the noise variance. Prove that ĝₙ and its relevant weak derivatives are consistent whenever Sₙ→σ², and separately determine its convergence rate under suitable quantitative conditions on Sₙ.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "nonparametric regression", "sobolev spaces", "smoothing splines", "derivative estimation", "convergence rate", "constrained optimization" ], "literature": [], "source": { "title": "Estimating a Function and Its Derivatives Under a Smoothness Condition", "authors": [ "Eunji Lim" ], "doi": "10.1287/moor.2020.0161", "url": "https://doi.org/10.1287/moor.2020.0161", "preprintUrl": "https://arxiv.org/pdf/2405.10126", "volume": "50", "issue": "2", "pages": "1112-1138" }, "number": 161 }, { "id": "mor-2024-W4396625954_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2024, "title": "Equilibrium of refined learning threshold with Coxian service times", "topic": "Load Balancing Queues", "publicationDate": "2024-05-03", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 21, Section 4.3 (Extensions)", "sourceQuote": "proving that the threshold eventually reaches an equilibrium in all sufficiently large systems would require non-trivial additional arguments.", "problemStatement": "A rate-nλ Poisson stream is dispatched among n infinite-server pools using a learned threshold ℓₙ(t), constrained to multiples of an integer step Δ; arrivals prefer pools below the threshold and then those below ℓₙ(t)+Δ. Service times have sequential Coxian phases, and ρ is the offered load per pool. Give conditions ensuring that, after a finite time and for all sufficiently large n, ℓₙ(t) almost surely stabilizes at a deterministic equilibrium, and characterize that level in terms of ρ and Δ.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "load balancing", "many-server asymptotics", "Coxian service times", "learning thresholds", "stability", "Markov processes" ], "literature": [], "source": { "title": "Learning and Balancing Unknown Loads in Large-Scale Systems", "authors": [ "Diego Goldsztajn", "Sem C. Borst", "Johan S. H. van Leeuwaarden" ], "doi": "10.1287/moor.2021.0212", "url": "https://doi.org/10.1287/moor.2021.0212", "volume": "50", "issue": "2", "pages": "1139-1172" }, "number": 162 }, { "id": "siopt-2024-027", "journal": "SIAM Journal on Optimization", "publicationYear": 2024, "title": "Single projection step bound", "topic": "Projection methods", "publicationDate": "2024-05-03", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 5, open question 1; extracted preprint lines 2037-2045", "sourceQuote": "Can Theorem 4.5 be used to estimate the number of steps required", "problemStatement": "Let A be closed and convex and consider min_(x∈A)⟨x*,x⟩. The set A is α-sharp relative to −x* when every nonoptimal normal cone stays at distance at least α from −x*; Theorem 4.5 says P_A(v) is optimal if ⟨x*,v⟩ 0", "problemStatement": "Every polyhedral convex set is alpha-sharp with respect to every unit direction for some positive alpha. The converse would turn a uniform geometric error bound into a characterization of polyhedrality. Prove that uniform directional sharpness forces a set to be polyhedral, or construct a nonpolyhedral counterexample.", "statusEvidence": "No forward-citing primary paper or exact-query result explicitly claiming a proof or counterexample was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "sharp sets", "polyhedrality", "error bounds", "convex geometry" ], "literature": [], "source": { "title": "Single-Projection Procedure for Infinite Dimensional Convex Optimization Problems", "authors": [ "Hoa T. Bui", "Regina S. Burachik", "Evgeni A. Nurminski", "Matthew K. Tam" ], "doi": "10.1137/22m1530173", "url": "https://doi.org/10.1137/22m1530173", "preprintUrl": "https://arxiv.org/abs/2210.11252", "volume": "34", "issue": "2", "pages": "1646-1678" }, "number": 164 }, { "id": "mor-2024-W4226226337_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2024, "title": "Determine whether Golden Parachutes arise with an impatient agent", "topic": "Principal Agent Optimal Stopping", "publicationDate": "2024-05-06", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 9, Remark 3.5(iii) (end of Section 3.1, before Section 3.2).", "sourceQuote": "it is unclear in general whether y_(gp) is finite or not", "problemStatement": "In Sannikov's continuous-time principal-agent model, V(y) is the principal's value at agent continuation utility y and F(y) is the retirement payoff. For an impatient agent with discount-rate ratio δ>1, the stopping set is 0∪[yGP,∞) if the upper retirement threshold yGP is finite. Determine whether such a golden-parachute threshold must be finite or can instead be infinite.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "continuous-time principal-agent", "optimal stopping", "viscosity solutions", "dynamic contracting", "discounting", "golden parachute" ], "literature": [], "source": { "title": "Is There a Golden Parachute in Sannikov’s Principal–Agent Problem?", "authors": [ "Dylan Possamaï", "Nizar Touzi" ], "doi": "10.1287/moor.2022.0305", "url": "https://doi.org/10.1287/moor.2022.0305", "volume": "50", "issue": "2", "pages": "1173-1203" }, "number": 165 }, { "id": "mor-2024-W4396690120_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2024, "title": "Linear-time Micali–Vazirani phase implementation on pointer machines via path compression", "topic": "Maximum Matching Algorithms", "publicationDate": "2024-05-07", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 9.3, Remark 9 (page 2044), and Section 10, final open question (page 2046).", "sourceQuote": "A question arising from Theorem 8 is whether there is a linear time implementation of bud^* in the pointer model.", "problemStatement": "The Micali–Vazirani maximum-matching algorithm proceeds in phases that build and search a layered blossom structure. The source asks whether path compression and the restricted union–find operations permit an O(m)-time phase on a pointer machine. Prove such a linear implementation or establish a lower obstruction; also determine whether the existing union–find implementation is already linear per phase.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "maximum matching", "micali vazirani algorithm", "pointer machine", "disjoint set union", "path compression", "amortized analysis" ], "literature": [], "source": { "title": "A Theory of Alternating Paths and Blossoms from the Perspective of Minimum Length", "authors": [ "Vijay V. Vazirani" ], "doi": "10.1287/moor.2020.0388", "url": "https://doi.org/10.1287/moor.2020.0388", "preprintUrl": "https://escholarship.org/content/qt3083d90j/qt3083d90j.pdf", "volume": "49", "issue": "3", "pages": "2009-2047" }, "number": 166 }, { "id": "mp-2024-first-018", "journal": "Mathematical Programming", "publicationYear": 2024, "title": "Linear time QSPP linearization with sparse costs", "topic": "Combinatorial optimization and algorithms", "publicationDate": "2024-05-09", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conclusion", "sourceQuote": "In the case of sparsely encoded input cost coefficients, it is an open question how to obtain a linear time algorithm.", "problemStatement": "The paper recognizes and linearizes quadratic and fixed higher-order shortest-path instances in time linear in the number of coefficients when the cost tensor is stored densely. Develop an algorithm whose running time is linear in the encoded input size when only the nonzero cost coefficients are listed. The analysis must avoid work proportional to the full dense tensor while preserving the exact linearizability decision and construction.", "statusEvidence": "No primary paper explicitly claiming a proof or counterexample to this precise formulation was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "quadratic shortest path", "linearization", "sparse encoding", "linear-time algorithm" ], "literature": [], "source": { "title": "A linear time algorithm for linearizing quadratic and higher-order shortest path problems", "authors": [ "Eranda Çela", "Bettina Klinz", "Stefan Lendl", "Gerhard J. Woeginger", "Lasse Wulf" ], "doi": "10.1007/s10107-024-02086-z", "url": "https://doi.org/10.1007/s10107-024-02086-z", "volume": "210", "issue": "1-2", "pages": "165-188" }, "number": 167 }, { "id": "mor-2024-W4396905249_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2024, "title": "Design learning guarantees that improve liquid welfare in sequential budgeted auctions", "topic": "Learning In Auctions", "publicationDate": "2024-05-14", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 18, Section 8 (Conclusion)", "sourceQuote": "While we prove positive liquid welfare guarantees under mild learning assumptions, we leave open the problem of whether stronger learning guarantees imply better liquid welfare.", "problemStatement": "Sequential budgeted first-price auctions sell one item per round; liquid welfare sums each bidder's total value capped by that bidder's budget. Existing guarantees assume each bidder learns against the best fixed multiplier λ∈[0,1] used to shade every value. Determine whether regret against richer Lipschitz value-to-bid rules, or a stochastic-values benchmark, yields a strictly better worst-case liquid-welfare approximation.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "sequential auctions", "liquid welfare", "budgets", "first-price auctions", "no-regret learning", "competitive ratio", "price of anarchy", "Lipschitz bidding rules" ], "literature": [], "source": { "title": "Liquid Welfare Guarantees for No-Regret Learning in Sequential Budgeted Auctions", "authors": [ "Giannis Fikioris", "Éva Tardos" ], "doi": "10.1287/moor.2023.0274", "url": "https://doi.org/10.1287/moor.2023.0274", "volume": "50", "issue": "2", "pages": "1233-1249" }, "number": 168 }, { "id": "focm-2024-max-filter-strong-separation-groups", "journal": "Foundations of Computational Mathematics", "publicationYear": 2024, "title": "Max Filter Strong Separation Groups", "topic": "Invariant signal processing", "publicationDate": "2024-05-17", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Problem 15(a)", "sourceQuote": "Which semialgebraic groups have a max filter map that k-strongly separates?", "problemStatement": "Characterize the semialgebraic groups whose orbit spaces admit a max-filter map that k-strongly separates points. The value and role of k are those defined in the source paper's separation framework. A max filter sends two orbits to their maximal inner product, and k-strong separation means generic level-set coincidences have codimension at least k.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "max filtering", "semialgebraic groups", "orbit separation", "invariants" ], "literature": [], "source": { "title": "Group-Invariant Max Filtering", "authors": [ "Jameson Cahill", "Joseph W. Iverson", "Dustin G. Mixon", "Daniel Packer" ], "doi": "10.1007/s10208-024-09656-9", "url": "https://doi.org/10.1007/s10208-024-09656-9", "volume": "25", "issue": "3", "pages": "1047-1084" }, "number": 169 }, { "id": "focm-2024-max-filter-template-count", "journal": "Foundations of Computational Mathematics", "publicationYear": 2024, "title": "Max Filter Template Count", "topic": "Invariant signal processing", "publicationDate": "2024-05-17", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Problem 15(b)", "sourceQuote": "How many templates are needed to separate orbits?", "problemStatement": "Determine the minimum or asymptotically optimal number of max-filter templates needed to separate group orbits. The question asks for bounds sensitive to the acting group and ambient dimension. Each template contributes one invariant coordinate obtained by maximizing an inner product over the group orbit; the paper proves general sufficient counts but not optimal ones.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "max filtering", "templates", "orbit separation", "embedding dimension" ], "literature": [], "source": { "title": "Group-Invariant Max Filtering", "authors": [ "Jameson Cahill", "Joseph W. Iverson", "Dustin G. Mixon", "Daniel Packer" ], "doi": "10.1007/s10208-024-09656-9", "url": "https://doi.org/10.1007/s10208-024-09656-9", "volume": "25", "issue": "3", "pages": "1047-1084" }, "number": 170 }, { "id": "focm-2024-separating-max-filter-bilipschitz", "journal": "Foundations of Computational Mathematics", "publicationYear": 2024, "title": "Separating Max Filter Bilipschitz", "topic": "Invariant signal processing", "publicationDate": "2024-05-17", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Problem 21", "sourceQuote": "Is every separating max filter bank bilipschitz?", "problemStatement": "Decide whether every separating max-filter bank on a finite-dimensional orthogonal-group quotient is automatically bilipschitz. A positive answer would turn generic separating constructions into stable embeddings. Bilipschitz means the feature map distorts quotient-space distances by bounded multiplicative factors, which is stronger than merely mapping distinct orbits to distinct vectors.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "max filtering", "bilipschitz embedding", "stability", "group quotients" ], "literature": [], "source": { "title": "Group-Invariant Max Filtering", "authors": [ "Jameson Cahill", "Joseph W. Iverson", "Dustin G. Mixon", "Daniel Packer" ], "doi": "10.1007/s10208-024-09656-9", "url": "https://doi.org/10.1007/s10208-024-09656-9", "volume": "25", "issue": "3", "pages": "1047-1084" }, "number": 171 }, { "id": "focm-2024-ill-posed-fem-optimal-alpha-tradeoff", "journal": "Foundations of Computational Mathematics", "publicationYear": 2024, "title": "Ill Posed FEM Optimal Alpha Tradeoff", "topic": "Numerical analysis of ill-posed problems", "publicationDate": "2024-05-20", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 4, discussion after error bound (27)", "sourceQuote": "The question of constructing such an optimal method with alpha1 > alpha is currently an open question.", "problemStatement": "In the elliptic unique-continuation problem Delta u=0 in Omega with interior observations u restricted to omega equal to q, reconstruction on B has conditional stability ||u||_L2(B) <= C||u||_L2(omega)^alpha||u||_L2(Omega)^(1-alpha). The paper's discretization analysis uses approximation exponents alpha1 and alpha2 and yields the effective exponent alpha1/(1+alpha1-alpha2), together with a noise-amplification restriction on the smallest mesh. Construct a method with alpha1>alpha while retaining effective exponent alpha, so it attains the optimal error bound with a larger admissible minimum mesh size and lower computational cost.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "finite elements", "ill-posed problem", "conditional stability", "optimal error" ], "literature": [], "source": { "title": "Optimal Approximation of Unique Continuation", "authors": [ "Erik Burman", "Mihai Nechita", "Lauri Oksanen" ], "doi": "10.1007/s10208-024-09655-w", "url": "https://doi.org/10.1007/s10208-024-09655-w", "volume": "25", "issue": "3", "pages": "1025-1045" }, "number": 172 }, { "id": "focm-2024-pnp-general-nonlinear-denoiser-regularization", "journal": "Foundations of Computational Mathematics", "publicationYear": 2024, "title": "PnP General Nonlinear Denoiser Regularization", "topic": "Inverse problems and imaging", "publicationDate": "2024-06-03", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conclusion", "sourceQuote": "the question of convergent regularization is still open for general non-linear denoisers", "problemStatement": "Develop a convergent regularization theory for plug-and-play inverse-problem methods with general nonlinear denoisers. The source paper proves such guarantees only for spectrally filtered learned linear denoisers. In a PnP method the denoiser replaces a proximal operator inside an iterative reconstruction scheme, so convergence alone does not automatically imply regularization as noise vanishes.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "plug-and-play", "nonlinear denoisers", "convergent regularization", "inverse problems" ], "literature": [], "source": { "title": "Convergent Regularization in Inverse Problems and Linear Plug-and-Play Denoisers", "authors": [ "Andreas Hauptmann", "Subhadip Mukherjee", "Carola-Bibiane Schönlieb", "Ferdia Sherry" ], "doi": "10.1007/s10208-024-09654-x", "url": "https://doi.org/10.1007/s10208-024-09654-x", "volume": "25", "issue": "4", "pages": "1087-1120" }, "number": 173 }, { "id": "mor-2024-W3217714619_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2024, "title": "Polynomial bound on degenerate simplex pivots for 0/1 polytope linear programs", "topic": "Simplex Pivot Rules", "publicationDate": "2024-06-05", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 28, Section 4 (Conclusions, Connections, and Comparisons)", "sourceQuote": "Is there a polynomial bound to the number of degenerate pivots performed by any of the pivot rules we presented when applied to 0/1-LPs?", "problemStatement": "For linear programs over 0/1 polytopes, the analyzed steepest-edge and shadow pivot rules take only polynomially many nondegenerate pivots. A degenerate pivot exchanges basis elements without moving to a different vertex, and the existing proof does not bound how many can occur. Determine whether each of these rules has a polynomial bound on degenerate pivots, or construct a superpolynomial example.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "simplex method", "pivot rules", "degeneracy", "0/1 polytopes", "degenerate pivots", "linear programming" ], "literature": [], "source": { "title": "On the Simplex Method for 0/1-Polytopes", "authors": [ "Alexander E. Black", "Jesús A. De Loera", "Sean Kafer", "Laura Sanità" ], "doi": "10.1287/moor.2021.0345", "url": "https://doi.org/10.1287/moor.2021.0345", "volume": "50", "issue": "2", "pages": "1398-1420" }, "number": 174 }, { "id": "siopt-2024-015", "journal": "SIAM Journal on Optimization", "publicationYear": 2024, "title": "Degree bounds for non SOS certificates", "topic": "Polynomial optimization", "publicationDate": "2024-06-06", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 3.3 discussion; extracted preprint lines 951-956", "sourceQuote": "A compelling line for future work is to compute degree bounds for this type of certificates.", "problemStatement": "The paper constructs sparse non-SOS Positivstellensatz certificates on semialgebraic sets using univariate multiplier polynomials. Their sparse structure makes fixed-degree relaxations small, but no quantitative degree bound is proved. Bound the degree required as a function of the input polynomials and positivity margin.", "statusEvidence": "No forward-citing primary paper or exact-query result explicitly claiming a proof or counterexample was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "Positivstellensatz", "degree bounds", "non-SOS certificates", "semialgebraic sets" ], "literature": [], "source": { "title": "Reducing Nonnegativity over General Semialgebraic Sets to Nonnegativity over Simple Sets", "authors": [ "Olga Kuryatnikova", "Juan C. Vera", "Luis F. Zuluaga" ], "doi": "10.1137/22m1501027", "url": "https://doi.org/10.1137/22m1501027", "preprintUrl": "https://arxiv.org/abs/1909.06689", "volume": "34", "issue": "2", "pages": "1970-2006" }, "number": 175 }, { "id": "mor-2024-W4399497388_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2024, "title": "Classify stable serial knockout competitions up to equivalence for larger powers of two", "topic": "Tournament Design", "publicationDate": "2024-06-10", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 13, end of Section 5.2 (after Theorem 3) and reiterated in Section 7 (Discussion), page 14.", "sourceQuote": "we do not know whether there exist stable SKCs that are inequivalent to the ones that we construct in this paper", "problemStatement": "On n=2ᵏ players, a serial knockout competition is a collection of n−1 balanced single-elimination brackets. It is stable when every player pair can meet in round i in exactly 2ⁱ⁻¹ of the brackets, for each round i. For n>8, determine whether every stable collection is equivalent under relabeling to the finite-field construction over GF(2ᵏ), or find an inequivalent example, especially at n=16.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "tournament design", "serial knockout competition", "stability", "finite fields", "combinatorial designs", "equivalence classes" ], "literature": [], "source": { "title": "How to Design a Stable Serial Knockout Competition", "authors": [ "Roel Lambers", "Rudi Pendavingh", "Frits Spieksma" ], "doi": "10.1287/moor.2022.0352", "url": "https://doi.org/10.1287/moor.2022.0352", "preprintUrl": "https://pure.tue.nl/ws/files/360156142/lambers-et-al-2024-how-to-design-a-stable-serial-knockout-competition.pdf", "volume": "50", "issue": "2", "pages": "1421-1432" }, "number": 176 }, { "id": "siopt-2024-031", "journal": "SIAM Journal on Optimization", "publicationYear": 2024, "title": "STOSQP unbounded noise", "topic": "Stochastic optimization", "publicationDate": "2024-06-11", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conclusion; extracted preprint lines 1662-1667", "sourceQuote": "how to establish global convergence without the assumption of bounded noise remains an open question.", "problemStatement": "The paper proves global convergence and complexity for a fully stochastic trust-region SQP method on equality-constrained problems, assuming almost-surely bounded oracle noise. That assumption excludes common Gaussian and heavy-tailed estimators. Establish global convergence under unbounded noise using moment, tail, or adaptive-sampling conditions.", "statusEvidence": "No forward-citing primary paper or exact-query result explicitly claiming a proof or counterexample was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "stochastic SQP", "trust region", "unbounded noise", "global convergence" ], "literature": [], "source": { "title": "Fully Stochastic Trust-Region Sequential Quadratic Programming for Equality-Constrained Optimization Problems", "authors": [ "Yuchen Fang", "Sen Na", "Michael W. Mahoney", "Mladen Kolar" ], "doi": "10.1137/22m1537862", "url": "https://doi.org/10.1137/22m1537862", "preprintUrl": "https://arxiv.org/abs/2211.15943", "volume": "34", "issue": "2", "pages": "2007-2037" }, "number": 177 }, { "id": "focm-2024-kl-subgradient-linear-convergence", "journal": "Foundations of Computational Mathematics", "publicationYear": 2024, "title": "KL Subgradient Linear Convergence", "topic": "Nonsmooth optimization", "publicationDate": "2024-06-14", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Introduction, discrete-time subgradient discussion", "sourceQuote": "we do not know whether the KL inequality alone guarantees locally linear convergence", "problemStatement": "Determine whether the Kurdyka-Lojasiewicz inequality alone yields locally linear convergence for a discrete-time subgradient method beyond the sharp convex or weakly convex case. The unresolved regime corresponds to nonzero KL exponents in the paper's framework.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "Kurdyka-Lojasiewicz", "subgradient method", "linear convergence", "nonsmooth optimization" ], "literature": [], "source": { "title": "A Local Nearly Linearly Convergent First-Order Method for Nonsmooth Functions with Quadratic Growth", "authors": [ "Damek Davis", "Liwei Jiang" ], "doi": "10.1007/s10208-024-09653-y", "url": "https://doi.org/10.1007/s10208-024-09653-y", "preprintUrl": "https://arxiv.org/abs/2205.00064", "volume": "25", "issue": "3", "pages": "943-1024" }, "number": 178 }, { "id": "mp-2024-first-019", "journal": "Mathematical Programming", "publicationYear": 2024, "title": "Slope attouch theorem in infinite dimensions", "topic": "Variational analysis and convex convergence", "publicationDate": "2024-06-20", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Main result, final comments and perspectives", "sourceQuote": "a natural question is whether Theorem 1.6 is true in Hilbert spaces, or more generally, in reflexive Banach spaces", "problemStatement": "The paper proves in finite-dimensional Euclidean spaces that, for proper lower-semicontinuous convex functions with a bounded-below limit, epigraphical convergence of slopes plus normalization is equivalent to epigraphical convergence of the functions. Extend this slope form of Attouch's theorem first to Hilbert spaces and, if possible, to reflexive or arbitrary Banach spaces. The finite-dimensional proof uses local compactness, so an infinite-dimensional proof needs a different mechanism.", "statusEvidence": "No primary paper explicitly claiming a proof or counterexample to this precise formulation was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "Attouch theorem", "slope convergence", "Hilbert space", "Banach space" ], "literature": [], "source": { "title": "A slope generalization of Attouch theorem", "authors": [ "Aris Daniilidis", "David Salas", "Sebastián Tapia-García" ], "doi": "10.1007/s10107-024-02108-w", "url": "https://doi.org/10.1007/s10107-024-02108-w", "volume": "212", "issue": "1-2", "pages": "319-348" }, "number": 179 }, { "id": "mor-2024-W4400196985_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2024, "title": "Prove NP-hardness of deciding convexity of coordinate projections of rotation matrices", "topic": "Computational Complexity", "publicationDate": "2024-07-01", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 29, Section 8 (Summary and open questions), paragraph 'Convex coordinate projections'", "sourceQuote": "The problem CONVEX is NP-hard.", "problemStatement": "For n ≥ 3, SO(n) is the set of n-by-n rotation matrices. A coordinate projection keeps only entries indexed by a chosen set S, and CONVEX asks whether the resulting image of SO(n) is a convex subset of Euclidean space. Determine the computational complexity of CONVEX, in particular whether deciding convexity from n and S is NP-hard.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "special orthogonal group", "hidden convexity", "coordinate projection", "convexity decision problem", "computational complexity", "np-hardness" ], "literature": [], "source": { "title": "Hidden Convexity, Optimization, and Algorithms on Rotation Matrices", "authors": [ "Akshay Ramachandran", "Kevin Shu", "Alex L. Wang" ], "doi": "10.1287/moor.2023.0114", "url": "https://doi.org/10.1287/moor.2023.0114", "volume": "50", "issue": "2", "pages": "1454-1477" }, "number": 180 }, { "id": "mp-2024-first-021", "journal": "Mathematical Programming", "publicationYear": 2024, "title": "Log rank conjecture", "topic": "Communication complexity and combinatorics", "publicationDate": "2024-07-05", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Introduction", "sourceQuote": "It states that for any Boolean function f of rank r, its deterministic communication complexity is bounded by polylog(r).", "problemStatement": "For every Boolean communication matrix of real rank r, prove that deterministic communication complexity is bounded by a fixed polynomial in the logarithm of r. The source improves the known general upper bound to order square root of r by finding large monochromatic submatrices. A 2025 paper gives new equivalent formulations but continues to describe the quantitative conjecture as open.", "statusEvidence": "No primary paper explicitly claiming a proof or counterexample to this precise formulation was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "log-rank conjecture", "communication complexity", "Boolean matrix", "matrix rank" ], "literature": [], "source": { "title": "Matrix discrepancy and the log-rank conjecture", "authors": [ "Benny Sudakov", "István Tomon" ], "doi": "10.1007/s10107-024-02117-9", "url": "https://doi.org/10.1007/s10107-024-02117-9", "volume": "212", "issue": "1-2", "pages": "567-579" }, "number": 181 }, { "id": "mp-2024-first-022", "journal": "Mathematical Programming", "publicationYear": 2024, "title": "Lovett sparse real matrix zero submatrix conjecture", "topic": "Extremal matrix theory and communication complexity", "publicationDate": "2024-07-05", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conjecture 5.1", "sourceQuote": "Then there exist X,Y⊂[n] such that Mₓ,ᵧ=0 for all (x,y)∈X×Y and |X|=|Y|≥n2^(-c√(εr)) for some absolute constant c>0.", "problemStatement": "For an n-by-n real matrix of rank at most r with at most epsilon times n squared nonzero entries, prove the existence of an all-zero square submatrix whose side length is at least n times two to a negative constant times the square root of epsilon r. The constant must be universal. This would extend the large-monochromatic-submatrix phenomenon beyond binary matrices and match the order suggested by known constructions.", "statusEvidence": "No primary paper explicitly claiming a proof or counterexample to this precise formulation was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "Lovett conjecture", "sparse real matrix", "zero submatrix", "low rank" ], "literature": [], "source": { "title": "Matrix discrepancy and the log-rank conjecture", "authors": [ "Benny Sudakov", "István Tomon" ], "doi": "10.1007/s10107-024-02117-9", "url": "https://doi.org/10.1007/s10107-024-02117-9", "volume": "212", "issue": "1-2", "pages": "567-579" }, "number": 182 }, { "id": "mp-2024-first-020", "journal": "Mathematical Programming", "publicationYear": 2024, "title": "Pseudo subregularity in directional asymptotic stationarity", "topic": "Variational analysis and constraint qualifications", "publicationDate": "2024-07-05", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Directional asymptotic stationarity in nonsmooth optimization", "sourceQuote": "We conjecture that the sufficient condition (2.12) can be weakened to just pseudo-subregularity of Φ in Corollary 4.2 (b).", "problemStatement": "At a local minimizer, consider a critical direction with first-order displacement in the decision variable and order-gamma displacement in the constraint residual. A multiplier whose order-gamma directional pseudo-coderivative contribution balances a directional subgradient of the objective is known to exist when that pseudo-coderivative has trivial kernel. Prove the same multiplier stationarity statement assuming only pseudo-subregularity of the constraint mapping; the case gamma equal to one is already known.", "statusEvidence": "No primary paper explicitly claiming a proof or counterexample to this precise formulation was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "pseudo-subregularity", "directional asymptotic stationarity", "coderivative", "constraint mapping" ], "literature": [], "source": { "title": "On the directional asymptotic approach in optimization theory", "authors": [ "Matúš Benko", "Patrick Mehlitz" ], "doi": "10.1007/s10107-024-02089-w", "url": "https://doi.org/10.1007/s10107-024-02089-w", "volume": "209", "issue": "1-2", "pages": "859-937" }, "number": 183 }, { "id": "mp-2024-second-conditioned-combinatorial-diameter", "journal": "Mathematical Programming", "publicationYear": 2024, "title": "Conditioned Polynomial Combinatorial Diameter", "topic": "Polyhedral geometry and linear optimization", "publicationDate": "2024-07-08", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Introduction, after the circuit-diameter theorems", "sourceQuote": "the next important step towards the polynomial Hirsch conjecture might be to show a poly(n,log κA) bound on the combinatorial diameter", "problemStatement": "For a standard-form polyhedron with n variables and circuit-imbalance parameter κA, bound the ordinary combinatorial diameter by a polynomial in n and log κA. The source obtains logarithmic dependence on κA for circuit diameter, whereas the best stated edge-diameter bound depends polynomially on κA and may be exponential in input length. The target would connect circuit conditioning to an actual vertex-edge walk.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact statement, title, authors and citations across publisher pages, arXiv and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper direct result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "circuit imbalance", "combinatorial diameter", "condition number", "standard-form polyhedron" ], "literature": [], "source": { "title": "On circuit diameter bounds via circuit imbalances", "authors": [ "Daniel Dadush", "Zhuan Khye Koh", "Bento Natura", "László A. Végh" ], "doi": "10.1007/s10107-024-02107-x", "url": "https://doi.org/10.1007/s10107-024-02107-x", "volume": "206", "issue": "1-2", "pages": "631-662" }, "number": 184 }, { "id": "mp-2024-second-polynomial-hirsch-combinatorial-diameter", "journal": "Mathematical Programming", "publicationYear": 2024, "title": "Polynomial Hirsch Bound for Combinatorial Diameter", "topic": "Polyhedral geometry and linear optimization", "publicationDate": "2024-07-08", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Introduction, opening paragraph", "sourceQuote": "finding a poly(f) bound on the combinatorial diameter remains a central question in the theory of linear programming.", "problemStatement": "Prove that the vertex-edge diameter of every polyhedron with f facets is bounded by a polynomial in f, the polynomial Hirsch conjecture. Quasipolynomial bounds are known in general and polynomial bounds hold for important structured classes, but no polynomial upper bound is known for arbitrary polyhedra. The question concerns ordinary edge walks, not the more permissive circuit walks studied in the source.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact statement, title, authors and citations across publisher pages, arXiv and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper direct result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "polynomial Hirsch conjecture", "combinatorial diameter", "polyhedron", "linear programming" ], "literature": [], "source": { "title": "On circuit diameter bounds via circuit imbalances", "authors": [ "Daniel Dadush", "Zhuan Khye Koh", "Bento Natura", "László A. Végh" ], "doi": "10.1007/s10107-024-02107-x", "url": "https://doi.org/10.1007/s10107-024-02107-x", "volume": "206", "issue": "1-2", "pages": "631-662" }, "number": 185 }, { "id": "mp-2024-second-circuit-hirsch-linear-bound", "journal": "Mathematical Programming", "publicationYear": 2024, "title": "Sharp Circuit Hirsch Bound", "topic": "Polyhedral geometry and linear optimization", "publicationDate": "2024-07-08", "recordType": "reviewed", "status": "partial_progress", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Introduction, circuit-diameter discussion", "sourceQuote": "asserts that the circuit diameter of d-dimensional polyhedron with f facets is at most f−d", "problemStatement": "Prove or disprove that every d-dimensional polyhedron with f facets has circuit diameter at most f−d, including the proposed extension to unbounded polyhedra. Circuit steps may follow any feasible circuit direction and must be maximal, so this is distinct from the disproved edge-walk Hirsch conjecture. A 2026 strongly polynomial circuit-diameter theorem is substantial progress but does not attain the conjectured linear bound.", "statusEvidence": "Proves a general O(m² log m) circuit-diameter bound for standard-form polyhedra, leaving the conjectured linear f−d bound open.", "verificationNote": "This status is tied to the cited primary-literature result and the precise scope stated above.", "keywords": [ "circuit diameter", "Hirsch bound", "circuit walk", "polyhedron" ], "literature": [ { "title": "Circuit Diameter of Polyhedra is Strongly Polynomial", "url": "https://arxiv.org/abs/2602.06958", "relation": "Proves a general O(m² log m) circuit-diameter bound for standard-form polyhedra, leaving the conjectured linear f−d bound open." } ], "source": { "title": "On circuit diameter bounds via circuit imbalances", "authors": [ "Daniel Dadush", "Zhuan Khye Koh", "Bento Natura", "László A. Végh" ], "doi": "10.1007/s10107-024-02107-x", "url": "https://doi.org/10.1007/s10107-024-02107-x", "volume": "206", "issue": "1-2", "pages": "631-662" }, "number": 186 }, { "id": "mor-2024-W4400454886_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2024, "title": "Recovery threshold for complete LP relaxation in semi-random rank-one Boolean tensors", "topic": "Boolean Tensor Factorization", "publicationDate": "2024-07-09", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 4, Section 1.4 (Our contribution).", "sourceQuote": "We believe that obtaining recovery guarantees for the complete LP is an interesting open question.", "problemStatement": "Rank-one binary tensor factorization seeks three binary vectors whose outer product explains an observed semi-randomly corrupted tensor. The paper proves exact-recovery thresholds for its standard and flower linear-programming relaxations, while a stronger complete LP adds the full local multilinear convex hull. Derive a high-probability recovery threshold for the complete LP in terms of factor densities and corruption level.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "boolean tensor factorization", "linear programming relaxation", "semi-random model", "recovery guarantee", "multilinear polytope", "planted rank-one tensor" ], "literature": [], "source": { "title": "Rank-One Boolean Tensor Factorization and the Multilinear Polytope", "authors": [ "Alberto Del Pia", "Aida Khajavirad" ], "doi": "10.1287/moor.2022.0201", "url": "https://doi.org/10.1287/moor.2022.0201", "volume": "50", "issue": "2", "pages": "1514-1554" }, "number": 187 }, { "id": "mp-2024-second-true-approximation-generalized-k-median", "journal": "Mathematical Programming", "publicationYear": 2024, "title": "True Approximation for Generalized k-Median", "topic": "Approximation algorithms and facility location", "publicationDate": "2024-07-10", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Concluding discussion", "sourceQuote": "It is an interesting open question to obtain a true approximation for GKM.", "problemStatement": "Design a polynomial-time constant-factor approximation algorithm for generalized k-median with multiple packing and covering constraints, rather than only a near-integral fractional solution. The source obtains a 6.387-approximate fractional solution with O(r₁+r₂) fractional facilities. The unresolved rounding step must open a facility in every required ball while satisfying all packing and covering constraints simultaneously.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact statement, title, authors and citations across publisher pages, arXiv and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper direct result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "generalized k-median", "approximation algorithm", "iterative rounding", "packing and covering" ], "literature": [], "source": { "title": "Structural iterative rounding for generalized k-median problems", "authors": [ "Anupam Gupta", "Benjamin Moseley", "Rudy Zhou" ], "doi": "10.1007/s10107-024-02119-7", "url": "https://doi.org/10.1007/s10107-024-02119-7", "volume": "212", "issue": "1-2", "pages": "581-634" }, "number": 188 }, { "id": "mor-2024-W4400583097_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2024, "title": "Determine the asymptotic order of maximum independent sets in recursive Markov random graphs", "topic": "Random Graphs", "publicationDate": "2024-07-12", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 2.2 Discussion, page 7", "sourceQuote": "Does there exist a function f(n) = Ω(n/ log n) such that f(n) = o(n) and α(G^r_(n,p)) = Θ(f(n)) w.h.p.?", "problemStatement": "Fix p∈(0,1] and r∈(0,1), and build G⁽ʳ⁾ₙ,ₚ by scanning the earlier neighbors of each new vertex: the first edge is Bernoulli(p), while a later edge has conditional probability q after a missing predecessor and rq after a present predecessor, where q is the preceding edge's marginal probability. The independence number is known only within logarithmically separated bounds. Find an explicit f(n) with f(n)=Ω(n/log n), f(n)=o(n), and α(G⁽ʳ⁾ₙ,ₚ)=Θ(f(n)) with high probability.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "random graphs", "independent set", "Markov dependence", "asymptotic order", "phase transition" ], "literature": [], "source": { "title": "Large Independent Sets in Recursive Markov Random Graphs", "authors": [ "Akshay Gupte", "Yiran Zhu" ], "doi": "10.1287/moor.2022.0215", "url": "https://doi.org/10.1287/moor.2022.0215", "volume": "50", "issue": "3", "pages": "1611-1634" }, "number": 189 }, { "id": "mor-2024-W4400583084_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2024, "title": "Order dependence of sequential issue-by-issue voting without LAMO bases", "topic": "Spatial Voting", "publicationDate": "2024-07-12", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 23, Section 5 (end of the proof of Proposition 3, immediately before Proposition 4).", "sourceQuote": "We conjecture that the outcome of sequential issue-by-issue voting is then necessarily order dependent.", "problemStatement": "Voters have ideal points in a normed policy space and decide coordinates sequentially by pairwise majority. A basis has local alignment of median outcomes (LAMO) when adding any multiple of one basis vector cannot shorten a vector in the span of all remaining basis vectors. If LAMO fails, prove that some preference profile and two issue orders produce different iteratively undominated equilibrium outcomes, or give a counterexample.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "sequential voting", "order independence", "normed spaces", "birkhoff-james orthogonality", "LAMO bases", "issue-by-issue median", "ex-post perfect equilibrium" ], "literature": [], "source": { "title": "Order Independence in Sequential, Issue-by-Issue Voting", "authors": [ "Alex Gershkov", "Benny Moldovanu", "Xianwen Shi" ], "doi": "10.1287/moor.2022.0342", "url": "https://doi.org/10.1287/moor.2022.0342", "volume": "50", "issue": "3", "pages": "1635-1653" }, "number": 190 }, { "id": "mor-2024-W4400685310_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2024, "title": "Derive an exact tractable convex program for robust Markov decision processes", "topic": "Robust Markov Decision Processes", "publicationDate": "2024-07-16", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 1 (Introduction), page 2; reiterated in Section 7 (Conclusion), page 22.", "sourceQuote": "First, it may be possible to obtain exact formulations for RMDPs, not based on regularization, for specific uncertainty sets such as ell_p-balls or φ-divergence balls", "problemStatement": "A robust Markov decision process optimizes discounted reward against all transition kernels in a rectangular uncertainty set. Unlike the nominal case, its Bellman inequalities do not directly produce a convex feasible region, and known formulations use approximations or regularization. Find conditions or a reformulation yielding an exact polynomial-size tractable convex program for the robust value and an optimal stationary policy.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "robust MDPs", "convex formulation", "conic optimization", "rectangular uncertainty", "Bellman operator", "exact reformulation" ], "literature": [], "source": { "title": "On the Convex Formulations of Robust Markov Decision Processes", "authors": [ "Julien Grand-Clément", "Marek Petrik" ], "doi": "10.1287/moor.2022.0284", "url": "https://doi.org/10.1287/moor.2022.0284", "volume": "50", "issue": "3", "pages": "1681-1706" }, "number": 191 }, { "id": "focm-2024-svgf-space-time-discretization-convergence", "journal": "Foundations of Computational Mathematics", "publicationYear": 2024, "title": "SVGF Space Time Discretization Convergence", "topic": "Computational optimal transport", "publicationDate": "2024-07-17", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conclusion", "sourceQuote": "A convergence analysis of the space-time discretization is an interesting and challenging open question.", "problemStatement": "Establish convergence of practical space-time discretizations of the stochastic variational gradient flow and its regularized variant. The analysis should connect the discrete particle and time scheme to the continuous SVGF or R-SVGF dynamics.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "SVGF", "space-time discretization", "particle methods", "convergence" ], "literature": [], "source": { "title": "Regularized Stein Variational Gradient Flow", "authors": [ "Ye He", "Krishnakumar Balasubramanian", "Bharath K. Sriperumbudur", "Jianfeng Lu" ], "doi": "10.1007/s10208-024-09663-w", "url": "https://doi.org/10.1007/s10208-024-09663-w", "volume": "25", "issue": "4", "pages": "1199-1257" }, "number": 192 }, { "id": "mp-2024-second-degeneracy-general-nonsmooth-optimization", "journal": "Mathematical Programming", "publicationYear": 2024, "title": "Degeneracy Without Slowdown in Nonsmooth Optimization", "topic": "First-order methods and nonsmooth optimization", "publicationDate": "2024-07-17", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 6, extensions to nonsmooth optimization", "sourceQuote": "How to extend this result to general non-smooth optimization can be an interesting future direction.", "problemStatement": "Extend the source's conclusion that degeneracy itself need not slow first-order convergence from linear programming to a general class of nonsmooth optimization problems. The desired theory should explain finite identification and the subsequent local rate without the usual nondegeneracy or irrepresentability condition. It must also isolate a quantitative near-degeneracy measure that replaces the linear-programming geometry used in the source.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact statement, title, authors and citations across publisher pages, arXiv and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper direct result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "nonsmooth optimization", "degeneracy", "manifold identification", "partial smoothness" ], "literature": [], "source": { "title": "On the geometry and refined rate of primal–dual hybrid gradient for linear programming", "authors": [ "Haihao Lu", "Jinwen Yang" ], "doi": "10.1007/s10107-024-02109-9", "url": "https://doi.org/10.1007/s10107-024-02109-9", "volume": "212", "issue": "1-2", "pages": "349-387" }, "number": 193 }, { "id": "mp-2024-second-two-stage-admm-linear-programming", "journal": "Mathematical Programming", "publicationYear": 2024, "title": "Two-Stage ADMM Theory for Linear Programming", "topic": "First-order methods and linear programming", "publicationDate": "2024-07-17", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 6, extensions to other primal-dual algorithms", "sourceQuote": "most of the theoretical results can be extended to ADMM.", "problemStatement": "Establish for ADMM on linear programs the two-stage complexity theory proved in the source for primal-dual hybrid gradient. The first stage should identify nonbasic and nondegenerate basic sets using a near-degeneracy metric, and the second should have a local linear rate governed by a local sharpness parameter rather than a global Hoffman constant. The analysis should cover algorithmic forms used by solvers such as OSQP and SCS.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact statement, title, authors and citations across publisher pages, arXiv and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper direct result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "ADMM", "linear programming", "two-stage convergence", "active-set identification" ], "literature": [], "source": { "title": "On the geometry and refined rate of primal–dual hybrid gradient for linear programming", "authors": [ "Haihao Lu", "Jinwen Yang" ], "doi": "10.1007/s10107-024-02109-9", "url": "https://doi.org/10.1007/s10107-024-02109-9", "volume": "212", "issue": "1-2", "pages": "349-387" }, "number": 194 }, { "id": "mp-2024-second-two-stage-infeasible-linear-programs", "journal": "Mathematical Programming", "publicationYear": 2024, "title": "Two-Stage Behavior on Infeasible Linear Programs", "topic": "First-order methods and linear programming", "publicationDate": "2024-07-17", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 6, extensions to infeasible problems", "sourceQuote": "How to theoretically analyze it without the non-degeneracy condition can be another interesting question.", "problemStatement": "Analyze the empirically observed two-stage behavior of primal-dual hybrid gradient on infeasible linear programs without assuming nondegeneracy. A satisfactory result should identify the relevant active or certificate structure, quantify the transition time, and give the later local rate for detecting infeasibility. Accelerated certificate recovery for a restarted variant does not establish those two stages or their transition and therefore does not resolve a special case of the retained task.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact statement, title, authors and citations across publisher pages, arXiv and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper direct result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "infeasible linear programming", "PDHG", "infeasibility certificate", "degeneracy" ], "literature": [], "source": { "title": "On the geometry and refined rate of primal–dual hybrid gradient for linear programming", "authors": [ "Haihao Lu", "Jinwen Yang" ], "doi": "10.1007/s10107-024-02109-9", "url": "https://doi.org/10.1007/s10107-024-02109-9", "volume": "212", "issue": "1-2", "pages": "349-387" }, "number": 195 }, { "id": "mp-2024-second-two-stage-restarted-primal-dual-methods", "journal": "Mathematical Programming", "publicationYear": 2024, "title": "Two-Stage Theory for Restarted Primal-Dual Methods", "topic": "First-order methods and linear programming", "publicationDate": "2024-07-17", "recordType": "reviewed", "status": "partial_progress", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 6, extensions to restarted algorithms", "sourceQuote": "We believe our theoretical results can be extended to the restarted algorithms, and we leave it as future work.", "problemStatement": "Extend the source's identification and local-sharpness analysis from unrestarted primal-dual hybrid gradient to restarted primal-dual algorithms for linear programming. The target is a rigorous two-stage bound that separates identification from the eventual local regime and explains the benefit of restart without reverting to a conservative global Hoffman constant. Later work proves this behavior for important restarted PDHG variants and unique-optimum instances, but not in the source's full generality.", "statusEvidence": "Proves accelerated two-stage convergence for restarted Halpern PDHG on feasible bounded linear programs. Analyzes basis identification and a second local stage for restarted PDHG under a unique-optimizer assumption.", "verificationNote": "This status is tied to the cited primary-literature result and the precise scope stated above.", "keywords": [ "restarted PDHG", "linear programming", "two-stage convergence", "local sharpness" ], "literature": [ { "title": "Restarted Halpern PDHG for Linear Programming", "url": "https://arxiv.org/abs/2407.16144", "relation": "Proves accelerated two-stage convergence for restarted Halpern PDHG on feasible bounded linear programs." }, { "title": "Accessible Complexity Bounds for Restarted PDHG on Linear Programs with a Unique Optimizer", "url": "https://arxiv.org/abs/2410.04043", "relation": "Analyzes basis identification and a second local stage for restarted PDHG under a unique-optimizer assumption." } ], "source": { "title": "On the geometry and refined rate of primal–dual hybrid gradient for linear programming", "authors": [ "Haihao Lu", "Jinwen Yang" ], "doi": "10.1007/s10107-024-02109-9", "url": "https://doi.org/10.1007/s10107-024-02109-9", "volume": "212", "issue": "1-2", "pages": "349-387" }, "number": 196 }, { "id": "mor-2024-W4400732905_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2024, "title": "Equivalence between discount-to-one limit and average optimality in risk-averse MDPs", "topic": "Risk Averse Markov Decision", "publicationDate": "2024-07-17", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 14, Remark 2.12 (problem formulation discussion)", "sourceQuote": "However, the equivalence of this limit in current context remains unclear.", "problemStatement": "A distributional dynamic risk measure evaluates a controlled Markov process by recursively applying one-step risk mappings rather than expected cost. In risk-neutral finite models, multiplying the discounted value by 1-γ and sending γ to 1 recovers an average-cost problem under standard assumptions. Establish an analogous discount-to-average limit and average-optimality equation for the risk-averse framework.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "risk-averse mdp", "dynamic risk measures", "discount factor limit", "average optimality", "Blackwell optimality" ], "literature": [], "source": { "title": "Risk-Averse Markov Decision Processes Through a Distributional Lens", "authors": [ "Ziteng Cheng", "Sebastian Jaimungal" ], "doi": "10.1287/moor.2023.0211", "url": "https://doi.org/10.1287/moor.2023.0211", "volume": "50", "issue": "3", "pages": "1707-1733" }, "number": 197 }, { "id": "siopt-2024-039", "journal": "SIAM Journal on Optimization", "publicationYear": 2024, "title": "Square root lasso assumptions and uniqueness", "topic": "Statistical optimization", "publicationDate": "2024-07-18", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conclusion and Appendix A preface; extracted preprint lines 1485-1491 and 1552-1553", "sourceQuote": "theoretically showing the validity of Assumption 1 for suitable classes of sampling matrices remains an open problem.", "problemStatement": "Square-root LASSO minimizes ‖Ax−b‖₂+λ‖x‖₁; let x̄ be a solution, I=supp(x̄), and y=(b−Ax̄)/‖Ax̄−b‖₂. Assumption 1 requires A_I to have full column rank, b∉range(A_I), and some z∈ker(A_Iᵀ)∩y^⊥ satisfying ‖A_(I^c)ᵀ(y+z)‖_∞<λ. Prove this condition with high probability for useful random sampling matrices and decide whether it is necessary for uniqueness. Determine separately whether the standing requirement b∉range(A_I) can be removed from the variational analysis.", "statusEvidence": "No forward-citing primary paper or exact-query result explicitly claiming a proof or counterexample was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "square-root LASSO", "random matrices", "Lipschitz stability", "uniqueness" ], "literature": [], "source": { "title": "Square Root LASSO: Well-Posedness, Lipschitz Stability, and the Tuning Trade-Off", "authors": [ "Aaron Berk", "Simone Brugiapaglia", "Tim Hoheisel" ], "doi": "10.1137/23m1561968", "url": "https://doi.org/10.1137/23m1561968", "preprintUrl": "https://arxiv.org/abs/2303.15588", "volume": "34", "issue": "3", "pages": "2609-2637" }, "number": 198 }, { "id": "siopt-2024-019", "journal": "SIAM Journal on Optimization", "publicationYear": 2024, "title": "Single versus multipoint zeroth order rates", "topic": "Derivative-free optimization", "publicationDate": "2024-07-19", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Introduction; extracted preprint lines 288-294", "sourceQuote": "whether single-point estimators enable equally fast convergence rates as multi-point estimators.", "problemStatement": "Zeroth-order algorithms estimate gradients from function values. Multi-point estimators achieve faster known rates but require multiple new evaluations, whereas a single-point estimator is cheaper per call. Determine whether a genuine single-point estimator can attain the best multi-point convergence rates without hiding extra evaluations in memory or batching.", "statusEvidence": "No forward-citing primary paper or exact-query result explicitly claiming a proof or counterexample was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "zeroth-order optimization", "single-point estimator", "multi-point estimator", "convergence rate" ], "literature": [], "source": { "title": "Small Errors in Random Zeroth-Order Optimization Are Imaginary", "authors": [ "Wouter Jongeneel", "Man-Chung Yue", "Daniel Kuhn" ], "doi": "10.1137/22m1510261", "url": "https://doi.org/10.1137/22m1510261", "preprintUrl": "https://arxiv.org/abs/2103.05478", "volume": "34", "issue": "3", "pages": "2638-2670" }, "number": 199 }, { "id": "siopt-2024-041", "journal": "SIAM Journal on Optimization", "publicationYear": 2024, "title": "Vector optimization rate general norm", "topic": "Vector optimization", "publicationDate": "2024-07-25", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conclusion; extracted preprint lines 1414-1416", "sourceQuote": "It is an open problem to prove the improved convergence rate for the proposed algorithm in the general setting", "problemStatement": "Consider the convex vector problem of minimizing a continuous C-convex map Γ:X→ℝ^q over compact convex X, where the ordering cone C⊊ℝ^q is closed, convex, solid, pointed, and polyhedral. The outer-approximation algorithm repeatedly solves min ‖z‖ subject to Γ(x)−z−v≤_C0 and a compactifying halfspace, using an arbitrary fixed norm on ℝ^q; for its Hausdorff error it proves O(k^(−1/(q−1))) for every norm but the sharper, best-possible O(k^(−2/(q−1))) only for the Euclidean norm. Prove the sharper rate for every norm under these assumptions, or characterize exactly which norms permit it.", "statusEvidence": "No forward-citing primary paper or exact-query result explicitly claiming a proof or counterexample was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "convex vector optimization", "scalarization", "general norm", "convergence rate" ], "literature": [], "source": { "title": "Convergence Analysis of a Norm Minimization-Based Convex Vector Optimization Algorithm", "authors": [ "Çağin Ararat", "Firdevs Ulus", "Muhammad Umer" ], "doi": "10.1137/23m1574580", "url": "https://doi.org/10.1137/23m1574580", "preprintUrl": "https://arxiv.org/abs/2302.08723", "volume": "34", "issue": "3", "pages": "2700-2728" }, "number": 200 }, { "id": "focm-2024-general-symplectic-resonance-integrators", "journal": "Foundations of Computational Mathematics", "publicationYear": 2024, "title": "General Symplectic Resonance Integrators", "topic": "Geometric numerical integration", "publicationDate": "2024-07-26", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 5, symplecticity discussion", "sourceQuote": "The construction of general symplectic resonance-based schemes remains open.", "problemStatement": "Develop a general construction of symplectic resonance-based time integrators beyond the one-dimensional NLS and KdV examples. The scheme should preserve symplectic structure while exactly incorporating the leading oscillations.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "symplectic integrators", "resonance", "dispersive PDEs", "structure preservation" ], "literature": [], "source": { "title": "Resonances as a Computational Tool", "authors": [ "Frédéric Rousset", "Katharina Schratz" ], "doi": "10.1007/s10208-024-09665-8", "url": "https://doi.org/10.1007/s10208-024-09665-8", "preprintUrl": "https://arxiv.org/abs/2405.10572", "volume": "25", "issue": "6", "pages": "1879-1906" }, "number": 201 }, { "id": "focm-2024-second-order-resonance-low-regularity-nls", "journal": "Foundations of Computational Mathematics", "publicationYear": 2024, "title": "Second Order Resonance Low Regularity NLS", "topic": "Numerical analysis of dispersive PDEs", "publicationDate": "2024-07-26", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Remark 2.4", "sourceQuote": "It remains open to obtain second-order convergence under the same regularity assumptions.", "problemStatement": "Prove second-order convergence of the resonance-based nonlinear-Schrodinger integrator under the low regularity sufficient for the first-order method. Existing second-order estimates require stronger Sobolev regularity. Resonance integrators exactly retain the dominant Fourier-frequency interaction; the first-order analysis loses fewer derivatives than classical splitting, but the second-order proof still loses two.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "resonance integrators", "nonlinear Schrodinger equation", "low regularity", "second order" ], "literature": [], "source": { "title": "Resonances as a Computational Tool", "authors": [ "Frédéric Rousset", "Katharina Schratz" ], "doi": "10.1007/s10208-024-09665-8", "url": "https://doi.org/10.1007/s10208-024-09665-8", "preprintUrl": "https://arxiv.org/abs/2405.10572", "volume": "25", "issue": "6", "pages": "1879-1906" }, "number": 202 }, { "id": "mp-2024-second-necessity-metric-subregularity-markov-operators", "journal": "Mathematical Programming", "publicationYear": 2024, "title": "Necessity of Metric Subregularity for Markov Convergence", "topic": "Stochastic fixed-point methods", "publicationDate": "2024-07-27", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Final remarks, first open technicality", "sourceQuote": "whether metric subregularity is necessary for quantitative convergence in some appropriate metric of Markov operators that are not paracontractions in measure.", "problemStatement": "Determine whether metric subregularity is necessary for quantitative convergence, in a suitable metric, of Markov operators generated by randomized fixed-point iterations that are not paracontractions in measure. The source develops sufficient conditions for convergence in distribution of partially separable algorithms and conjectures necessity in this broader setting. A resolution may be a necessity theorem with a precise rate metric or a counterexample that converges quantitatively without subregularity.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact statement, title, authors and citations across publisher pages, arXiv and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper direct result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "metric subregularity", "Markov operator", "randomized iteration", "convergence in distribution" ], "literature": [], "source": { "title": "Convergence in distribution of randomized algorithms: the case of partially separable optimization", "authors": [ "D. Russell Luke" ], "doi": "10.1007/s10107-024-02124-w", "url": "https://doi.org/10.1007/s10107-024-02124-w", "volume": "212", "issue": "1-2", "pages": "763-798" }, "number": 203 }, { "id": "mp-2024-second-supports-invariant-measures-inconsistent-fixed-points", "journal": "Mathematical Programming", "publicationYear": 2024, "title": "Supports of Invariant Measures in Inconsistent Fixed-Point Problems", "topic": "Stochastic fixed-point methods", "publicationDate": "2024-07-27", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Final remarks, third open technicality", "sourceQuote": "Characterization of the supports of invariant measures in the inconsistent case is quite challenging and essential", "problemStatement": "Characterize the supports of invariant probability measures for inconsistent stochastic fixed-point problems. Such a characterization is needed to connect the limiting distributions of randomized algorithms with solutions or approximate solutions of the underlying optimization model. The open task is to state verifiable conditions on the update maps and sampling law that determine, bound, or otherwise meaningfully describe those supports.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact statement, title, authors and citations across publisher pages, arXiv and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper direct result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "invariant measure", "inconsistent feasibility", "support characterization", "Markov chain" ], "literature": [], "source": { "title": "Convergence in distribution of randomized algorithms: the case of partially separable optimization", "authors": [ "D. Russell Luke" ], "doi": "10.1007/s10107-024-02124-w", "url": "https://doi.org/10.1007/s10107-024-02124-w", "volume": "212", "issue": "1-2", "pages": "763-798" }, "number": 204 }, { "id": "mp-2024-second-weak-convergence-pointwise-fne-expectation", "journal": "Mathematical Programming", "publicationYear": 2024, "title": "Weak Convergence Under Firm Nonexpansiveness in Expectation", "topic": "Stochastic fixed-point methods", "publicationDate": "2024-07-27", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Final remarks, second open technicality", "sourceQuote": "at issue here is whether this holds when Ti is pointwise α-fne in expectation at invariant measures of the corresponding Markov operator.", "problemStatement": "Decide whether almost-sure weak convergence persists when every random update Ti is only pointwise α-firmly nonexpansive in expectation at invariant measures of its Markov operator. The analogous conclusion holds when each update is itself α-firmly nonexpansive, but the source expects a counterexample under the weaker averaged property. The task is to construct that counterexample or prove convergence under explicit supplementary assumptions.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact statement, title, authors and citations across publisher pages, arXiv and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper direct result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "firm nonexpansiveness", "weak convergence", "invariant measure", "randomized fixed points" ], "literature": [], "source": { "title": "Convergence in distribution of randomized algorithms: the case of partially separable optimization", "authors": [ "D. Russell Luke" ], "doi": "10.1007/s10107-024-02124-w", "url": "https://doi.org/10.1007/s10107-024-02124-w", "volume": "212", "issue": "1-2", "pages": "763-798" }, "number": 205 }, { "id": "mor-2024-W4401117667_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2024, "title": "Largest achievable domination factor for feasible self-maximizing polynomial-time shares under additivity", "topic": "Fair Division", "publicationDate": "2024-07-30", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 7, Section 1.2 (Our Results), end of paragraph discussing [GT20] and Theorem 3; reiterated Page 26, Section 7 (Discussion), final paragraph.", "sourceQuote": "the largest value of ρ for which there is a ρ-dominating share that is feasible, self maximizing, and poly-time computable for additive valuations.", "problemStatement": "A share rule for indivisible goods is self-maximizing if an agent cannot improve her guarantee by misreporting, and its domination factor measures how strongly it improves on a baseline fair share. The paper studies feasible, polynomial-time share rules under additive values. Determine the largest domination factor achievable while retaining feasibility, self-maximization, and polynomial-time computability.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "fair division", "indivisible goods", "maximin share", "truthful reporting", "self-maximizing shares", "approximation ratio", "polynomial time" ], "literature": [], "source": { "title": "Fair Shares: Feasibility, Domination, and Incentives", "authors": [ "Moshe Babaioff", "Uriel Feige" ], "doi": "10.1287/moor.2022.0257", "url": "https://doi.org/10.1287/moor.2022.0257", "volume": "50", "issue": "3", "pages": "1901-1934" }, "number": 206 }, { "id": "mp-2024-second-constant-competitive-online-configuration-balancing", "journal": "Mathematical Programming", "publicationYear": 2024, "title": "Constant-Competitive Online Configuration Balancing", "topic": "Online algorithms and scheduling", "publicationDate": "2024-08-08", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conclusion", "sourceQuote": "It remains an interesting open question whether the online setting admits an O(1)-competitive algorithm.", "problemStatement": "Design an O(1)-competitive online algorithm for stochastic configuration balancing on related machines. Jobs arrive sequentially with stochastic processing requirements, and the algorithm must assign each request without seeing future requests while controlling the expected maximum load. The source gives an O(log log m)-competitive online guarantee and a constant-factor offline algorithm, leaving the dependence on the number m of machines open.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact statement, title, authors and citations across publisher pages, arXiv and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper direct result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "configuration balancing", "online scheduling", "competitive ratio", "stochastic requests" ], "literature": [], "source": { "title": "Configuration balancing for stochastic requests", "authors": [ "Franziska Eberle", "Anupam Gupta", "Nicole Megow", "Benjamin Moseley", "Rudy Zhou" ], "doi": "10.1007/s10107-024-02132-w", "url": "https://doi.org/10.1007/s10107-024-02132-w", "volume": "210", "issue": "1-2", "pages": "243-279" }, "number": 207 }, { "id": "mor-2024-W4401514485_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2024, "title": "Characterize simultaneous ordered spectral decompositions needed for spectral-function critical cones", "topic": "Spectral Convex Analysis", "publicationDate": "2024-08-12", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 5 (discussion immediately after Proposition 5.4), page 25.", "sourceQuote": "when the matrices Λ(Y)_(alpha_malpha_m) and U_(alpha_m)^top H U_(alpha_m) in Proposition 5.4 have a simultaneous ordered spectral decomposition", "problemStatement": "For a convex spectral function, parabolic regularity reduces to eigenvalue-multiplicity blocks of a symmetric matrix. The missing condition asks when the dual-eigenvalue block and a projected direction matrix admit a common eigenbasis with the required eigenvalue order. Give necessary and sufficient, checkable conditions for that simultaneous ordered spectral decomposition.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "spectral functions", "critical cone", "simultaneous diagonalization", "ordered eigenvalues", "second-order variational analysis" ], "literature": [], "source": { "title": "Parabolic Regularity of Spectral Functions", "authors": [ "Ashkan Mohammadi", "Ebrahim Sarabi" ], "doi": "10.1287/moor.2023.0010", "url": "https://doi.org/10.1287/moor.2023.0010", "volume": "50", "issue": "3", "pages": "2017-2046" }, "number": 208 }, { "id": "mor-2024-W4401542271_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2024, "title": "Enumerate minimal balanced collections on n players in cooperative games", "topic": "Cooperative Game Theory", "publicationDate": "2024-08-13", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Introduction, page 2 (discussion of minimal balanced collections and enumeration); also referenced indirectly in Section 5.5 via the number of facets depending on the number of minimal balanced collections.", "sourceQuote": "Indeed, the concept of minimal balanced collection generalizes the concept of partitions, and their enumeration remains a problem.", "problemStatement": "A balanced collection is a family of coalitions that can be assigned nonnegative weights so every player has total weight one; it is minimal if no proper subfamily is balanced. Such collections characterize balanced cooperative games, and every minimal collection on n players has at most n coalitions. Enumerate all minimal balanced collections or give an explicit formula for their number as a function of n.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "balanced collections", "core nonemptiness", "Bondareva-Shapley theorem", "polyhedral cones", "facet enumeration", "set systems" ], "literature": [], "source": { "title": "On the Set of Balanced Games", "authors": [ "Pedro Garcia-Segador", "Michel Grabisch", "Pedro Miranda" ], "doi": "10.1287/moor.2023.0379", "url": "https://doi.org/10.1287/moor.2023.0379", "preprintUrl": "https://arxiv.org/pdf/2501.14341", "volume": "50", "issue": "3", "pages": "2047-2072" }, "number": 209 }, { "id": "siopt-2024-035", "journal": "SIAM Journal on Optimization", "publicationYear": 2024, "title": "MGPROX multilevel acceleration", "topic": "Multilevel optimization", "publicationDate": "2024-08-13", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Numerical discussion and conclusion; extracted preprint lines 1419-1422", "sourceQuote": "How the multi-level process enhances convergence speed remains an open problem.", "problemStatement": "MGProx applies multigrid restriction and prolongation to proximal-gradient steps for strongly convex nonsmooth problems and is empirically much faster than its proved descent bound predicts. The multilevel correction may act as a variable metric or preconditioner, but no rate quantifies this effect. Derive a convergence theory that explains when and by how much the multilevel process accelerates the method.", "statusEvidence": "No forward-citing primary paper or exact-query result explicitly claiming a proof or counterexample was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "multigrid", "proximal gradient", "nonsmooth optimization", "convergence rate" ], "literature": [], "source": { "title": "MGProx: A Nonsmooth Multigrid Proximal Gradient Method with Adaptive Restriction for Strongly Convex Optimization", "authors": [ "Andersen Ang", "Hans De Sterck", "Stephen Vavasis" ], "doi": "10.1137/23m1552140", "url": "https://doi.org/10.1137/23m1552140", "preprintUrl": "https://arxiv.org/abs/2302.04077", "volume": "34", "issue": "3", "pages": "2788-2820" }, "number": 210 }, { "id": "focm-2024-physical-space-form-low-regularity-integrators", "journal": "Foundations of Computational Mathematics", "publicationYear": 2024, "title": "Physical Space Form Low Regularity Integrators", "topic": "Numerical analysis of dispersive PDEs", "publicationDate": "2024-08-15", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 5, discussion after the general exact-integration construction", "sourceQuote": "low regualrity schemes can always be rewritten in physical space. It is not clear how to prove such a statement in full generality", "problemStatement": "Find a general theorem deciding when low-regularity time integrators obtained by exact Fourier-space integration can be rewritten in physical space. The result should go beyond the individual dispersive equations treated in the paper.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "low-regularity integrators", "Fourier analysis", "physical-space formulation" ], "literature": [], "source": { "title": "Approximations of Dispersive PDEs in the Presence of Low-Regularity Randomness", "authors": [ "Yvonne Alama Bronsard", "Yvain Bruned", "Katharina Schratz" ], "doi": "10.1007/s10208-023-09625-8", "url": "https://doi.org/10.1007/s10208-023-09625-8", "preprintUrl": "https://arxiv.org/abs/2205.02156", "volume": "24", "issue": "6", "pages": "1819-1869" }, "number": 211 }, { "id": "focm-2024-ramsey-c34-curve", "journal": "Foundations of Computational Mathematics", "publicationYear": 2024, "title": "Ramsey c(3,4) Curve", "topic": "Extremal combinatorics", "publicationDate": "2024-08-26", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 1.2, the c_(3,4)(x) problem", "sourceQuote": "Let us take a closer look at the smallest open case, that is s=3 and t=4.", "problemStatement": "Determine the missing extremal behavior of the feasible curve relating triangle density to complementary K4 density. In the paper's notation this is the smallest open off-diagonal density problem c_(3,4)(x). The function c_(3,4)(x) is the minimum complementary K4 density among graph limits with triangle density x, so determining it is a constrained extremal graphon problem.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "Ramsey multiplicity", "triangle density", "K4 density", "graphons" ], "literature": [], "source": { "title": "New Ramsey Multiplicity Bounds and Search Heuristics", "authors": [ "Olaf Parczyk", "Sebastian Pokutta", "Christoph Spiegel", "Tibor Szabó" ], "doi": "10.1007/s10208-024-09675-6", "url": "https://doi.org/10.1007/s10208-024-09675-6", "volume": "25", "issue": "5", "pages": "1777-1814" }, "number": 212 }, { "id": "focm-2024-ramsey-c4-weighted-blowup-optimality", "journal": "Foundations of Computational Mathematics", "publicationYear": 2024, "title": "Ramsey C4 Weighted Blowup Optimality", "topic": "Extremal combinatorics", "publicationDate": "2024-08-26", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Introduction, discussion of weighted and iterated blow-ups", "sourceQuote": "whether a tight upper bound can be achieved through the sequence of (possibly weighted or iterated) blow-ups of a finite-sized graph", "problemStatement": "Determine whether the best known upper bound for the K4 Ramsey multiplicity constant c4 is attained by a weighted or iterated blow-up of a finite graph. The analogous structural description is known in the smaller cases discussed by the authors.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "Ramsey multiplicity", "K4", "weighted blow-up", "graph limits" ], "literature": [], "source": { "title": "New Ramsey Multiplicity Bounds and Search Heuristics", "authors": [ "Olaf Parczyk", "Sebastian Pokutta", "Christoph Spiegel", "Tibor Szabó" ], "doi": "10.1007/s10208-024-09675-6", "url": "https://doi.org/10.1007/s10208-024-09675-6", "volume": "25", "issue": "5", "pages": "1777-1814" }, "number": 213 }, { "id": "mor-2024-W3216294154_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2024, "title": "Prove convexity and monotonicity of the optimal robust ratio in acceptance probability", "topic": "Secretary Problem", "publicationDate": "2024-08-29", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 9 (Concluding Remarks), page 24, paragraph beginning “We empirically observe …”", "sourceQuote": "the robust ratio gamma_n^*(p) is convex and decreasing as a function of p", "problemStatement": "In the secretary problem with uncertain acceptance, n candidates arrive in random order, only relative ranks are observed, and each offer is independently accepted with probability p. The robust ratio compares, in the worst case over top-k benchmarks, the probability of successfully hiring a top-k candidate with the probability that some top-k candidate would accept. Prove or refute that the optimal finite-horizon ratio and its infinite-horizon limit are both decreasing and convex functions of p.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "secretary problem", "robust ratio", "uncertain acceptance", "convexity", "monotonicity", "optimal stopping" ], "literature": [], "source": { "title": "Robust Online Selection with Uncertain Offer Acceptance", "authors": [ "Sebastian Perez-Salazar", "Mohit Singh", "Alejandro Toriello" ], "doi": "10.1287/moor.2023.0210", "url": "https://doi.org/10.1287/moor.2023.0210", "volume": "50", "issue": "3", "pages": "2226-2260" }, "number": 214 }, { "id": "mor-2024-W4402128674_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2024, "title": "Construct natural valuation counterexamples to envy-free multi-divisions with long knives", "topic": "Fair Division", "publicationDate": "2024-09-02", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 25, Section 6 (Concluding remarks), paragraph beginning “Limitation of the approach based on Volovikov’s theorem.”", "sourceQuote": "However, it is still an open question whether or not a counterexample exists for some natural valuation functions.", "problemStatement": "A multi-layered cake consists of parallel unit intervals, and a feasible contiguous piece uses one interval per layer without overlapping positions across layers. For a division into q pieces, q−1 long knives cut every layer at the same q−1 positions; an envy-free division assigns every agent to a most-preferred piece and uses every piece. Construct natural valuations for which no feasible contiguous envy-free q-piece division arises from q−1 long knives, or prove existence for a broad natural valuation class.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "fair division", "envy-free cake cutting", "multi-layered cake", "long knives", "valuation functions", "topological methods" ], "literature": [], "source": { "title": "Envy-Free Division of Multilayered Cakes", "authors": [ "Ayumi Igarashi", "Frédéric Meunier" ], "doi": "10.1287/moor.2022.0350", "url": "https://doi.org/10.1287/moor.2022.0350", "volume": "50", "issue": "3", "pages": "2261-2286" }, "number": 215 }, { "id": "siopt-2024-014", "journal": "SIAM Journal on Optimization", "publicationYear": 2024, "title": "Parameter free optimal type II convex method", "topic": "First-order complexity", "publicationDate": "2024-09-04", "recordType": "reviewed", "status": "resolved", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Introduction, convex setting; extracted preprint lines 129-133", "sourceQuote": "Whether a parameter-free complexity-optimal type-II method exists in the convex setting is still unknown.", "problemStatement": "A type-II method terminates using a stationarity measure such as the norm of a gradient or projected gradient, rather than a function-value gap. At publication, known optimal convex type-II methods required problem constants or distance information. Later work produced parameter-free algorithms matching the convex gradient-norm lower bound, so this problem is marked resolved.", "statusEvidence": "Constructs parameter-free convex methods attaining the optimal gradient-norm complexity and extends them to constrained and composite settings.", "verificationNote": "This status is tied to the cited primary-literature result and the precise scope stated above.", "keywords": [ "parameter-free methods", "gradient norm", "convex optimization", "oracle complexity" ], "literature": [ { "title": "Optimal and parameter-free gradient minimization methods for convex and nonconvex optimization", "url": "https://doi.org/10.1007/s10107-026-02352-2", "relation": "Constructs parameter-free convex methods attaining the optimal gradient-norm complexity and extends them to constrained and composite settings." } ], "source": { "title": "Complexity-Optimal and Parameter-Free First-Order Methods for Finding Stationary Points of Composite Optimization Problems", "authors": [ "Weiwei Kong" ], "doi": "10.1137/22m1498826", "url": "https://doi.org/10.1137/22m1498826", "preprintUrl": "https://arxiv.org/abs/2205.13055", "volume": "34", "issue": "3", "pages": "3005-3032" }, "number": 216 }, { "id": "siopt-2024-043", "journal": "SIAM Journal on Optimization", "publicationYear": 2024, "title": "PEP gradient method exact formula", "topic": "Performance estimation", "publicationDate": "2024-09-05", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conjecture 4.2; extracted preprint lines 886-922", "sourceQuote": "For all 0 < μ_g ≤ 1 and 0 ≤ μ_M ≤ 1, we have w(D_(μ_g)^(μ_M);h)=1/2 maxμ_gα*²/[μ_g−1+(1−μ_gα*²h)^(−2N)],(1−h)^(2N).", "problemStatement": "Let D_(μ_g)^(μ_M) contain F=g∘M where g is 1-smooth and μ_g-strongly convex and M is symmetric with eigenvalues in [μ_M,1], and let w(D_(μ_g)^(μ_M);h) be the normalized objective gap after N gradient steps of size h. Define α*=proj_[μ_M,1](√(h_0/h)), where h_0∈[0,1/μ_g] solves (1−μ_g)(1−μ_gh_0)^(2N+1)=1−(2N+1)μ_gh_0. Conjecture 4.2 gives w=1/2 maxμ_gα*²/[μ_g−1+(1−μ_gα*²h)^(−2N)],(1−h)^(2N). Prove this identity for all N, h, μ_g, and μ_M; the paper proves only the case N=1, h=1, μ_M=0 in a restricted μ_g range.", "statusEvidence": "No forward-citing primary paper or exact-query result explicitly claiming a proof or counterexample was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "gradient method", "exact worst case", "performance estimation", "conditioned operator" ], "literature": [], "source": { "title": "Interpolation Conditions for Linear Operators and Applications to Performance Estimation Problems", "authors": [ "Nizar Bousselmi", "Julien M. Hendrickx", "François Glineur" ], "doi": "10.1137/23m1575391", "url": "https://doi.org/10.1137/23m1575391", "preprintUrl": "https://arxiv.org/abs/2302.08781", "volume": "34", "issue": "3", "pages": "3033-3063" }, "number": 217 }, { "id": "siopt-2024-042", "journal": "SIAM Journal on Optimization", "publicationYear": 2024, "title": "PEP linear operator worst cases", "topic": "Performance estimation", "publicationDate": "2024-09-05", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conjecture 4.1; extracted preprint lines 843-848", "sourceQuote": "Worst-case performances of C and D are reached by scaling operator M = alpha I", "problemStatement": "For F=g∘M with g 1-smooth and μ_g-strongly convex, class C permits an arbitrary linear M with singular values in [0,1], while class D restricts M to be symmetric with eigenvalues in [μ_M,1]. For N fixed-step gradient iterations with normalized step h, w(C;h) and w(D;h) are the suprema of F(x_N)−F* over functions in the respective class and initial points with ‖x_0−x*‖≤1. Prove that each worst case is attained by a scaling M=αI, or exhibit a nonscalar operator with strictly worse performance.", "statusEvidence": "No forward-citing primary paper or exact-query result explicitly claiming a proof or counterexample was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "linear operators", "performance estimation", "worst case", "scaling operator" ], "literature": [], "source": { "title": "Interpolation Conditions for Linear Operators and Applications to Performance Estimation Problems", "authors": [ "Nizar Bousselmi", "Julien M. Hendrickx", "François Glineur" ], "doi": "10.1137/23m1575391", "url": "https://doi.org/10.1137/23m1575391", "preprintUrl": "https://arxiv.org/abs/2302.08781", "volume": "34", "issue": "3", "pages": "3033-3063" }, "number": 218 }, { "id": "siopt-2024-050", "journal": "SIAM Journal on Optimization", "publicationYear": 2024, "title": "Alpha MCMGP NP hardness", "topic": "Combinatorial optimization", "publicationDate": "2024-09-10", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Complexity discussion; extracted preprint line 1237", "sourceQuote": "We conjecture that, in general, the alpha-MCMGP is NP-hard.", "problemStatement": "For a rational weight vector α=(s_1,…,s_d)/ŝ, an α-mediated graph contains anchor nodes 0,ŝe_1,…,ŝe_(d−1), the target b_α=(s_1,…,s_(d−1)), and auxiliary nodes X such that every x∈X is the midpoint of two distinct graph nodes. The Minimum Cardinality α-Mediated Graph Problem minimizes |X|; by a one-to-one correspondence, this also minimizes the number of second-order cones in an extended representation of the rational power cone x≤z^α. Prove that the general α-MCMGP is NP-hard, or give a polynomial-time algorithm.", "statusEvidence": "No forward-citing primary paper or exact-query result explicitly claiming a proof or counterexample was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "maximum cardinality", "minimum gain", "graph optimization", "NP-hardness" ], "literature": [], "source": { "title": "On Minimal Extended Representations of Generalized Power Cones", "authors": [ "Víctor Blanco", "Miguel Martínez-Antón" ], "doi": "10.1137/23m1617205", "url": "https://doi.org/10.1137/23m1617205", "preprintUrl": "https://arxiv.org/abs/2311.10470", "volume": "34", "issue": "3", "pages": "3088-3111" }, "number": 219 }, { "id": "focm-2024-gw-monge-map-broader-measures", "journal": "Foundations of Computational Mathematics", "publicationYear": 2024, "title": "GW Monge Map Broader Measures", "topic": "Optimal transport", "publicationDate": "2024-09-16", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Question 4", "sourceQuote": "Can the existence of Monge optimal couplings be extended to more general measures?", "problemStatement": "A Monge coupling is a GW optimizer supported on the graph of a measurable map. Extend the paper's Monge-existence results from ambient absolutely continuous measures and its proved equatorial optimizer between uniform sphere measures to broader singular source and target measures. The extension should state conditions guaranteeing a map optimizer despite failure of absolute continuity in the surrounding Euclidean space.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "Gromov-Wasserstein", "Monge map", "singular measures", "sphere" ], "literature": [], "source": { "title": "The Gromov–Wasserstein Distance Between Spheres", "authors": [ "Shreya Arya", "Arnab Auddy", "Ranthony A. Clark", "Sunhyuk Lim", "Facundo Mémoli", "Daniel Packer" ], "doi": "10.1007/s10208-024-09678-3", "url": "https://doi.org/10.1007/s10208-024-09678-3", "volume": "26", "issue": "1", "pages": "75-130" }, "number": 220 }, { "id": "focm-2024-gw-polynomial-time-classes", "journal": "Foundations of Computational Mathematics", "publicationYear": 2024, "title": "GW Polynomial Time Classes", "topic": "Optimal transport", "publicationDate": "2024-09-16", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Question 1", "sourceQuote": "Are there meaningful classes of metric measure spaces for which the (p,q)-GW distance is computable in polynomial time?", "problemStatement": "Identify nontrivial classes of metric-measure spaces on which the (p,q)-Gromov-Wasserstein distance can be computed in polynomial time. The result should give an exact algorithm rather than a generic nonconvex heuristic.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "Gromov-Wasserstein", "computational complexity", "polynomial time", "metric measure spaces" ], "literature": [], "source": { "title": "The Gromov–Wasserstein Distance Between Spheres", "authors": [ "Shreya Arya", "Arnab Auddy", "Ranthony A. Clark", "Sunhyuk Lim", "Facundo Mémoli", "Daniel Packer" ], "doi": "10.1007/s10208-024-09678-3", "url": "https://doi.org/10.1007/s10208-024-09678-3", "volume": "26", "issue": "1", "pages": "75-130" }, "number": 221 }, { "id": "focm-2024-gw-sphere-equatorial-coupling", "journal": "Foundations of Computational Mathematics", "publicationYear": 2024, "title": "GW Sphere Equatorial Coupling", "topic": "Optimal transport", "publicationDate": "2024-09-16", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Question 2", "sourceQuote": "Are there other values of p and q for which the equatorial coupling is optimal?", "problemStatement": "Determine all further parameter pairs (p,q) for which the equatorial coupling is optimal between spheres. Also determine whether any such optimality persists when Euclidean chordal distances are replaced by geodesic distances.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "Gromov-Wasserstein", "spheres", "equatorial coupling", "geodesic distance" ], "literature": [], "source": { "title": "The Gromov–Wasserstein Distance Between Spheres", "authors": [ "Shreya Arya", "Arnab Auddy", "Ranthony A. Clark", "Sunhyuk Lim", "Facundo Mémoli", "Daniel Packer" ], "doi": "10.1007/s10208-024-09678-3", "url": "https://doi.org/10.1007/s10208-024-09678-3", "volume": "26", "issue": "1", "pages": "75-130" }, "number": 222 }, { "id": "mor-2024-W3214544052_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2024, "title": "Finite-sample bounds for multivariate shortfall risk estimators with streaming samples", "topic": "Risk Measure Estimation", "publicationDate": "2024-09-16", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 33, Section 9 (Concluding Remarks and Future Work).", "sourceQuote": "Third, it remains open to extend our finite sample bounds to the estimator of multivariate shortfall risk measure (MSRM) [2].", "problemStatement": "A multivariate shortfall risk measure asks for a vector of capital additions that makes the expected value of a prescribed loss function fall below a risk threshold. The paper studies an online estimator built from bounded IID observations arriving one at a time, but does not give nonasymptotic accuracy guarantees. Derive explicit finite-sample error or concentration bounds as functions of the sample size, analogous to those known for univariate shortfall-risk estimators.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "shortfall risk", "multivariate risk measures", "stochastic approximation", "finite-sample bounds", "streaming estimation", "concentration inequalities" ], "literature": [], "source": { "title": "Online Estimation and Optimization of Utility-Based Shortfall Risk", "authors": [ "Vishwajit Hegde", "Arvind Satish Menon", "L.A. Prashanth", "Krishna Jagannathan" ], "doi": "10.1287/moor.2022.0266", "url": "https://doi.org/10.1287/moor.2022.0266", "volume": "50", "issue": "4", "pages": "2470-2501" }, "number": 223 }, { "id": "mp-2024-second-higher-order-stationary-point-complexity", "journal": "Mathematical Programming", "publicationYear": 2024, "title": "Higher-Order Complexity of Stationary Points", "topic": "Nonconvex optimization and oracle complexity", "publicationDate": "2024-09-27", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 1.2, Open Problem 2", "sourceQuote": "Characterize the query and computational complexity of higher order methods, both in the unconstrained and in the constrained setting.", "problemStatement": "Characterize both query complexity and computational complexity for higher-order methods that seek stationary points of smooth nonconvex functions. The classification should address unconstrained problems and constrained domains under explicit access to derivatives above first order. It should determine whether higher-order information changes the dimension-dependent tractability and hardness boundaries established in the source for first-order oracles.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact statement, title, authors and citations across publisher pages, arXiv and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper direct result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "higher-order methods", "stationary point", "query complexity", "computational complexity" ], "literature": [], "source": { "title": "The computational complexity of finding stationary points in non-convex optimization", "authors": [ "Alexandros Hollender", "Manolis Zampetakis" ], "doi": "10.1007/s10107-024-02139-3", "url": "https://doi.org/10.1007/s10107-024-02139-3", "volume": "213", "issue": "1-2", "pages": "281-341" }, "number": 224 }, { "id": "mp-2024-second-stationary-points-fixed-dimension", "journal": "Mathematical Programming", "publicationYear": 2024, "title": "Stationary-Point Complexity in Fixed Dimension", "topic": "Nonconvex optimization and oracle complexity", "publicationDate": "2024-09-27", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 1.2, Open Problem 1", "sourceQuote": "Characterize the query complexity of the problem (both in its unconstrained and constrained versions) in fixed dimension d≥3.", "problemStatement": "Characterize the first-order query complexity of finding an approximate stationary point of a smooth nonconvex function in each fixed dimension d≥3, for both unconstrained and constrained domains. The source emphasizes that even dimension three is unresolved after establishing sharp classifications in lower-dimensional or variable-dimensional regimes. The target is matching upper and lower bounds under the paper's oracle and smoothness models.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact statement, title, authors and citations across publisher pages, arXiv and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper direct result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "stationary point", "query complexity", "fixed dimension", "nonconvex optimization" ], "literature": [], "source": { "title": "The computational complexity of finding stationary points in non-convex optimization", "authors": [ "Alexandros Hollender", "Manolis Zampetakis" ], "doi": "10.1007/s10107-024-02139-3", "url": "https://doi.org/10.1007/s10107-024-02139-3", "volume": "213", "issue": "1-2", "pages": "281-341" }, "number": 225 }, { "id": "mp-2024-second-linear-support-p-adic-lp", "journal": "Mathematical Programming", "publicationYear": 2024, "title": "Linear Support Bounds for p-Adic Linear Programs", "topic": "Linear optimization and arithmetic structure", "publicationDate": "2024-10-03", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 5, support-size discussion", "sourceQuote": "our upper bound is O(m ln m). Closing the gap in this case remains an intriguing open question.", "problemStatement": "Close the support-size gap for optimal solutions of [p]-adic linear programs when the constraint matrix has maximum entry norm one. The source has a linear lower-bound example and an O(m log m) upper bound in the number m of constraints, while a fractional T-join subclass admits O(m). The question is whether a universal O(m) upper bound holds or whether a superlinear construction is possible.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact statement, title, authors and citations across publisher pages, arXiv and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper direct result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "p-adic linear programming", "support size", "sparse optimum", "dyadic linear programming" ], "literature": [], "source": { "title": "Dyadic linear programming and extensions", "authors": [ "Ahmad Abdi", "Gérard Cornuéjols", "Bertrand Guenin", "Levent Tunçel" ], "doi": "10.1007/s10107-024-02146-4", "url": "https://doi.org/10.1007/s10107-024-02146-4", "volume": "213", "issue": "1-2", "pages": "473-516" }, "number": 226 }, { "id": "mp-2024-second-linear-support-dyadic-generating-cones", "journal": "Mathematical Programming", "publicationYear": 2024, "title": "Linear Support for Dyadic Generating Cones", "topic": "Linear optimization and arithmetic structure", "publicationDate": "2024-10-03", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 5, dyadic generating sets", "sourceQuote": "we conjecture that there is a O(m) upper bound on k.", "problemStatement": "Let a finite set of integer vectors be a dyadic generating set for a cone, so every integral vector in the cone has a dyadic conic representation. Prove that every such vector has a representation using only O(m) nonzero coefficients, where m is the ambient dimension. The source proves O(m log(m‖A‖∞²)) and motivates the linear conjecture by comparison with support bounds for Hilbert bases.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact statement, title, authors and citations across publisher pages, arXiv and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper direct result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "dyadic generating set", "conic representation", "support bound", "integer cone" ], "literature": [], "source": { "title": "Dyadic linear programming and extensions", "authors": [ "Ahmad Abdi", "Gérard Cornuéjols", "Bertrand Guenin", "Levent Tunçel" ], "doi": "10.1007/s10107-024-02146-4", "url": "https://doi.org/10.1007/s10107-024-02146-4", "volume": "213", "issue": "1-2", "pages": "473-516" }, "number": 227 }, { "id": "mp-2024-second-seymour-dyadic-optimum-ideal-matrices", "journal": "Mathematical Programming", "publicationYear": 2024, "title": "Seymour's Dyadic Optimum Conjecture", "topic": "Combinatorial optimization and dyadic linear programming", "publicationDate": "2024-10-03", "recordType": "reviewed", "status": "partial_progress", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Introduction, ideal set-covering systems", "sourceQuote": "Seymour conjectured that the dual linear program max 1⊤y:A⊤y≤c,y≥0 always has a dyadic optimal solution y.", "problemStatement": "Prove that for every ideal zero-one set-covering matrix A and every nonnegative integral cost vector c, the dual packing linear program has an optimal solution with dyadic coordinates. Idealness guarantees integral primal optima but does not automatically control denominators in the dual. A 2026 local formulation proves an important special case, while the unrestricted conjecture remains open.", "statusEvidence": "Proves an important local special case while leaving the dyadic-optimum conjecture for arbitrary ideal clutters open.", "verificationNote": "This status is tied to the cited primary-literature result and the precise scope stated above.", "keywords": [ "Seymour conjecture", "dyadic solution", "ideal matrix", "set covering" ], "literature": [ { "title": "The Local Dyadic Conjecture", "url": "https://doi.org/10.1007/978-3-032-28691-8_10", "relation": "Proves an important local special case while leaving the dyadic-optimum conjecture for arbitrary ideal clutters open." } ], "source": { "title": "Dyadic linear programming and extensions", "authors": [ "Ahmad Abdi", "Gérard Cornuéjols", "Bertrand Guenin", "Levent Tunçel" ], "doi": "10.1007/s10107-024-02146-4", "url": "https://doi.org/10.1007/s10107-024-02146-4", "volume": "213", "issue": "1-2", "pages": "473-516" }, "number": 228 }, { "id": "mp-2024-second-bounded-denominator-fractional-t-join-packing", "journal": "Mathematical Programming", "publicationYear": 2024, "title": "Small Dyadic Denominators for Fractional T-Join Packing", "topic": "Combinatorial optimization and dyadic linear programming", "publicationDate": "2024-10-03", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Concluding remarks, fractional T-join packing", "sourceQuote": "it has been conjectured by Seymour that k≤2", "problemStatement": "For the fractional packing of T-joins with integral capacities, prove that an optimal dual solution is 1/2ᵏ-integral with k≤2. A weaker explicit target left open by the same passage is a universal logarithmic bound k≤c log |E|. Existing dyadic optimality gives some finite denominator but no quantitative bound, so either estimate would make the arithmetic complexity uniform in the graph size.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact statement, title, authors and citations across publisher pages, arXiv and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper direct result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "fractional T-join packing", "dyadic denominator", "Seymour conjecture", "graph optimization" ], "literature": [], "source": { "title": "Dyadic linear programming and extensions", "authors": [ "Ahmad Abdi", "Gérard Cornuéjols", "Bertrand Guenin", "Levent Tunçel" ], "doi": "10.1007/s10107-024-02146-4", "url": "https://doi.org/10.1007/s10107-024-02146-4", "volume": "213", "issue": "1-2", "pages": "473-516" }, "number": 229 }, { "id": "mp-2024-second-tau-indexed-p-adic-dual-optimum", "journal": "Mathematical Programming", "publicationYear": 2024, "title": "Tau-Indexed p-Adic Dual Optima", "topic": "Combinatorial optimization and dyadic linear programming", "publicationDate": "2024-10-03", "recordType": "reviewed", "status": "partial_progress", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Concluding remarks, Conjecture 6.1", "sourceQuote": "Then max 1⊤y:A⊤y≤c,y≥0 has a [p]-adic optimal solution.", "problemStatement": "Let A be an ideal zero-one matrix, let τc be the integral optimum of its set-covering problem for cost c, and let p be the largest prime not exceeding τc. Prove that the dual packing problem has an optimal solution whose denominator uses only primes at most p. The conjecture is verified for dijoins, but the proposed arithmetic bound for arbitrary ideal matrices remains open.", "statusEvidence": "Proves the required dyadic dual optimum for the clutter of dijoins, a direct restricted family of the conjecture for arbitrary ideal matrices.", "verificationNote": "This status is tied to the cited primary-literature result and the precise scope stated above.", "keywords": [ "p-adic optimum", "ideal matrix", "set covering", "dual packing" ], "literature": [ { "title": "Dyadic Packing of Dijoins", "url": "https://doi.org/10.1137/24M1647692", "relation": "Proves the required dyadic dual optimum for the clutter of dijoins, a direct restricted family of the conjecture for arbitrary ideal matrices." } ], "source": { "title": "Dyadic linear programming and extensions", "authors": [ "Ahmad Abdi", "Gérard Cornuéjols", "Bertrand Guenin", "Levent Tunçel" ], "doi": "10.1007/s10107-024-02146-4", "url": "https://doi.org/10.1007/s10107-024-02146-4", "volume": "213", "issue": "1-2", "pages": "473-516" }, "number": 230 }, { "id": "siopt-2024-049", "journal": "SIAM Journal on Optimization", "publicationYear": 2024, "title": "Generic optimality conditions moment SOS", "topic": "Polynomial optimization", "publicationDate": "2024-10-03", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 6, before Conjecture 6.1; extracted preprint lines 1721-1728", "sourceQuote": "it is an open question whether or not the LICQC, SCC, SOSC hold at every minimizer", "problemStatement": "Finite convergence of the paper's moment-SOS hierarchy follows when LICQC, strict complementarity, and second-order sufficiency hold at every minimizer. For generic polynomial optimization these conditions are expected, but generalized moment problems include optimizer-dependent dual coefficients that are not themselves generic. Prove that the three conditions hold outside a measure-zero set of input polynomials, or identify a generic obstruction.", "statusEvidence": "No forward-citing primary paper or exact-query result explicitly claiming a proof or counterexample was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "moment-SOS", "finite convergence", "genericity", "constraint qualification" ], "literature": [], "source": { "title": "Finite Convergence of Moment-SOS Relaxations with Nonreal Radical Ideals", "authors": [ "Lei Huang", "Jiawang Nie", "Ya-Xiang Yuan" ], "doi": "10.1137/23m1605430", "url": "https://doi.org/10.1137/23m1605430", "preprintUrl": "https://arxiv.org/abs/2309.15398", "volume": "34", "issue": "4", "pages": "3399-3428" }, "number": 231 }, { "id": "mp-2024-second-exact-second-lifting-complex-cut-four", "journal": "Mathematical Programming", "publicationYear": 2024, "title": "Exact Second Lifting for the Four-Variable Complex Cut Polytope", "topic": "Semidefinite optimization and complex cuts", "publicationDate": "2024-10-09", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 6, Conjecture 1", "sourceQuote": "The second semidefinite lifting is exact for CUT∞⁴", "problemStatement": "Prove that the paper's second semidefinite moment lifting equals the complex cut polytope CUT∞⁴. The source shows that this lifting excludes every rank-two extreme point of the four-by-four complex elliptope and satisfies an additional valid cut, but those facts do not prove exactness. A proof needs to exclude all remaining non-cut matrices, while a counterexample would identify the next obstruction.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact statement, title, authors and citations across publisher pages, arXiv and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper direct result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "complex cut polytope", "semidefinite lifting", "moment matrix", "exact relaxation" ], "literature": [], "source": { "title": "Cuts and semidefinite liftings for the complex cut polytope", "authors": [ "Lennart Sinjorgo", "Renata Sotirov", "Miguel F. Anjos" ], "doi": "10.1007/s10107-024-02147-3", "url": "https://doi.org/10.1007/s10107-024-02147-3" }, "number": 232 }, { "id": "mor-2024-W4403279582_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2024, "title": "Remove the log factor in residual complexity for locally Lipschitz monotone inclusions", "topic": "Monotone Inclusion", "publicationDate": "2024-10-09", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 7 (Concluding remarks), page 23", "sourceQuote": "whether the aforementioned operation complexity of O(ε^(-1)log ε^(-1)) can be improved to O(ε^(-1))", "problemStatement": "The problem is to find a point with residual at most ε for the monotone inclusion 0∈F(x)+B(x), where F is monotone and locally Lipschitz and B is maximal monotone with a computable resolvent. The best cited method needs O(ε⁻¹log(1/ε)) evaluations. Design a method using O(ε⁻¹) evaluations, or show that the logarithmic factor is unavoidable.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "monotone inclusions", "local Lipschitz continuity", "operator splitting", "iteration complexity", "backtracking line search", "residual certificate" ], "literature": [], "source": { "title": "Primal-Dual Extrapolation Methods for Monotone Inclusions Under Local Lipschitz Continuity", "authors": [ "Zhaosong Lu", "Sanyou Mei" ], "doi": "10.1287/moor.2024.0407", "url": "https://doi.org/10.1287/moor.2024.0407", "volume": "50", "issue": "4", "pages": "2577-2599" }, "number": 233 }, { "id": "siopt-2024-048", "journal": "SIAM Journal on Optimization", "publicationYear": 2024, "title": "QRI implies face relative interior", "topic": "Convex analysis", "publicationDate": "2024-10-10", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conjecture 3.7; extracted preprint lines 618-623", "sourceQuote": "The face relative interior of C is nonempty whenever the quasi-relative interior of C is nonempty.", "problemStatement": "For a convex set C in a topological vector space, qri(C) consists of x∈C for which the closure of cone(C−x) is a linear subspace. The face-relative interior fri(C) consists of x∈C for which the closure of the minimal face F_min(x,C) contains all of C. Prove that qri(C)≠∅ implies fri(C)≠∅, or construct a convex counterexample.", "statusEvidence": "No forward-citing primary paper or exact-query result explicitly claiming a proof or counterexample was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "quasi-relative interior", "face-relative interior", "topological vector space", "convex sets" ], "literature": [], "source": { "title": "Face Relative Interior of Convex Sets in Topological Vector Spaces", "authors": [ "R. Díaz Millán", "Vera Roshchina" ], "doi": "10.1137/23m1602814", "url": "https://doi.org/10.1137/23m1602814", "preprintUrl": "https://arxiv.org/abs/2309.06699", "volume": "34", "issue": "4", "pages": "3456-3476" }, "number": 234 }, { "id": "mor-2024-W4403453300_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2024, "title": "Achieve faster residual decay for cocoercive-operator Newton-corrected inertial dynamics", "topic": "Inertial Monotone Operator Dynamics", "publicationDate": "2024-10-16", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 26, Section 5.1 (Remark 8).", "sourceQuote": "It would be interesting to know if we can achieve the same convergence rate using time scaling and perturbation techniques.", "problemStatement": "For a cocoercive operator M with a nonempty zero set, consider the inertial trajectory y″+(α/s)y′+[s/(α+1)](M(y))′+M(y)=0. Existing analysis proves ‖M(y(s))‖=o(1/s), while a nearby correction coefficient yields o(1/s²) and finite integral ∫s³‖M(y(s))‖²ds. Determine whether time scaling and perturbation can establish these sharper conclusions for this trajectory or for an explicitly modified coefficient.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "cocoercive operator", "inertial dynamics", "newton correction term", "time scaling", "convergence rate", "residual decay" ], "literature": [], "source": { "title": "Fast Convex Optimization via Time Scale and Averaging of the Steepest Descent", "authors": [ "Hedy Attouch", "Radu Ioan Boţ", "Dang-Khoa Nguyen" ], "doi": "10.1287/moor.2023.0186", "url": "https://doi.org/10.1287/moor.2023.0186", "volume": "50", "issue": "4", "pages": "2633-2665" }, "number": 235 }, { "id": "mor-2024-W4403683873_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2024, "title": "Convergence of single-valued fixed-point iteration for Gibbs-form equilibrium operators", "topic": "Entropy Regularized Mdp", "publicationDate": "2024-10-23", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 10, Remark 2.6 (end of Section 2.2).", "sourceQuote": "it remains to show the theoretic convergence of the single-valued fixed-point iteration", "problemStatement": "An entropy-regularized time-inconsistent Markov decision process has a Gibbs-form best-response operator, so equilibria are its fixed points. The regularization makes the operator single-valued, but the paper does not prove that repeated application converges. Give conditions ensuring convergence of the fixed-point or policy iteration to a relaxed equilibrium, or characterize failures.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "time-inconsistent MDP", "entropy regularization", "relaxed equilibrium", "fixed-point iteration", "convergence analysis" ], "literature": [], "source": { "title": "Relaxed Equilibria for Time-Inconsistent Markov Decision Processes", "authors": [ "Erhan Bayraktar", "Yu-Jui Huang", "Zhenhua Wang", "Zhou Zhou" ], "doi": "10.1287/moor.2023.0209", "url": "https://doi.org/10.1287/moor.2023.0209", "volume": "50", "issue": "4", "pages": "2666-2687" }, "number": 236 }, { "id": "mp-2024-second-polynomial-time-delta-modular-ip", "journal": "Mathematical Programming", "publicationYear": 2024, "title": "Polynomial-Time Delta-Modular Integer Programming", "topic": "Integer programming and bounded subdeterminants", "publicationDate": "2024-10-30", "recordType": "reviewed", "status": "partial_progress", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Introduction, Conjecture 1", "sourceQuote": "For constant Δ∈Z≥0, Δ-modular IPs can be solved in polynomial time.", "problemStatement": "Prove that integer programs whose full-rank constraint matrix has every maximal subdeterminant bounded in absolute value by a fixed constant Δ can be solved in polynomial time. The conjecture is known for Δ=2 and for several structural subclasses, but not for arbitrary fixed Δ. Recent tree-decomposition work adds another bounded-domain subclass without settling the general optimization problem.", "statusEvidence": "Gives a polynomial algorithm for a bounded-domain, bounded-treewidth structural subclass rather than arbitrary fixed-Delta modular integer programs.", "verificationNote": "This status is tied to the cited primary-literature result and the precise scope stated above.", "keywords": [ "Delta-modular matrix", "integer programming", "bounded subdeterminants", "polynomial time" ], "literature": [ { "title": "Totally Δ-Modular Tree Decompositions of Graphic Matrices for Integer Programming", "url": "https://arxiv.org/abs/2602.01499", "relation": "Gives a polynomial algorithm for a bounded-domain, bounded-treewidth structural subclass rather than arbitrary fixed-Delta modular integer programs." } ], "source": { "title": "Advances on strictly Δ-modular IPs", "authors": [ "Martin Nägele", "Christian Nöbel", "Richard Santiago", "Rico Zenklusen" ], "doi": "10.1007/s10107-024-02148-2", "url": "https://doi.org/10.1007/s10107-024-02148-2", "volume": "210", "issue": "1-2", "pages": "731-760" }, "number": 237 }, { "id": "mp-2024-second-polynomial-time-strict-delta-modular-ip", "journal": "Mathematical Programming", "publicationYear": 2024, "title": "Polynomial-Time Strict Delta-Modular Integer Programming", "topic": "Integer programming and bounded subdeterminants", "publicationDate": "2024-10-30", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Introduction, Conjecture 2", "sourceQuote": "Strictly Δ-modular IPs can be solved in polynomial time for constant Δ∈Z≥0.", "problemStatement": "Prove polynomial-time solvability for integer programs whose full-rank constraint matrix has every maximal minor in −Δ,0,Δ, for each fixed Δ. This strict case already contains congruency-constrained cut and matching problems and remains open beyond limited values and tasks. The source's randomized strongly polynomial feasibility result for Δ≤4 advances a separate decision subproblem but does not prove the requested optimization theorem for any new fixed-Δ class.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact statement, title, authors and citations across publisher pages, arXiv and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper direct result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "strictly Delta-modular matrix", "integer programming", "congruency constraints", "polynomial time" ], "literature": [], "source": { "title": "Advances on strictly Δ-modular IPs", "authors": [ "Martin Nägele", "Christian Nöbel", "Richard Santiago", "Rico Zenklusen" ], "doi": "10.1007/s10107-024-02148-2", "url": "https://doi.org/10.1007/s10107-024-02148-2", "volume": "210", "issue": "1-2", "pages": "731-760" }, "number": 238 }, { "id": "mp-2024-second-composite-modulus-congruency-submodular-minimization", "journal": "Mathematical Programming", "publicationYear": 2024, "title": "Strongly Polynomial Congruency Optimization for Composite Moduli", "topic": "Combinatorial optimization with congruency constraints", "publicationDate": "2024-10-30", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 3, congruency-constrained lattice feasibility", "sourceQuote": "the case of general constant composite moduli remains open", "problemStatement": "Give a strongly polynomial algorithm for optimization over congruency-constrained lattices or, equivalently in the stated reduction, congruency-constrained submodular minimization with an arbitrary fixed composite modulus. Strongly polynomial algorithms are known for fixed prime-power moduli, and the source settles the pure feasibility analogue for finite abelian groups. The unresolved target retains an objective function and must handle composite moduli without prime-field tools.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact statement, title, authors and citations across publisher pages, arXiv and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper direct result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "composite modulus", "submodular minimization", "lattice optimization", "strongly polynomial algorithm" ], "literature": [], "source": { "title": "Advances on strictly Δ-modular IPs", "authors": [ "Martin Nägele", "Christian Nöbel", "Richard Santiago", "Rico Zenklusen" ], "doi": "10.1007/s10107-024-02148-2", "url": "https://doi.org/10.1007/s10107-024-02148-2", "volume": "210", "issue": "1-2", "pages": "731-760" }, "number": 239 }, { "id": "mp-2024-second-deterministic-commutative-edmonds-problem", "journal": "Mathematical Programming", "publicationYear": 2024, "title": "Deterministic Polynomial-Time Edmonds Rank", "topic": "Algebraic algorithms and derandomization", "publicationDate": "2024-11-02", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Introduction, commutative symbolic rank", "sourceQuote": "A deterministic polynomial-time algorithm for Edmonds’ problem is not known", "problemStatement": "Design a deterministic polynomial-time algorithm to compute the rank of a commutative symbolic matrix A₁x₁+⋯+Aₘxₘ over its rational-function field. Random substitution gives a polynomial-time randomized algorithm over sufficiently large fields, but derandomization is a central polynomial-identity-testing problem and would have major circuit-complexity consequences. The source solves a noncommutative degree generalization, not this commutative rank problem.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact statement, title, authors and citations across publisher pages, arXiv and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper direct result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "Edmonds problem", "symbolic matrix rank", "polynomial identity testing", "derandomization" ], "literature": [], "source": { "title": "Algebraic combinatorial optimization on the degree of determinants of noncommutative symbolic matrices", "authors": [ "Hiroshi Hirai", "Yuni Iwamasa", "Taihei Oki", "Tasuku Soma" ], "doi": "10.1007/s10107-024-02158-0", "url": "https://doi.org/10.1007/s10107-024-02158-0", "volume": "213", "issue": "1-2", "pages": "941-984" }, "number": 240 }, { "id": "mp-2024-second-strong-membership-brascamp-lieb-polytope", "journal": "Mathematical Programming", "publicationYear": 2024, "title": "Strong Membership for General Brascamp–Lieb Polytopes", "topic": "Algebraic optimization and convex polytopes", "publicationDate": "2024-11-02", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 4.2, Brascamp–Lieb inequality", "sourceQuote": "It is open if one can improve the time complexity to poly(N, log(1/ε)).", "problemStatement": "Develop a strong weak-membership algorithm for a general Brascamp–Lieb polytope with running time polynomial in the input bit length N and log(1/ε). The known general algorithm is polynomial in N and 1/ε, while the source obtains exact polynomial-time membership only when every defining matrix has rank two. The open problem is to replace the precision dependence for unrestricted Brascamp–Lieb data without assuming that special rank structure.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact statement, title, authors and citations across publisher pages, arXiv and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper direct result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "Brascamp-Lieb polytope", "weak membership", "bit complexity", "operator scaling" ], "literature": [], "source": { "title": "Algebraic combinatorial optimization on the degree of determinants of noncommutative symbolic matrices", "authors": [ "Hiroshi Hirai", "Yuni Iwamasa", "Taihei Oki", "Tasuku Soma" ], "doi": "10.1007/s10107-024-02158-0", "url": "https://doi.org/10.1007/s10107-024-02158-0", "volume": "213", "issue": "1-2", "pages": "941-984" }, "number": 241 }, { "id": "mp-2024-second-local-lm-convergence-alternative-growth", "journal": "Mathematical Programming", "publicationYear": 2024, "title": "Local Levenberg–Marquardt Convergence Under Alternative Growth", "topic": "Nonlinear optimization and Levenberg–Marquardt methods", "publicationDate": "2024-11-04", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 5, discussion after condition (5.4)", "sourceQuote": "whether our algorithm exhibits favorable local convergence under (5.4), in place of Assumption 3", "problemStatement": "Determine whether the proposed generalized Levenberg–Marquardt algorithm has a favorable local convergence rate under condition (5.4) instead of the paper's Hölderian growth Assumption 3. Condition (5.4) lower-bounds the squared residual difference by a constant times squared distance to the solution set. The two conditions are not ordered in general, so the existing local quadratic analysis cannot simply be transferred.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact statement, title, authors and citations across publisher pages, arXiv and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper direct result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "Levenberg-Marquardt method", "local convergence", "quadratic growth", "composite optimization" ], "literature": [], "source": { "title": "Accelerated-gradient-based generalized Levenberg–Marquardt method with oracle complexity bound and local quadratic convergence", "authors": [ "Naoki Marumo", "Takayuki Okuno", "Akiko Takeda" ], "doi": "10.1007/s10107-024-02154-4", "url": "https://doi.org/10.1007/s10107-024-02154-4", "volume": "213", "issue": "1-2", "pages": "771-822" }, "number": 242 }, { "id": "mp-2024-second-stochastic-lm-complexity-local-convergence", "journal": "Mathematical Programming", "publicationYear": 2024, "title": "Stochastic Levenberg–Marquardt Complexity and Local Convergence", "topic": "Stochastic nonlinear optimization", "publicationDate": "2024-11-04", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conclusion, final future-work sentence", "sourceQuote": "whether oracle complexity bound and local convergence can still be achieved with stochastic LM methods.", "problemStatement": "Construct a stochastic Levenberg–Marquardt method for large finite-sum composite problems that simultaneously has a nonasymptotic oracle-complexity bound and a provable local convergence rate. The deterministic method in the source has both properties, but evaluating all component maps and derivatives can be too expensive when the number of summands is large. The challenge is to control sampling error closely enough to preserve the fast local regime.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact statement, title, authors and citations across publisher pages, arXiv and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper direct result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "stochastic Levenberg-Marquardt", "oracle complexity", "local convergence", "finite-sum optimization" ], "literature": [], "source": { "title": "Accelerated-gradient-based generalized Levenberg–Marquardt method with oracle complexity bound and local quadratic convergence", "authors": [ "Naoki Marumo", "Takayuki Okuno", "Akiko Takeda" ], "doi": "10.1007/s10107-024-02154-4", "url": "https://doi.org/10.1007/s10107-024-02154-4", "volume": "213", "issue": "1-2", "pages": "771-822" }, "number": 243 }, { "id": "mp-2024-second-unrestricted-memory-heap-selection-lower-bound", "journal": "Mathematical Programming", "publicationYear": 2024, "title": "Superlinear Heap-Selection Lower Bound Without Memory Limits", "topic": "Randomized algorithms and data structures", "publicationDate": "2024-11-05", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Introduction, concluding open problem", "sourceQuote": "whether a superlinear lower bound also holds without any restriction on the memory usage.", "problemStatement": "Prove or refute a superlinear expected travel-cost lower bound for explorable heap selection when an algorithm has unrestricted working memory. The source proves a lower bound of n logₛ n for algorithms with s memory units and gives a randomized near-linear algorithm using logarithmic space. Removing the memory restriction would show whether the logarithmic overhead is intrinsic to exploration rather than a space-time tradeoff.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact statement, title, authors and citations across publisher pages, arXiv and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper direct result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "heap selection", "explorable heap", "lower bound", "memory tradeoff" ], "literature": [], "source": { "title": "A nearly optimal randomized algorithm for explorable heap selection", "authors": [ "Sander Borst", "Daniel Dadush", "Sophie Huiberts", "Danish Kashaev" ], "doi": "10.1007/s10107-024-02145-5", "url": "https://doi.org/10.1007/s10107-024-02145-5", "volume": "210", "issue": "1-2", "pages": "75-96" }, "number": 244 }, { "id": "mp-2024-second-finite-exact-bilevel-lagrange-multiplier", "journal": "Mathematical Programming", "publicationYear": 2024, "title": "Finite Exact Lagrange Multiplier for Convex Bilevel Optimization", "topic": "Bilevel and hierarchical optimization", "publicationDate": "2024-11-05", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 2, Lagrangian reformulation", "sourceQuote": "Hence, we do not know whether there exists a finite λopt≥0 such that minx∈X L(x,λopt)=fopt and Xopt⊆arg minx∈X L(x,λopt).", "problemStatement": "Determine whether the convex simple bilevel problem in the source always admits a finite multiplier λopt such that minimizing its penalized Lagrangian attains the bilevel optimum and contains every bilevel-optimal point among its minimizers. Strong duality gives equality only in the supremum over multipliers, while the dual need not attain its supremum. A counterexample or verifiable conditions guaranteeing finite exactness would settle the issue.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact statement, title, authors and citations across publisher pages, arXiv and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper direct result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "convex bilevel optimization", "Lagrange multiplier", "exact penalty", "dual attainment" ], "literature": [], "source": { "title": "A projection-free method for solving convex bilevel optimization problems", "authors": [ "Khanh-Hung Giang-Tran", "Nam Ho-Nguyen", "Dabeen Lee" ], "doi": "10.1007/s10107-024-02157-1", "url": "https://doi.org/10.1007/s10107-024-02157-1", "volume": "213", "issue": "1-2", "pages": "907-940" }, "number": 245 }, { "id": "mp-2024-second-replication-conjecture-clutters", "journal": "Mathematical Programming", "publicationYear": 2024, "title": "Replication Conjecture for Clutters", "topic": "Clutters and integer programming", "publicationDate": "2024-11-05", "recordType": "reviewed", "status": "partial_progress", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 8, Replication Conjecture", "sourceQuote": "the Replication Conjecture predicts that the packing property implies the max-flow min-cut property.", "problemStatement": "Prove that every clutter with the packing property has the max-flow min-cut property. The source verifies the Replication Conjecture for multipartite uniform clutters arising from coordinate subspaces. A 2026 primary preprint further verifies cuboids of degree at most nine and all clutters on at most ten elements, but the unrestricted implication remains open.", "statusEvidence": "Verifies the Replication Conjecture for multipartite uniform clutters arising from coordinate subspaces. Verifies cuboids of degree at most nine and all clutters on at most ten elements, but not the unrestricted conjecture.", "verificationNote": "This status is tied to the cited primary-literature result and the precise scope stated above.", "keywords": [ "Replication Conjecture", "clutter", "packing property", "max-flow min-cut" ], "literature": [ { "title": "From coordinate subspaces over finite fields to ideal multipartite uniform clutters", "url": "https://doi.org/10.1007/s10107-024-02155-3", "relation": "Verifies the Replication Conjecture for multipartite uniform clutters arising from coordinate subspaces." }, { "title": "Testing the max-flow min-cut property and the replication conjecture", "url": "https://arxiv.org/abs/2606.16543", "relation": "Verifies cuboids of degree at most nine and all clutters on at most ten elements, but not the unrestricted conjecture." } ], "source": { "title": "From coordinate subspaces over finite fields to ideal multipartite uniform clutters", "authors": [ "Ahmad Abdi", "Dabeen Lee" ], "doi": "10.1007/s10107-024-02155-3", "url": "https://doi.org/10.1007/s10107-024-02155-3", "volume": "213", "issue": "1-2", "pages": "823-861" }, "number": 246 }, { "id": "mp-2024-second-tau-two-conjecture-clutters", "journal": "Mathematical Programming", "publicationYear": 2024, "title": "Tau-Two Conjecture for Minimally Nonpacking Clutters", "topic": "Clutters and integer programming", "publicationDate": "2024-11-05", "recordType": "reviewed", "status": "partial_progress", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 8, tau=2 Conjecture", "sourceQuote": "if a clutter C is ideal and minimally non-packing, then its covering number is two.", "problemStatement": "Prove that every ideal minimally nonpacking clutter has covering number two. This tau-two statement is stronger than the Replication Conjecture and would imply it through known reductions. The source verifies the claim for multipartite uniform clutters produced by coordinate subspaces, where the only relevant obstruction is Q₆, but no proof is known for arbitrary clutters.", "statusEvidence": "Proves the conjecture for multipartite uniform clutters generated by coordinate subspaces, a restricted family.", "verificationNote": "This status is tied to the cited primary-literature result and the precise scope stated above.", "keywords": [ "tau-two conjecture", "ideal clutter", "minimally nonpacking", "covering number" ], "literature": [ { "title": "From coordinate subspaces over finite fields to ideal multipartite uniform clutters", "url": "https://doi.org/10.1007/s10107-024-02155-3", "relation": "Proves the conjecture for multipartite uniform clutters generated by coordinate subspaces, a restricted family." } ], "source": { "title": "From coordinate subspaces over finite fields to ideal multipartite uniform clutters", "authors": [ "Ahmad Abdi", "Dabeen Lee" ], "doi": "10.1007/s10107-024-02155-3", "url": "https://doi.org/10.1007/s10107-024-02155-3", "volume": "213", "issue": "1-2", "pages": "823-861" }, "number": 247 }, { "id": "mp-2024-second-measurable-chromatic-number-tiling-polytopes", "journal": "Mathematical Programming", "publicationYear": 2024, "title": "Measurable Chromatic Number of Tiling-Polytope Distance Graphs", "topic": "Discrete geometry and harmonic optimization", "publicationDate": "2024-11-05", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 6, Euclidean set-avoiding graphs", "sourceQuote": "χm(Rⁿ,∂P)≤2ⁿ whenever P tiles Rⁿ and equality is conjectured.", "problemStatement": "Prove that χm(Rⁿ,∂P)=2ⁿ whenever the centrally symmetric convex polytope P tiles Rⁿ by translations. Here vertices are all points of Euclidean space and two points are adjacent when their difference lies on the boundary of P, while χm denotes the measurable chromatic number. The upper bound 2ⁿ is known; the open direction is a matching measurable lower bound for every tiling polytope.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact statement, title, authors and citations across publisher pages, arXiv and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper direct result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "measurable chromatic number", "tiling polytope", "distance graph", "Euclidean space" ], "literature": [], "source": { "title": "Optimization of trigonometric polynomials with crystallographic symmetry and spectral bounds for set avoiding graphs", "authors": [ "Evelyne Hubert", "Tobias Metzlaff", "Philippe Moustrou", "Cordian Riener" ], "doi": "10.1007/s10107-024-02149-1", "url": "https://doi.org/10.1007/s10107-024-02149-1", "volume": "213", "issue": "1-2", "pages": "517-573" }, "number": 248 }, { "id": "mor-2024-W4404109711_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2024, "title": "Prove divergence of last-iterate mixed strategies without monotonicity in 2x2 games", "topic": "No Regret Learning", "publicationDate": "2024-11-06", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 3.5 (\"Beyond the monotonicity assumption: A conjecture\"), page 24-25; Conjecture 3.12. Also reiterated as a main future-work question in Section 5 (Conclusion), page 29.", "sourceQuote": "We conjecture below that last-iterate oscillations will continue to hold for mean-based strategies that are not monotonic.", "problemStatement": "Consider an infinitely repeated competitive 2×2 game with a unique completely mixed Nash equilibrium. Each player's mixed action at round t depends only on the opponent's empirical frequency of realized actions, and both mean-based strategies have uniform regret O(√T). Prove or refute that the pair of last-iterate mixed actions fails to converge almost surely.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "no-regret learning", "last-iterate convergence", "repeated games", "mean-based strategies", "mixed nash equilibrium", "stochastic feedback", "2x2 competitive games" ], "literature": [], "source": { "title": "On the Impossibility of Convergence of Mixed Strategies with Optimal No-Regret Learning", "authors": [ "Vidya Muthukumar", "Soham Phade", "Anant Sahai" ], "doi": "10.1287/moor.2022.0016", "url": "https://doi.org/10.1287/moor.2022.0016", "volume": "50", "issue": "4", "pages": "2794-2833" }, "number": 249 }, { "id": "focm-2024-classify-groups-five-prime-factors", "journal": "Foundations of Computational Mathematics", "publicationYear": 2024, "title": "Classify Groups Five Prime Factors", "topic": "Computational group theory", "publicationDate": "2024-11-12", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 7, Problem 5", "sourceQuote": "Classify the groups whose order factorizes into at most five primes.", "problemStatement": "Classify finite groups whose orders have at most five prime factors, counted with multiplicity. The desired result should be uniform across the possible prime choices and divisibility relations. The four-prime-factor case is classified, whereas the fifth factor introduces many additional Sylow-subgroup and extension configurations that are not covered by one symbolic construction.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "finite groups", "order factorization", "classification" ], "literature": [], "source": { "title": "Classification of Finite Groups: Recent Developements and Open Problems", "authors": [ "Bettina Eick" ], "doi": "10.1007/s10208-024-09688-1", "url": "https://doi.org/10.1007/s10208-024-09688-1", "volume": "25", "issue": "6", "pages": "1907-1919" }, "number": 250 }, { "id": "focm-2024-classify-groups-order-1024", "journal": "Foundations of Computational Mathematics", "publicationYear": 2024, "title": "Classify Groups Order 1024", "topic": "Computational group theory", "publicationDate": "2024-11-12", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 7, Problem 4", "sourceQuote": "Classify the groups of order 1024.", "problemStatement": "Complete the isomorphism classification of all finite groups of order 1024. The task includes constructing a usable, nonredundant catalogue rather than only counting the groups. Order 1024 equals 2^10 and has an enormous number of isomorphism types, so existing SmallGroups coverage and lower-order p-group classifications do not provide the requested complete catalogue.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "finite groups", "order 1024", "classification", "small groups" ], "literature": [], "source": { "title": "Classification of Finite Groups: Recent Developements and Open Problems", "authors": [ "Bettina Eick" ], "doi": "10.1007/s10208-024-09688-1", "url": "https://doi.org/10.1007/s10208-024-09688-1", "volume": "25", "issue": "6", "pages": "1907-1919" }, "number": 251 }, { "id": "focm-2024-classify-groups-p8", "journal": "Foundations of Computational Mathematics", "publicationYear": 2024, "title": "Classify Groups p^8", "topic": "Computational group theory", "publicationDate": "2024-11-12", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 7, Problem 7", "sourceQuote": "Classify the groups of order p^8.", "problemStatement": "Classify the finite groups of order p^8 for an arbitrary prime p. The goal is a parametrized isomorphism classification valid beyond individually enumerated small primes. Here p is an arbitrary prime, so enumerations for isolated small primes do not constitute the desired symbolic classification by parametrized presentations and isomorphism conditions.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "p-groups", "order p^8", "classification", "isomorphism" ], "literature": [], "source": { "title": "Classification of Finite Groups: Recent Developements and Open Problems", "authors": [ "Bettina Eick" ], "doi": "10.1007/s10208-024-09688-1", "url": "https://doi.org/10.1007/s10208-024-09688-1", "volume": "25", "issue": "6", "pages": "1907-1919" }, "number": 252 }, { "id": "focm-2024-classify-groups-pnq-n7", "journal": "Foundations of Computational Mathematics", "publicationYear": 2024, "title": "Classify Groups p^n q n <= 7", "topic": "Computational group theory", "publicationDate": "2024-11-12", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 7, Problem 6", "sourceQuote": "Classify the groups of order p^n q for n at most 7.", "problemStatement": "Classify all finite groups of order p^n q for distinct primes p and q and n at most seven. The classification must cover the unresolved prime and exponent configurations uniformly. Earlier work counts, but does not explicitly list, the cases n <= 5; the source asks for a constructive symbolic classification extending through exponents six and seven.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "finite groups", "p-groups", "semidirect products", "classification" ], "literature": [], "source": { "title": "Classification of Finite Groups: Recent Developements and Open Problems", "authors": [ "Bettina Eick" ], "doi": "10.1007/s10208-024-09688-1", "url": "https://doi.org/10.1007/s10208-024-09688-1", "volume": "25", "issue": "6", "pages": "1907-1919" }, "number": 253 }, { "id": "focm-2024-proximal-galerkin-strict-complementarity-order", "journal": "Foundations of Computational Mathematics", "publicationYear": 2024, "title": "Proximal Galerkin Strict Complementarity Order", "topic": "Numerical variational inequalities", "publicationDate": "2024-11-20", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 4.8, numerical experiment 2", "sourceQuote": "We conjecture that this improved convergence order holds in general whenever a strict complementarity condition is satisfied.", "problemStatement": "For the obstacle problem minimizing one-half ||grad u||^2-(f,u) subject to u>=phi, the entropic proximal-Galerkin iteration uses the latent variable psi=log(u-phi) and positive step sizes alpha_k. Strict complementarity means there is no biactive region: on the contact set u=phi the associated multiplier is positive. Prove the experimentally observed improvement over the general Bregman bound, including linear iteration convergence for fixed alpha_k and the faster order observed for geometrically increasing step sizes, under this strict-complementarity hypothesis.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "proximal Galerkin", "strict complementarity", "iteration complexity", "variational inequalities" ], "literature": [], "source": { "title": "Proximal Galerkin: A Structure-Preserving Finite Element Method for Pointwise Bound Constraints", "authors": [ "Brendan Keith", "Thomas M. Surowiec" ], "doi": "10.1007/s10208-024-09681-8", "url": "https://doi.org/10.1007/s10208-024-09681-8", "volume": "26", "issue": "1", "pages": "385-481" }, "number": 254 }, { "id": "mor-2024-W4404566852_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2024, "title": "Determine the exact master route ratio for a priori TSP with depot", "topic": "Stochastic Tsp", "publicationDate": "2024-11-20", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 5, Section 1.2 (after Theorem 6); reiterated in Page 38, Section 9 (Discussion).", "sourceQuote": "We conjecture that the master route ratio is exactly frac11-e^(-1/2).", "problemStatement": "In the a priori traveling-salesman problem, a tour is chosen before independent customer activations are observed. A master route visits a fixed subset and inserts active off-route customers locally, and its ratio compares the best such policy with the optimal a priori tour. Prove the conjecture that the exact worst-case metric master-route ratio with a depot is 1/(1−e^(−1/2)), or construct an instance with a different ratio.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "a priori tsp", "stochastic optimization", "master route solutions", "worst-case ratio", "metric tsp", "random activations" ], "literature": [], "source": { "title": "Improved Guarantees for the A Priori TSP", "authors": [ "Jannis Blauth", "Meike Neuwohner", "Luise Puhlmann", "Jens Vygen" ], "doi": "10.1287/moor.2023.0322", "url": "https://doi.org/10.1287/moor.2023.0322", "preprintUrl": "https://researchonline.lse.ac.uk/id/eprint/126218/1/Improved_guarantees_for_the_a_priori_TSP.pdf", "volume": "50", "issue": "4", "pages": "2909-2940" }, "number": 255 }, { "id": "mp-2024-second-optimal-silver-stepsize-exponent", "journal": "Mathematical Programming", "publicationYear": 2024, "title": "Optimality of the Silver Stepsize Exponent", "topic": "First-order convex optimization", "publicationDate": "2024-11-25", "recordType": "reviewed", "status": "partial_progress", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Discussion after Theorem 1", "sourceQuote": "We conjecture that, among all possible stepsize schedules, the exponent log₂ρ in the rate is exactly optimal", "problemStatement": "Prove that no stepsize schedule for gradient descent on smooth convex functions can improve the Silver schedule's worst-case exponent log₂ρ≈1.2716, and determine the optimal multiplicative constant up to the conjectured modest factor. Later composition theory improves finite-horizon constants and conjectures minimax schedules, while a 2026 anytime lower bound limits positive schedules to exponent at most 1.334. Neither result closes the exponent gap in the source's formulation.", "statusEvidence": "Proves the exact Silver exponent for the paper's basic composable schedule class and constructs improved compositions, without establishing unrestricted stepsize optimality. Shows that no positive anytime stepsize schedule has function-value rate o(n^−1.334), a direct lower bound for the anytime subclass rather than the unrestricted source formulation.", "verificationNote": "This status is tied to the cited primary-literature result and the precise scope stated above.", "keywords": [ "gradient descent", "stepsize schedule", "Silver schedule", "optimal convergence rate" ], "literature": [ { "title": "Composing Optimized Stepsize Schedules for Gradient Descent", "url": "https://doi.org/10.1287/moor.2024.0764", "relation": "Proves the exact Silver exponent for the paper's basic composable schedule class and constructs improved compositions, without establishing unrestricted stepsize optimality." }, { "title": "Lower Bounds for Anytime Acceleration of Gradient Descent", "url": "https://arxiv.org/abs/2607.02053", "relation": "Shows that no positive anytime stepsize schedule has function-value rate o(n^−1.334), a direct lower bound for the anytime subclass rather than the unrestricted source formulation." } ], "source": { "title": "Acceleration by stepsize hedging: Silver Stepsize Schedule for smooth convex optimization", "authors": [ "Jason M. Altschuler", "Pablo A. Parrilo" ], "doi": "10.1007/s10107-024-02164-2", "url": "https://doi.org/10.1007/s10107-024-02164-2", "volume": "213", "issue": "1-2", "pages": "1105-1118" }, "number": 256 }, { "id": "mor-2024-W4404717018_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2024, "title": "Designing polynomial-time 1/4-competitive secretary matching algorithm under edge arrivals", "topic": "Online Matching", "publicationDate": "2024-11-26", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 6, Section 1.1 (Edge arrival techniques discussion); and reiterated Page 23, Remark after Theorem 5.2.", "sourceQuote": "It remains an interesting problem whether there is a poly-time online algorithm that matches this bound of 1/4.", "problemStatement": "In edge-arrival secretary matching, weighted graph edges arrive in uniformly random order and accepted edges must form a matching. The paper proves an information-theoretic 1/4-competitive rule, but implementing it seems to require conditional availability probabilities that are not efficiently computable. Find a polynomial-time online algorithm that attains the same 1/4 ratio on every weighted graph.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "secretary problem", "online matching", "edge arrivals", "competitive ratio", "polynomial-time algorithms", "random-order model" ], "literature": [], "source": { "title": "Tight Bounds for Secretary Matching in General Graphs", "authors": [ "Tomer Ezra", "Michal Feldman", "Nikolai Gravin", "Zhihao (Gavin) Tang" ], "doi": "10.1287/moor.2022.0206", "url": "https://doi.org/10.1287/moor.2022.0206", "volume": "50", "issue": "4", "pages": "3039-3054" }, "number": 257 }, { "id": "mp-2024-second-multi-unit-auction-polyhedral-framework", "journal": "Mathematical Programming", "publicationYear": 2024, "title": "Polyhedral Framework for Multi-Unit Auctions", "topic": "Algorithmic game theory and polyhedral geometry", "publicationDate": "2024-11-28", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 6, Question 25", "sourceQuote": "How does our approach generalize to allocation mechanisms in a setting with multiple copies of items?", "problemStatement": "Extend the paper's indifference-complex and regular-subdivision description of truthful allocation mechanisms from single copies to auctions with multiple copies of each item. The generalization should identify the correct type space, allocation polytope, and compatibility conditions for incentive-compatible mechanisms. It should also recover a useful sensitivity or structural characterization rather than only restating the mechanism-design constraints.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact statement, title, authors and citations across publisher pages, arXiv and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper direct result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "multi-unit auction", "truthful mechanism", "regular subdivision", "allocation polytope" ], "literature": [], "source": { "title": "The polyhedral geometry of truthful auctions", "authors": [ "Michael Joswig", "Max Klimm", "Sylvain Spitz" ], "doi": "10.1007/s10107-024-02168-y", "url": "https://doi.org/10.1007/s10107-024-02168-y", "volume": "210", "issue": "1-2", "pages": "539-566" }, "number": 258 }, { "id": "mp-2024-second-auction-sensitivity-bound-gap", "journal": "Mathematical Programming", "publicationYear": 2024, "title": "Tight Sensitivity Bounds for One-Player Auctions", "topic": "Algorithmic game theory and polyhedral geometry", "publicationDate": "2024-11-28", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 6, Question 24", "sourceQuote": "How can the sensitivity bounds for one-player mechanisms and affine maximizers be improved?", "problemStatement": "Close the gap in sensitivity bounds for truthful one-player allocation mechanisms and affine maximizers. In the paper's Hamming-sensitivity model, the stated bounds leave the extremal value Mh(m) between 2 and m−1, depending on the number m of alternatives. The task is to derive tighter universal upper or lower bounds, ideally an exact asymptotic or exact formula, using the geometry of regular subdivisions.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact statement, title, authors and citations across publisher pages, arXiv and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper direct result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "truthful auction", "sensitivity", "affine maximizer", "regular subdivision" ], "literature": [], "source": { "title": "The polyhedral geometry of truthful auctions", "authors": [ "Michael Joswig", "Max Klimm", "Sylvain Spitz" ], "doi": "10.1007/s10107-024-02168-y", "url": "https://doi.org/10.1007/s10107-024-02168-y", "volume": "210", "issue": "1-2", "pages": "539-566" }, "number": 259 }, { "id": "focm-2024-hierarchical-spline-commuting-projections", "journal": "Foundations of Computational Mathematics", "publicationYear": 2024, "title": "Hierarchical Spline Commuting Projections", "topic": "Isogeometric analysis", "publicationDate": "2024-12-04", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conclusion", "sourceQuote": "The existence of commuting projection operators onto the exact spline subcomplexes is still open.", "problemStatement": "Construct bounded commuting projection operators from the continuous de Rham complex onto the exact hierarchical spline subcomplexes developed in the paper. Such projections are a central missing ingredient for a full stability and approximation theory.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "commuting projections", "hierarchical splines", "finite element exterior calculus" ], "literature": [], "source": { "title": "Locally-Verifiable Sufficient Conditions for Exactness of the Hierarchical B-spline Discrete de Rham Complex in ℝⁿ", "authors": [ "Kendrick Shepherd", "Deepesh Toshniwal" ], "doi": "10.1007/s10208-024-09659-6", "url": "https://doi.org/10.1007/s10208-024-09659-6", "volume": "26", "issue": "1", "pages": "525-567" }, "number": 260 }, { "id": "focm-2024-hierarchical-spline-mesh-repair", "journal": "Foundations of Computational Mathematics", "publicationYear": 2024, "title": "Hierarchical Spline Mesh Repair", "topic": "Isogeometric analysis", "publicationDate": "2024-12-04", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conclusion", "sourceQuote": "finding a repair with a minimal number of added degrees of freedom remains an open problem", "problemStatement": "In the hierarchical B-spline de Rham complex, Assumption 3 requires any two refined level-ell zero-form B-splines sharing an (n-1,ell+1)-intersection to be joined by a shortest index-translation chain whose members are all refined. It also requires every finer j-form supported by a union of such zero-form supports to fit inside one refined zero-form support within their smallest axis-aligned bounding box. Given a refinement violating these conditions, add the minimum possible degrees of freedom so Assumption 3 holds and the hierarchical complex is exact.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "hierarchical splines", "mesh refinement", "degrees of freedom", "de Rham complex" ], "literature": [], "source": { "title": "Locally-Verifiable Sufficient Conditions for Exactness of the Hierarchical B-spline Discrete de Rham Complex in ℝⁿ", "authors": [ "Kendrick Shepherd", "Deepesh Toshniwal" ], "doi": "10.1007/s10208-024-09659-6", "url": "https://doi.org/10.1007/s10208-024-09659-6", "volume": "26", "issue": "1", "pages": "525-567" }, "number": 261 }, { "id": "focm-2024-conley-finite-spaces-larger-multivalued-map-class", "journal": "Foundations of Computational Mathematics", "publicationYear": 2024, "title": "Conley Finite Spaces Larger Multivalued Map Class", "topic": "Computational topology and dynamics", "publicationDate": "2024-12-09", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 8, final open problem", "sourceQuote": "whether our theory could be generalized to allow for a larger class of multivalued maps", "problemStatement": "Generalize the finite-topological-space Conley-index theory to a larger class of multivalued maps than lower-semicontinuous maps with closed acyclic values. The extension should capture more combinatorial vector fields while retaining meaningful isolated invariant sets and Morse graphs.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "Conley index", "finite topological spaces", "multivalued maps", "combinatorial dynamics" ], "literature": [], "source": { "title": "Conley Index for Multivalued Maps on Finite Topological Spaces", "authors": [ "Jonathan Barmak", "Marian Mrozek", "Thomas Wanner" ], "doi": "10.1007/s10208-024-09685-4", "url": "https://doi.org/10.1007/s10208-024-09685-4", "volume": "26", "issue": "2", "pages": "647-687" }, "number": 262 }, { "id": "focm-2024-rpd-end-to-end-floating-point-stability", "journal": "Foundations of Computational Mathematics", "publicationYear": 2024, "title": "RPD End To End Floating Point Stability", "topic": "Numerical linear algebra", "publicationDate": "2024-12-09", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conclusion", "sourceQuote": "An end-to-end floating point stability analysis of RPD remains open.", "problemStatement": "Derive a rigorous end-to-end floating-point error bound for the randomized pivoted decomposition algorithm. The analysis should propagate rounding effects through every stage rather than treating only isolated kernels. RPD regularizes a matrix pencil by randomized perturbation and recursively separates its pseudospectrum; the paper proves backward accuracy in exact arithmetic and only componentwise finite-precision bounds.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "randomized pivoted decomposition", "floating point", "backward stability", "roundoff" ], "literature": [], "source": { "title": "Generalized Pseudospectral Shattering and Inverse-Free Matrix Pencil Diagonalization", "authors": [ "James Demmel", "Ioana Dumitriu", "Ryan Schneider" ], "doi": "10.1007/s10208-024-09682-7", "url": "https://doi.org/10.1007/s10208-024-09682-7", "volume": "26", "issue": "2", "pages": "569-645" }, "number": 263 }, { "id": "mor-2024-W4405512051_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2024, "title": "Polynomial-time optimal or approximate strategies for Pandora problem with budget-additive costs", "topic": "Pandoras Box", "publicationDate": "2024-12-18", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 6 (Conclusion and Future Directions), page 25.", "sourceQuote": "For example, it is not clear whether there exists a poly-time algorithm that finds an optimal (or an approximately optimal) strategy under budget-additive costs.", "problemStatement": "Pandora's problem adaptively opens boxes with independent random values and pays a set-dependent inspection cost. Here the cost is the smaller of a budget B and the sum of the opened boxes' weights, and it is available only through value queries. Give a polynomial-query, polynomial-time algorithm for an optimal adaptive strategy or one with a proved approximation ratio.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "pandora's problem", "budget-additive costs", "submodular costs", "adaptive search", "oracle complexity", "approximation" ], "literature": [], "source": { "title": "Pandora’s Problem with Combinatorial Cost", "authors": [ "Ben Berger", "Tomer Ezra", "Michal Feldman", "Federico Fusco" ], "doi": "10.1287/moor.2023.0248", "url": "https://doi.org/10.1287/moor.2023.0248", "volume": "50", "issue": "4", "pages": "3161-3189" }, "number": 264 }, { "id": "mor-2024-W4405624370_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2024, "title": "Designing ADMM methods for nonsmooth Riemannian optimization without Moreau smoothing", "topic": "Manifold Admm", "publicationDate": "2024-12-20", "recordType": "reviewed", "status": "resolved", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 20, Section 5 (Conclusion)", "sourceQuote": "How to design ADMM for solving (1) without smoothing remains an open question for future work.", "problemStatement": "Riemannian ADMM minimizes f(x)+g(Ax) over an embedded manifold, where g may be nonsmooth. The paper smooths g by a Moreau envelope before applying its alternating-direction method. Construct an ADMM-type algorithm for the original unsmoothed problem and prove convergence and iteration complexity; later work provides such an algorithm.", "statusEvidence": "ARADMM handles the original nonsmooth term without smoothing and proves O(epsilon^-3) complexity for approximate KKT points.", "verificationNote": "This status is tied to the cited primary-literature result and the precise scope stated above.", "keywords": [ "riemannian optimization", "admm", "nonsmooth optimization", "manifold constraints", "augmented lagrangian", "moreau envelope" ], "literature": [ { "title": "Adaptive Riemannian ADMM for Nonsmooth Optimization: Optimal Complexity without Smoothing", "url": "https://arxiv.org/abs/2510.18617", "relation": "ARADMM handles the original nonsmooth term without smoothing and proves O(epsilon^-3) complexity for approximate KKT points." } ], "source": { "title": "A Riemannian Alternating Direction Method of Multipliers", "authors": [ "Jiaxiang Li", "Shiqian Ma", "Tejes Srivastava" ], "doi": "10.1287/moor.2023.0068", "url": "https://doi.org/10.1287/moor.2023.0068", "volume": "50", "issue": "4", "pages": "3222-3242" }, "number": 265 }, { "id": "mor-2024-W4405639491_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2024, "title": "Prove sharp decay rates for nonlinear comparison ODE driven by SDE noise", "topic": "Nonlinear Differential Inequalities", "publicationDate": "2024-12-20", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 23, Section 4.2 (Remark 4.10 and Conjecture 4.11).", "sourceQuote": "If p_δ=O(sigma_∞^2) and sigma_∞^2(t)=Obig((t+1)^(-(b)/(b-1))big) (for constants independent of δ and t̂_δ), then y(t)=Obig((t+1)^(-(1)/(b-1))big).", "problemStatement": "For a>0 and b>1, consider y′=−ayᵇ+pδ(t), with positive initial value at a localization time t̂δ. Suppose the forcing is uniformly bounded by a noise variance that decays at rate (t+1) to the power −b/(b−1), with constants independent of δ and t̂δ. Prove that y decays at rate (t+1) to the power −1/(b−1), uniformly in those localization parameters.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "stochastic differential equations", "lojasiewicz inequality", "comparison principle", "nonlinear ode", "convergence rates", "noise decay" ], "literature": [], "source": { "title": "An Stochastic Differential Equation Perspective on Stochastic Convex Optimization", "authors": [ "Maulén S. Rodrigo", "Jalal Fadili", "Hedy Attouch" ], "doi": "10.1287/moor.2022.0162", "url": "https://doi.org/10.1287/moor.2022.0162", "volume": "50", "issue": "4", "pages": "3190-3221" }, "number": 266 }, { "id": "mp-2024-second-sgd-homogenization-batch-o-n", "journal": "Mathematical Programming", "publicationYear": 2024, "title": "SGD Homogenization for Every Sublinear Batch Size", "topic": "High-dimensional stochastic optimization", "publicationDate": "2024-12-24", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Introduction, discussion of prior dynamical equivalence", "sourceQuote": "It is conjectured that this holds up to β=o(n).", "problemStatement": "Extend the high-dimensional dynamical equivalence between stochastic gradient descent and its homogenized process from batch sizes β≤n^(1/5−δ) to every β=o(n). The data matrix is assumed left-orthogonally invariant and n is the ambient sample-scale parameter used in the source. The open task is to control the accumulated approximation error throughout the larger sublinear-batch regime without changing the limiting dynamics.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact statement, title, authors and citations across publisher pages, arXiv and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper direct result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "stochastic gradient descent", "homogenization", "batch size", "random matrix theory" ], "literature": [], "source": { "title": "Homogenization of SGD in high-dimensions: exact dynamics and generalization properties", "authors": [ "Courtney Paquette", "Elliot Paquette", "Ben Adlam", "Jeffrey Pennington" ], "doi": "10.1007/s10107-024-02171-3", "url": "https://doi.org/10.1007/s10107-024-02171-3", "volume": "214", "issue": "1-2", "pages": "1-90" }, "number": 267 }, { "id": "mor-2024-W3171990983_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2024, "title": "Random-order contention resolution for rank-1 matroids with negative correlation", "topic": "Contention Resolution Schemes", "publicationDate": "2024-12-24", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 6, end of Subsection 1.2 (\"Reinterpretation as contention resolution under negative correlation\")", "sourceQuote": "We leave it as an open question whether a (1- 1/e)-selectable Random-order Contention Resolution Scheme is possible under property (2).", "problemStatement": "A random-order contention-resolution scheme for a rank-one matroid may select at most one active element. Here the activity indicators satisfy strong negative-cylinder inequalities for both active and inactive events rather than independence. Determine whether a (1-1/e)-selectable scheme exists under exactly that negative-correlation assumption.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "contention resolution schemes", "rank-1 matroid", "random-order model", "negative correlation", "prophet inequalities", "stochastic matching" ], "literature": [], "source": { "title": "Improved Guarantees for Offline Stochastic Matching via New Ordered Contention Resolution Schemes", "authors": [ "Brian Brubach", "Nathaniel Grammel", "Will Ma", "Aravind Srinivasan" ], "doi": "10.1287/moor.2022.0256", "url": "https://doi.org/10.1287/moor.2022.0256", "volume": "50", "issue": "4", "pages": "3243-3256" }, "number": 268 }, { "title": "Characterize Frank–Wolfe acceleration on the nuclear-norm ball", "topic": "Convex & nonlinear optimization", "series": "A", "publicationDate": "2025-01-06", "sourceLocation": "Section 6, Other numerical experiments", "sourceQuote": "This raises the open question to characterize the acceleration FW enjoys when optimizing over the nuclear norm ball.", "problemStatement": "Frank–Wolfe methods minimize a smooth convex function over a compact feasible set using only a linear-optimization oracle; on the nuclear-norm ball, each oracle call returns a low-rank direction. The source paper proves accelerated affine-invariant rates for open-loop step sizes under several geometric conditions and observes still faster behavior for matrix problems. Characterize precisely when this acceleration occurs on the nuclear-norm ball, including the attainable primal-error, Frank–Wolfe-gap, and primal–dual rates.", "status": "partial_progress", "statusEvidence": "Garber (2025) gives accelerated Frank–Wolfe-family algorithms for nuclear-norm and spectrahedral domains under complementarity and sparsity conditions, but does not fully characterize the source algorithm’s observed acceleration.", "literature": [ { "title": "Accelerated Frank-Wolfe Algorithms: Complementarity Conditions and Sparsity", "url": "https://arxiv.org/abs/2511.02821", "relation": "Later partial progress" } ], "keywords": [ "Frank–Wolfe", "nuclear norm", "open-loop step size", "accelerated rate" ], "source": { "title": "Accelerated affine-invariant convergence rates of the Frank–Wolfe algorithm with open-loop step-sizes", "authors": [ "Elias Wirth", "Javier Peña", "Sebastian Pokutta" ], "doi": "10.1007/s10107-024-02180-2", "url": "https://doi.org/10.1007/s10107-024-02180-2", "preprintUrl": "https://arxiv.org/abs/2310.04096" }, "id": "mp-2025-001", "number": 269, "recordType": "reviewed", "sourceClaim": "explicit_open_problem", "statusCheckedThrough": "2026-07-13", "verificationNote": "This status is tied to the cited primary-literature result and the precise scope stated above.", "journal": "Mathematical Programming", "publicationYear": 2025 }, { "id": "mor-2025-W3133568468_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2025, "title": "Compute interim envy-free lotteries maximizing expected average Nash welfare in matching instances", "topic": "Fair Division Lotteries", "publicationDate": "2025-01-10", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 19, Section 7 (problems)", "sourceQuote": "Computing iEF lotteries of maximum expected average Nash welfare is elusive at this point.", "problemStatement": "A lottery over one-to-one agent-item matchings is interim envy-free when, conditional on receiving an item, each agent weakly prefers her own induced lottery to another agent's. The objective is expected average Nash social welfare of the realized matching. Determine the computational complexity of finding an interim-envy-free lottery maximizing that objective, and design an algorithm if tractable.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "fair division", "interim envy-freeness", "random allocations", "Nash social welfare", "average Nash welfare", "matching instances" ], "literature": [], "source": { "title": "On Interim Envy-Free Allocation Lotteries", "authors": [ "Ioannis Caragiannis", "Panagiotis Kanellopoulos", "Maria Kyropoulou" ], "doi": "10.1287/moor.2023.0203", "url": "https://doi.org/10.1287/moor.2023.0203", "preprintUrl": "http://repository.essex.ac.uk/30538/23/On_Interim_Envy_Free_Allocation_Lotteries.pdf", "volume": "50", "issue": "4", "pages": "3257-3282" }, "number": 270 }, { "title": "Evolution of solution-norm bounds in quantum-inspired refinement", "topic": "Matrix & semidefinite optimization", "series": "B", "publicationDate": "2025-01-11", "sourceLocation": "Conclusion", "sourceQuote": "An interesting question … is how ω⁽ᵏ⁾ evolves through iterations of the IR methods.", "problemStatement": "The IR–IF–QIPM framework solves a semidefinite program through iterative refinement, with each refinement subproblem handled by an inexact-feasible quantum-inspired interior-point method. Its total complexity depends on a bound ω⁽ᵏ⁾ for the norm of the solution to the kth refinement subproblem, but the analysis does not control how this quantity changes between iterations. Determine the evolution of ω⁽ᵏ⁾ and derive a bound strong enough to sharpen the end-to-end complexity guarantee.", "keywords": [ "iterative refinement", "quantum interior point", "semidefinite optimization", "norm bound" ], "source": { "title": "Quantum computing inspired iterative refinement for semidefinite optimization", "authors": [ "Mohammadhossein Mohammadisiahroudi", "Brandon Augustino", "Pouya Sampourmahani", "Tamás Terlaky" ], "doi": "10.1007/s10107-024-02183-z", "url": "https://doi.org/10.1007/s10107-024-02183-z", "preprintUrl": "https://arxiv.org/abs/2312.11253" }, "id": "mp-2025-002", "number": 271, "recordType": "reviewed", "sourceClaim": "explicit_open_problem", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "statusEvidence": "Targeted title, exact-statement, and forward-literature searches located no later paper claiming a complete resolution.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "literature": [], "journal": "Mathematical Programming", "publicationYear": 2025 }, { "title": "Matroid Secretary Conjecture", "topic": "Combinatorial optimization", "series": "B", "publicationDate": "2025-01-15", "sourceLocation": "Introduction, Conjecture 1.1", "sourceQuote": "Whereas the MSP Conjecture remains open.", "problemStatement": "In the Matroid Secretary Problem, elements of an arbitrary matroid arrive in uniformly random order with adversarial nonnegative weights, and an online algorithm must irrevocably select an independent set. Constant competitive ratios are known for important matroid classes, but no universal constant is known for all matroids. Prove or disprove that a single O(1)-competitive online algorithm exists for every matroid.", "status": "partial_progress", "statusEvidence": "A May 2026 technical report still describes the general conjecture as open. Later work improves restricted matroid classes and related formulations, but does not settle arbitrary matroids.", "literature": [ { "title": "Matroid Secretary via Labeling Schemes", "url": "https://egres.elte.hu/www/tr-25-09.html", "relation": "May 2026 source still describing the general conjecture as open" }, { "title": "Beating Competitive Ratio 4 for Graphic Matroid Secretary", "url": "https://doi.org/10.4230/LIPIcs.ESA.2025.52", "relation": "Later progress for a restricted matroid class" } ], "keywords": [ "matroid secretary", "online algorithms", "competitive analysis", "random order" ], "source": { "title": "Constant-competitiveness for random assignment Matroid secretary without knowing the Matroid", "authors": [ "Richard Santiago", "Ivan Sergeev", "Rico Zenklusen" ], "doi": "10.1007/s10107-024-02177-x", "url": "https://doi.org/10.1007/s10107-024-02177-x", "preprintUrl": "https://arxiv.org/abs/2305.05353" }, "id": "mp-2025-003", "number": 272, "recordType": "reviewed", "sourceClaim": "explicit_open_problem", "statusCheckedThrough": "2026-07-13", "verificationNote": "This status is tied to the cited primary-literature result and the precise scope stated above.", "journal": "Mathematical Programming", "publicationYear": 2025 }, { "id": "focm-2025-radiative-transfer-certified-initial-gap", "journal": "Foundations of Computational Mathematics", "publicationYear": 2025, "title": "Radiative Transfer Certified Initial Gap", "topic": "Validated numerical analysis", "publicationDate": "2025-01-21", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Introduction, discussion of the initial lower spectral-gap estimate", "sourceQuote": "How to rigorously certify such an initial spectral estimate remains open.", "problemStatement": "Turn a numerically guessed lower bound for the spectral gap in the radiative-transfer eigensolver into a rigorous certificate. The certificate should initialize the paper's subsequent a posteriori eigenvalue bounds without assuming the desired gap.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "radiative transfer", "spectral gap", "eigenvalue certification", "a posteriori bounds" ], "literature": [], "source": { "title": "Accuracy Controlled Schemes for the Eigenvalue Problem of the Radiative Transfer Equation", "authors": [ "Wolfgang Dahmen", "Olga Mula" ], "doi": "10.1007/s10208-025-09694-x", "url": "https://doi.org/10.1007/s10208-025-09694-x", "volume": "26", "issue": "2", "pages": "689-746" }, "number": 273 }, { "id": "focm-2025-noisy-subgradient-without-regularity", "journal": "Foundations of Computational Mathematics", "publicationYear": 2025, "title": "Noisy Subgradient Without Regularity", "topic": "Nonsmooth optimization", "publicationDate": "2025-01-22", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Introduction, after Theorem 1.1", "sourceQuote": "It is an open question whether Theorem 1.1 holds without subdifferential regularity.", "problemStatement": "For a definable locally Lipschitz objective f, consider x_(k+1)=x_k-alpha_k g_k+alpha_k nu_k, where g_k is a subgradient, alpha_k is asymptotic to k^-gamma with gamma in (1/2,1), and nu_k is mean-zero absolutely continuous noise. For almost every linear perturbation f_v(x)=f(x)-, the theorem says the iterates either diverge or converge to a local minimizer when f is subdifferentially regular. Determine whether that same generic avoidance-and-convergence conclusion holds without subdifferential regularity while preserving the other assumptions.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "subgradient dynamics", "definable functions", "subdifferential regularity", "noise" ], "literature": [], "source": { "title": "Active Manifolds, Stratifications, and Convergence to Local Minima in Nonsmooth Optimization", "authors": [ "Damek Davis", "Dmitriy Drusvyatskiy", "Liwei Jiang" ], "doi": "10.1007/s10208-025-09691-0", "url": "https://doi.org/10.1007/s10208-025-09691-0", "preprintUrl": "https://arxiv.org/abs/2108.11832", "volume": "26", "issue": "2", "pages": "779-861" }, "number": 274 }, { "id": "focm-2025-general-p-star-body-variational-minimizer", "journal": "Foundations of Computational Mathematics", "publicationYear": 2025, "title": "General P Star Body Variational Minimizer", "topic": "Convex and integral geometry", "publicationDate": "2025-01-27", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 4, discussion of variational problem (6)", "sourceQuote": "For general P, it remains an open problem to characterize the unique minimizer.", "problemStatement": "A star body K has a positive continuous radial function and gauge ||x||_K=inft>0:x in tK. For a probability distribution P, characterize the unique unit-volume minimizer of E_P[||x||_K] over star bodies K. The paper identifies the normalized radial solution when P has the required positive continuous directional density and proves only restricted existence for general P, so the requested characterization must cover arbitrary admissible distributions.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "star bodies", "variational problem", "Minkowski theory", "unique minimizer" ], "literature": [], "source": { "title": "Optimal Regularization for a Data Source", "authors": [ "Oscar Leong", "Eliza O’ Reilly", "Yong Sheng Soh", "Venkat Chandrasekaran" ], "doi": "10.1007/s10208-025-09693-y", "url": "https://doi.org/10.1007/s10208-025-09693-y", "preprintUrl": "https://arxiv.org/abs/2212.13597", "volume": "26", "issue": "2", "pages": "889-938" }, "number": 275 }, { "id": "focm-2025-ci-problem-weakening-simple-solution", "journal": "Foundations of Computational Mathematics", "publicationYear": 2025, "title": "CI Problem Weakening Simple Solution", "topic": "Computational algebra and topology", "publicationDate": "2025-01-27", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conjecture 6.10", "sourceQuote": "There is an odd c such that every solvable CI problem has a simple solution after c-weakening.", "problemStatement": "Prove that every solvable constrained-invertibility problem becomes simply solvable after weakening by one universal odd constant c. The paper shows this is equivalent to its upset-module refinement and matching conjectures.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "constrained invertibility", "weakening", "simple solution", "persistence modules" ], "literature": [], "source": { "title": "Stabilizing Decomposition of Multiparameter Persistence Modules", "authors": [ "Håvard Bakke Bjerkevik" ], "doi": "10.1007/s10208-025-09695-w", "url": "https://doi.org/10.1007/s10208-025-09695-w", "volume": "26", "issue": "2", "pages": "939-998" }, "number": 276 }, { "id": "focm-2025-persistence-common-refinement", "journal": "Foundations of Computational Mathematics", "publicationYear": 2025, "title": "Persistence Common Refinement", "topic": "Topological data analysis", "publicationDate": "2025-01-27", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conjecture 6.1", "sourceQuote": "There is a constant c such that any epsilon-interleaved modules have a common c-epsilon-refinement.", "problemStatement": "Prove that epsilon-interleaved pointwise finite-dimensional persistence modules admit a common c-epsilon refinement for a universal constant c. Determine the smallest possible constant or produce a counterexample. A refinement splits a module decomposition into smaller summands while staying within an interleaving tolerance; one-parameter barcode stability has such control, but multiparameter decompositions can be unstable.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "multiparameter persistence", "interleaving", "common refinement", "decomposition stability" ], "literature": [], "source": { "title": "Stabilizing Decomposition of Multiparameter Persistence Modules", "authors": [ "Håvard Bakke Bjerkevik" ], "doi": "10.1007/s10208-025-09695-w", "url": "https://doi.org/10.1007/s10208-025-09695-w", "volume": "26", "issue": "2", "pages": "939-998" }, "number": 277 }, { "id": "focm-2025-persistence-pruning-interleaving-equivalence", "journal": "Foundations of Computational Mathematics", "publicationYear": 2025, "title": "Persistence Pruning Interleaving Equivalence", "topic": "Topological data analysis", "publicationDate": "2025-01-27", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conjecture 4.20", "sourceQuote": "We conjecture that dP is Lipschitz equivalent to dI with constants depending only on r.", "problemStatement": "For persistence modules of pointwise dimension at most r, prove the proposed comparison dP <= dI <= 2r dP between pruning and interleaving distances. A counterexample or sharp replacement constants would also resolve the conjecture.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "multiparameter persistence", "pruning distance", "interleaving distance", "stability" ], "literature": [], "source": { "title": "Stabilizing Decomposition of Multiparameter Persistence Modules", "authors": [ "Håvard Bakke Bjerkevik" ], "doi": "10.1007/s10208-025-09695-w", "url": "https://doi.org/10.1007/s10208-025-09695-w", "volume": "26", "issue": "2", "pages": "939-998" }, "number": 278 }, { "id": "focm-2025-persistence-upset-matching", "journal": "Foundations of Computational Mathematics", "publicationYear": 2025, "title": "Persistence Upset Matching", "topic": "Topological data analysis", "publicationDate": "2025-01-27", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conjecture 6.5", "sourceQuote": "There is a constant c such that epsilon-interleaved upset-decomposable modules admit a c-epsilon-matching.", "problemStatement": "Prove a universal matching theorem for epsilon-interleaved pointwise finite-dimensional upset-decomposable persistence modules. Equivalently, determine whether their barcodes always admit a c-epsilon matching for some fixed c.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "upset-decomposable modules", "barcode matching", "multiparameter persistence" ], "literature": [], "source": { "title": "Stabilizing Decomposition of Multiparameter Persistence Modules", "authors": [ "Håvard Bakke Bjerkevik" ], "doi": "10.1007/s10208-025-09695-w", "url": "https://doi.org/10.1007/s10208-025-09695-w", "volume": "26", "issue": "2", "pages": "939-998" }, "number": 279 }, { "id": "siopt-2025-058", "journal": "SIAM Journal on Optimization", "publicationYear": 2025, "title": "Dimension free sign stiefel error bound", "topic": "Error bounds", "publicationDate": "2025-01-27", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Introduction; extracted preprint lines 98-104", "sourceQuote": "the explicit value of nu and the value of q that is independent of the dimension are still unknown", "problemStatement": "The sign-constrained Stiefel manifold consists of orthonormal matrices satisfying prescribed entrywise signs. The paper proves local error bounds with dimension-dependent exponents and nonconstructive constants on compact sets. Find explicit constants and a dimension-independent exponent, and decide whether comparable error bounds hold globally on unbounded ambient sets.", "statusEvidence": "No forward-citing primary paper or exact-query result explicitly claiming a proof or counterexample was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "Stiefel manifold", "sign constraints", "error bound", "dimension-free" ], "literature": [], "source": { "title": "Tight Error Bounds for the Sign-Constrained Stiefel Manifold", "authors": [ "Xiaojun Chen", "Yifan He", "Zaikun Zhang" ], "doi": "10.1137/24m1659030", "url": "https://doi.org/10.1137/24m1659030", "preprintUrl": "https://arxiv.org/abs/2210.05164", "volume": "35", "issue": "1", "pages": "302-329" }, "number": 280 }, { "id": "mor-2025-W4226364838_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2025, "title": "Prove consecutive-service-rate optimality for sojourn-time partitioning under moment monotonicity", "topic": "Queueing Optimization", "publicationDate": "2025-02-06", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 34, Section 5 (Extension: Analysis of Sojourn Time), near the discussion following Theorem 8; reiterated in Section 6 (Concluding Remarks) on page 35.", "sourceQuote": "we conjecture that under Assumption 1 it is optimal to assign customers with consecutive service rates to the same queue.", "problemStatement": "A normalized service capacity is divided among k parallel first-come-first-served queues serving Poisson customer types with service rates μ₁>⋯>μₙ. Assume the adjusted variability E[Sᵢ²]/E[Sᵢ] increases with mean service time. Prove or refute that an expected-sojourn-time minimizer assigns types to queues in consecutive blocks of this service-rate ordering.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "queue partitioning", "FCFS queues", "sojourn time minimization", "capacity allocation", "routing by types", "structural optimality", "moment conditions" ], "literature": [], "source": { "title": "Optimal Partition for a Multi-Type Queueing System", "authors": [ "Shengyu Cao", "Simai He", "Zizhuo Wang", "Yifan Feng" ], "doi": "10.1287/moor.2023.0035", "url": "https://doi.org/10.1287/moor.2023.0035", "volume": "51", "issue": "1", "pages": "277-298" }, "number": 281 }, { "id": "focm-2025-symmetric-group-decomposition-sharp-complexity", "journal": "Foundations of Computational Mathematics", "publicationYear": 2025, "title": "Symmetric Group Decomposition Sharp Complexity", "topic": "Computational representation theory", "publicationDate": "2025-02-10", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 9, future work", "sourceQuote": "These bounds are unlikely to be sharp, and in experiments we often achieve better complexity, though proving these sharper complexity results remains open", "problemStatement": "Given an orthogonal d-dimensional representation rho of S_n through adjacent-transposition generators, Algorithm 1 finds the irreducible types and multiplicities and Algorithm 2 constructs a full repeated-block decomposition. Prove sharper bounds than the current O(n^2 d^3) operations for Algorithm 1 and the additional O(n d^4) for Algorithm 2, in particular the empirically observed O(n d^3) cost for Algorithm 2. The proof should exploit the zero-row and sparse structure of the Young-Jucys-Murphy irreducible generators rather than assume dense linear algebra.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "symmetric group", "representation decomposition", "algorithmic complexity", "sparsity" ], "literature": [], "source": { "title": "Representations of the Symmetric Group are Decomposable in Polynomial Time", "authors": [ "Sheehan Olver" ], "doi": "10.1007/s10208-025-09697-8", "url": "https://doi.org/10.1007/s10208-025-09697-8", "volume": "26", "issue": "2", "pages": "1069-1092" }, "number": 282 }, { "title": "Core-membership complexity in partitioned matching games", "topic": "Game theory & matching", "series": "A", "publicationDate": "2025-02-11", "sourceLocation": "Conclusions", "sourceQuote": "We do not know if P2 … is co-NP-complete.", "problemStatement": "A partitioned matching game models coalitions whose value is the maximum feasible matching subject to vertex capacities and partition constraints, as in international kidney exchange. The source paper studies problem P2: deciding whether a proposed payoff vector belongs to the cooperative game’s core, and settles several bounded-width or bounded-capacity cases. Determine whether P2 is co-NP-complete for b-matching games with capacities not bounded by two and for partitioned matching games of width at least three.", "keywords": [ "matching games", "core membership", "co-NP-completeness", "kidney exchange" ], "source": { "title": "Partitioned matching games for international kidney exchange", "authors": [ "Márton Benedek", "Péter Biró", "Walter Kern", "Dömötör Pálvölgyi", "Daniel Paulusma" ], "doi": "10.1007/s10107-025-02200-9", "url": "https://doi.org/10.1007/s10107-025-02200-9" }, "id": "mp-2025-004", "number": 283, "recordType": "reviewed", "sourceClaim": "explicit_open_problem", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "statusEvidence": "Targeted title, exact-statement, and forward-literature searches located no later paper claiming a complete resolution.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "literature": [], "journal": "Mathematical Programming", "publicationYear": 2025 }, { "title": "Deterministic complexity of Exact Perfect Matching", "topic": "Combinatorial optimization", "series": "A", "publicationDate": "2025-02-11", "sourceLocation": "Section 1.4, p. 10", "sourceQuote": "The complexity status of Exact Perfect Matching is a longstanding open problem.", "problemStatement": "Exact Perfect Matching asks whether a graph whose edges are colored red or blue has a perfect matching containing exactly k red edges. Randomized polynomial-time algorithms are known through algebraic techniques, but no deterministic polynomial-time algorithm or matching hardness classification is known. Resolve the deterministic complexity of the decision problem.", "status": "partial_progress", "statusEvidence": "A 2025 paper gives parameterized and subexponential results for structured instances. A June 2026 primary source still describes even the bipartite deterministic case as open.", "literature": [ { "title": "Exact Matching and Top-k Perfect Matching Parameterized by Neighborhood Diversity or Bandwidth", "url": "https://arxiv.org/abs/2510.12552", "relation": "Parameterized partial progress" }, { "title": "Matroids are Equitable", "url": "https://doi.org/10.1007/s00493-026-00217-y", "relation": "June 2026 source still describing the deterministic question as open" } ], "keywords": [ "exact matching", "derandomization", "perfect matching", "complexity" ], "source": { "title": "Partitioned matching games for international kidney exchange", "authors": [ "Márton Benedek", "Péter Biró", "Walter Kern", "Dömötör Pálvölgyi", "Daniel Paulusma" ], "doi": "10.1007/s10107-025-02200-9", "url": "https://doi.org/10.1007/s10107-025-02200-9" }, "id": "mp-2025-027", "number": 284, "recordType": "reviewed", "sourceClaim": "explicit_open_problem", "statusCheckedThrough": "2026-07-13", "verificationNote": "This status is tied to the cited primary-literature result and the precise scope stated above.", "journal": "Mathematical Programming", "publicationYear": 2025 }, { "title": "Nucleolus complexity for matching-game variants", "topic": "Game theory & matching", "series": "A", "publicationDate": "2025-02-11", "sourceLocation": "Conclusions, p. 34", "sourceQuote": "Determining the complexity of computing the nucleolus is still open for the following games.", "problemStatement": "The nucleolus is the unique cooperative-game payoff vector that lexicographically minimizes coalition dissatisfaction, but computing it may require solving many coalition-separation problems. Its complexity remains unclassified for four matching-game families: b-matching games with capacity b ≤ 2, unrestricted partitioned matching games, partitioned matching games of width at most two, and partitioned matching games of width at most three. Determine separately whether the nucleolus can be computed in polynomial time for each family or prove the corresponding hardness.", "status": "partial_progress", "statusEvidence": "Ebert and Ellerbrock (2026) reduce the b ≤ 2 case to Shortest Non-Zero Cycle and solve certain subclasses, but do not settle all families listed by the source.", "literature": [ { "title": "Nucleolus Computation by Non-Zero-Constrained Optimization", "url": "https://arxiv.org/abs/2605.29571", "relation": "Direct 2026 partial progress" } ], "keywords": [ "nucleolus", "cooperative games", "b-matching", "computational complexity" ], "source": { "title": "Partitioned matching games for international kidney exchange", "authors": [ "Márton Benedek", "Péter Biró", "Walter Kern", "Dömötör Pálvölgyi", "Daniel Paulusma" ], "doi": "10.1007/s10107-025-02200-9", "url": "https://doi.org/10.1007/s10107-025-02200-9" }, "id": "mp-2025-028", "number": 285, "recordType": "reviewed", "sourceClaim": "explicit_open_problem", "statusCheckedThrough": "2026-07-13", "verificationNote": "This status is tied to the cited primary-literature result and the precise scope stated above.", "journal": "Mathematical Programming", "publicationYear": 2025 }, { "id": "mor-2025-W4407408247_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2025, "title": "Improve minimax regret dependence on states and horizon in episodic CTMDPs", "topic": "Reinforcement Learning", "publicationDate": "2025-02-12", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 14, Section 4 (Main Results), discussion immediately after Theorem 2.", "sourceQuote": "whether one can improve the dependence of these bounds on S, H and lambda_(max) to narrow down the gap between the upper and lower bounds.", "problemStatement": "The paper studies regret in an episodic finite-horizon continuous-time Markov decision process with finite state and action sets. Its upper bound has an extra state factor and worse horizon dependence than the lower bound, which scales essentially as H√(SAK) in the relevant regime. Design a reinforcement-learning algorithm with sharper state, horizon, and jump-rate dependence, or prove matching lower bounds.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "continuous-time MDPs", "episodic reinforcement learning", "minimax regret", "regret lower bound", "value iteration", "upper confidence bounds", "finite horizon" ], "literature": [], "source": { "title": "Square-Root Regret Bounds for Continuous-Time Episodic Markov Decision Processes", "authors": [ "Xuefeng Gao", "Xunyu Zhou" ], "doi": "10.1287/moor.2022.0283", "url": "https://doi.org/10.1287/moor.2022.0283", "volume": "51", "issue": "1", "pages": "333-357" }, "number": 286 }, { "id": "siopt-2025-052", "journal": "SIAM Journal on Optimization", "publicationYear": 2025, "title": "Split LMO rate and inexact oracles", "topic": "Projection-free optimization", "publicationDate": "2025-02-13", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conclusion; extracted preprint lines 1198-1201 and 1233-1239", "sourceQuote": "whether or not O(1/t) convergence is possible in the split-LMO setting without additional assumptions remains open.", "problemStatement": "A split linear-minimization-oracle method handles a sum or intersection of constraint blocks by querying separate LMOs rather than one global oracle. The paper proves asymptotic convergence and slower penalized-gap rates. Determine whether the primal gap can reach O(1/t) without extra assumptions and develop convergence guarantees for approximate or block-iterative LMO calls.", "statusEvidence": "No forward-citing primary paper or exact-query result explicitly claiming a proof or counterexample was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "conditional gradient", "split LMO", "inexact oracle", "convergence rate" ], "literature": [], "source": { "title": "Splitting the Conditional Gradient Algorithm", "authors": [ "Zev Woodstock", "Sebastian Pokutta" ], "doi": "10.1137/24m1638008", "url": "https://doi.org/10.1137/24m1638008", "preprintUrl": "https://arxiv.org/abs/2311.05381", "volume": "35", "issue": "1", "pages": "347-368" }, "number": 287 }, { "title": "The four-thirds conjecture for the Subtour LP", "topic": "Combinatorial optimization", "series": "B", "publicationDate": "2025-02-17", "sourceLocation": "Abstract and Introduction", "sourceQuote": "A long-standing conjecture … states that the integrality gap … is at most 4/3.", "problemStatement": "For metric traveling-salesman instances, the Subtour LP (also called the Held–Karp relaxation) imposes degree and subtour-elimination constraints and gives a lower bound on the optimal tour cost. Its worst-case integrality gap is at least 4/3, while the best general upper bound remains larger. Prove that every metric instance has a tour costing at most 4/3 times the LP optimum, or exhibit a counterexample.", "statusEvidence": "Two 2026 papers still call this a conjecture: one verifies additional small instances, and one improves the max-entropy analysis on half-integral cycle-cut instances without settling the general gap.", "literature": [ { "title": "Extending Exact Integrality Gap Computations for the Metric TSP", "url": "https://arxiv.org/abs/2603.12995", "relation": "2026 computational evidence; conjecture remains" }, { "title": "Maximum Entropy is a 10/7-Approximation Algorithm for the TSP on Half-Integral Cycle Cut Instances", "url": "https://arxiv.org/abs/2607.01536", "relation": "2026 progress on a special class" } ], "keywords": [ "traveling salesman", "Subtour LP", "integrality gap", "four-thirds conjecture" ], "source": { "title": "A 4/3-approximation algorithm for half-integral cycle cut instances of the TSP", "authors": [ "Billy Jin", "Nathan Klein", "David P. Williamson" ], "doi": "10.1007/s10107-025-02193-5", "url": "https://doi.org/10.1007/s10107-025-02193-5", "preprintUrl": "https://arxiv.org/abs/2211.04639" }, "id": "mp-2025-005", "number": 288, "recordType": "reviewed", "sourceClaim": "explicit_open_problem", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "journal": "Mathematical Programming", "publicationYear": 2025 }, { "id": "siopt-2025-053", "journal": "SIAM Journal on Optimization", "publicationYear": 2025, "title": "Facial signature lower bound tightness", "topic": "Convex algebraic geometry", "publicationDate": "2025-02-18", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 3.3; extracted preprint lines 671-675", "sourceQuote": "it is an open question whether this lower bound is always achieved.", "problemStatement": "The facial dimension signature I of a closed convex set is the set of dimensions of its nonempty faces, and n=max I. The lower bound is the smallest k for which nonnegative integers d_1≥⋯≥d_k exist such that every i∈I other than n lies, for some m≤k, between Σ_(j=1)^m d_j−(m−1)n and d_m. Determine whether every signature I can be realized by exactly k convex quadratic inequalities; if not, quantify the largest possible gap above this bound.", "statusEvidence": "No forward-citing primary paper or exact-query result explicitly claiming a proof or counterexample was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "facial dimension signature", "convex quadratic inequalities", "representation complexity" ], "literature": [], "source": { "title": "Everything Is Possible: Constructing Spectrahedra with Prescribed Facial Dimensions", "authors": [ "Vera Roshchina", "Levent Tunçel" ], "doi": "10.1137/24m164344x", "url": "https://doi.org/10.1137/24m164344x", "preprintUrl": "https://arxiv.org/abs/2312.04419", "volume": "35", "issue": "1", "pages": "400-418" }, "number": 289 }, { "id": "siopt-2025-056", "journal": "SIAM Journal on Optimization", "publicationYear": 2025, "title": "Measured facial signature characterization", "topic": "Convex geometry", "publicationDate": "2025-02-18", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 4.3, open question 2; extracted preprint pages 16-17", "sourceQuote": "characterize the set of M such that there exists a convex set with measured facial dimension signature (I, M).", "problemStatement": "A measured facial dimension signature augments the set I of face dimensions with M_i, the Hausdorff dimension of the union of all i-dimensional faces. This captures geometry invisible to I alone. For a fixed finite I, determine the sharp coordinatewise bounds on M and characterize exactly which vectors M are realizable by convex sets.", "statusEvidence": "No forward-citing primary paper or exact-query result explicitly claiming a proof or counterexample was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "measured facial signature", "Hausdorff dimension", "face unions", "realizability" ], "literature": [], "source": { "title": "Everything Is Possible: Constructing Spectrahedra with Prescribed Facial Dimensions", "authors": [ "Vera Roshchina", "Levent Tunçel" ], "doi": "10.1137/24m164344x", "url": "https://doi.org/10.1137/24m164344x", "preprintUrl": "https://arxiv.org/abs/2312.04419", "volume": "35", "issue": "1", "pages": "400-418" }, "number": 290 }, { "id": "siopt-2025-055", "journal": "SIAM Journal on Optimization", "publicationYear": 2025, "title": "Optimal lift size for facial signature", "topic": "Extended formulations", "publicationDate": "2025-02-18", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 4.2, open questions 1-3; extracted preprint page 16", "sourceQuote": "What are the sizes of smallest spectrahedral representations for a given facial dimension signature?", "problemStatement": "Many convex sets share a facial dimension signature but admit representations of very different algebraic sizes. The paper measures direct quadratic descriptions, while lifted spectrahedra, spectrahedral shadows, and hyperbolicity cones may be much smaller. Given a signature, determine the minimum matrix or lift size for each representation class and compare their separation.", "statusEvidence": "No forward-citing primary paper or exact-query result explicitly claiming a proof or counterexample was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "spectrahedral representation", "spectrahedral shadow", "hyperbolicity cone", "lift complexity" ], "literature": [], "source": { "title": "Everything Is Possible: Constructing Spectrahedra with Prescribed Facial Dimensions", "authors": [ "Vera Roshchina", "Levent Tunçel" ], "doi": "10.1137/24m164344x", "url": "https://doi.org/10.1137/24m164344x", "preprintUrl": "https://arxiv.org/abs/2312.04419", "volume": "35", "issue": "1", "pages": "400-418" }, "number": 291 }, { "id": "siopt-2025-057", "journal": "SIAM Journal on Optimization", "publicationYear": 2025, "title": "Polar facial signature realizability", "topic": "Convex geometry", "publicationDate": "2025-02-18", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 4.4, open questions 1-2; extracted preprint pages 17-18", "sourceQuote": "Is it true that for any such pair I, J there exists a compact convex set C", "problemStatement": "Polarity reverses the face structure of a compact convex set but can destroy facial exposedness. The paper shows that every low-dimensional admissible pair of primal and polar signatures is realizable. Determine whether every dimension signature has a facially exposed compact realization with facially exposed polar, and whether every admissible pair (I,J) can occur as the signatures of C and its polar C°.", "statusEvidence": "No forward-citing primary paper or exact-query result explicitly claiming a proof or counterexample was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "polarity", "facial exposedness", "facial dimension signature", "realizability" ], "literature": [], "source": { "title": "Everything Is Possible: Constructing Spectrahedra with Prescribed Facial Dimensions", "authors": [ "Vera Roshchina", "Levent Tunçel" ], "doi": "10.1137/24m164344x", "url": "https://doi.org/10.1137/24m164344x", "preprintUrl": "https://arxiv.org/abs/2312.04419", "volume": "35", "issue": "1", "pages": "400-418" }, "number": 292 }, { "id": "siopt-2025-054", "journal": "SIAM Journal on Optimization", "publicationYear": 2025, "title": "Special facial signature representation sizes", "topic": "Convex algebraic geometry", "publicationDate": "2025-02-18", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 4.1, open questions 1-3; extracted preprint pages 15-16", "sourceQuote": "What is the minimum number of convex quadratic inequalities whose solution set has this facial dimension signature?", "problemStatement": "The paper's general construction may be inefficient for indecomposable signatures such as primes, squares, or triangular numbers, which also occur as signatures of positive-semidefinite cones. For each of these families, determine the minimum number of convex quadratic inequalities needed to realize the signature. In particular, decide whether constructions exploiting algebraic structure beat the generic cardinality-based construction.", "statusEvidence": "No forward-citing primary paper or exact-query result explicitly claiming a proof or counterexample was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "facial signatures", "quadratic representation", "PSD cones", "indecomposable sequences" ], "literature": [], "source": { "title": "Everything Is Possible: Constructing Spectrahedra with Prescribed Facial Dimensions", "authors": [ "Vera Roshchina", "Levent Tunçel" ], "doi": "10.1137/24m164344x", "url": "https://doi.org/10.1137/24m164344x", "preprintUrl": "https://arxiv.org/abs/2312.04419", "volume": "35", "issue": "1", "pages": "400-418" }, "number": 293 }, { "id": "mor-2025-W4408054337_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2025, "title": "Extend dimension-reduction integral-equation method to Wonham-filter drift uncertainty", "topic": "Optimal Stopping Under Partialinformation", "publicationDate": "2025-02-28", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 18, Section 6 (Conclusions)", "sourceQuote": "There remain some problems, for example, the current approach can not be extended if the drift coefficient is a hidden two-state Markov chain.", "problemStatement": "An investor controls a portfolio and stopping time while observing an asset whose drift is a hidden two-state Markov chain, estimated through the Wonham filter. The dual optimal-stopping formulation has a free boundary; for a simpler Bayesian filter, dimension reduction yields a tractable integral equation. Extend that reduction to Wonham-filter uncertainty without requiring an unknown joint transition density, or derive another numerically tractable characterization.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "optimal stopping", "partial information", "Wonham filter", "hidden Markov model", "free boundary", "integral equations", "dimension reduction" ], "literature": [], "source": { "title": "A Simple Integral Equation Approach for Optimal Investment Stopping Problems with Partial Information", "authors": [ "Jie Xing", "Jingtang Ma", "Harry Zheng" ], "doi": "10.1287/moor.2023.0268", "url": "https://doi.org/10.1287/moor.2023.0268", "volume": "51", "issue": "1", "pages": "542-567" }, "number": 294 }, { "id": "focm-2025-n-body-turing-completeness", "journal": "Foundations of Computational Mathematics", "publicationYear": 2025, "title": "N Body Turing Completeness", "topic": "Dynamical systems and computability", "publicationDate": "2025-03-13", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 6, Open problem 1", "sourceQuote": "Is the n-body problem in celestial mechanics Turing complete for some n and some values of masses?", "problemStatement": "Determine whether a Newtonian n-body system is Turing complete for some finite n and some choice of masses. A solution must give an explicit universal-computation encoding in the Hamiltonian dynamics. Turing completeness here means reducing a universal machine's halting behavior to whether one explicitly specified gravitational trajectory reaches an explicitly specified region.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "n-body problem", "Turing completeness", "celestial mechanics", "undecidability" ], "literature": [], "source": { "title": "Towards a Fluid Computer", "authors": [ "Robert Cardona", "Eva Miranda", "Daniel Peralta-Salas" ], "doi": "10.1007/s10208-025-09699-6", "url": "https://doi.org/10.1007/s10208-025-09699-6", "volume": "25", "issue": "6", "pages": "1921-1937" }, "number": 295 }, { "id": "focm-2025-turing-complete-euler-standard-metric", "journal": "Foundations of Computational Mathematics", "publicationYear": 2025, "title": "Turing Complete Euler Standard Metric", "topic": "Fluid dynamics and computability", "publicationDate": "2025-03-13", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 6, Open problem 3", "sourceQuote": "Does there exist a Turing complete stationary Euler flow on the round sphere or flat three-torus?", "problemStatement": "Construct a Turing-complete stationary Euler flow on either the round three-sphere or the flat three-torus, or prove none exists. Existing universality constructions use a nonprescribed metric on compact manifolds or the Euclidean metric on noncompact three-space.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "stationary Euler", "Turing completeness", "round sphere", "flat torus" ], "literature": [], "source": { "title": "Towards a Fluid Computer", "authors": [ "Robert Cardona", "Eva Miranda", "Daniel Peralta-Salas" ], "doi": "10.1007/s10208-025-09699-6", "url": "https://doi.org/10.1007/s10208-025-09699-6", "volume": "25", "issue": "6", "pages": "1921-1937" }, "number": 296 }, { "id": "focm-2025-turing-complete-steady-navier-stokes", "journal": "Foundations of Computational Mathematics", "publicationYear": 2025, "title": "Turing Complete Steady Navier Stokes", "topic": "Fluid dynamics and computability", "publicationDate": "2025-03-13", "recordType": "reviewed", "status": "resolved", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 6, Open problem 4", "sourceQuote": "Does there exist a Turing complete steady Navier-Stokes flow on some compact Riemannian 3-manifold?", "problemStatement": "Determine whether a steady Navier-Stokes flow on some compact Riemannian three-manifold can be Turing complete. This asks whether viscosity is compatible with universal computation in a compact fluid system. At publication the known fluid-computer constructions applied to inviscid Euler flow, and the status evidence records the later 2026 construction that answers this precise existence question.", "statusEvidence": "Constructs stationary Navier-Stokes solutions on suitable compact Riemannian three-manifolds that perform universal computation, answering the existence question affirmatively.", "verificationNote": "This status is tied to the cited primary-literature result and the precise scope stated above.", "keywords": [ "Navier-Stokes", "Turing completeness", "steady flow", "compact manifold" ], "literature": [ { "title": "Turing complete Navier-Stokes steady states via cosymplectic geometry", "url": "https://doi.org/10.1093/pnasnexus/pgag131", "relation": "Constructs stationary Navier-Stokes solutions on suitable compact Riemannian three-manifolds that perform universal computation, answering the existence question affirmatively." } ], "source": { "title": "Towards a Fluid Computer", "authors": [ "Robert Cardona", "Eva Miranda", "Daniel Peralta-Salas" ], "doi": "10.1007/s10208-025-09699-6", "url": "https://doi.org/10.1007/s10208-025-09699-6", "volume": "25", "issue": "6", "pages": "1921-1937" }, "number": 297 }, { "id": "focm-2025-universal-turing-machine-slow-entropy", "journal": "Foundations of Computational Mathematics", "publicationYear": 2025, "title": "Universal Turing Machine Slow Entropy", "topic": "Dynamical systems and computability", "publicationDate": "2025-03-13", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 6, Open problem 2", "sourceQuote": "Given a universal Turing machine, is there some polynomial or subexponential growth entropy which is positive?", "problemStatement": "For every universal Turing machine with zero topological entropy, determine whether some polynomial or other subexponential slow entropy is positive. The entropy invariant should detect its computational universality despite zero exponential growth.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "universal Turing machine", "slow entropy", "polynomial entropy", "subexponential growth" ], "literature": [], "source": { "title": "Towards a Fluid Computer", "authors": [ "Robert Cardona", "Eva Miranda", "Daniel Peralta-Salas" ], "doi": "10.1007/s10208-025-09699-6", "url": "https://doi.org/10.1007/s10208-025-09699-6", "volume": "25", "issue": "6", "pages": "1921-1937" }, "number": 298 }, { "id": "mor-2025-W4408446556_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2025, "title": "Equality between semi-strict and strict subgradient derivatives for C2-decomposable functions", "topic": "Variational Analysis", "publicationDate": "2025-03-14", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 18, end of Section 3 (immediately after Theorem 3.9).", "sourceQuote": "The main question that remains open here is whether C^2-decomposable functions satisfy in (3.14).", "problemStatement": "A convex function is C²-decomposable near u when it equals a sublinear convex function composed with a C² smooth map. Its graphical subgradient derivative fixes the base subgradient y, whereas the semi-strict derivative lets that base subgradient approach y; let K be the critical cone where the directional derivative of g equals ⟨y,w⟩. Determine whether, in every direction w, the semi-strict derivative always equals the closure of the graphical derivative minus the polar cone K*, or give a counterexample.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "augmented Lagrangian", "C2-decomposable functions", "subgradient mapping", "graphical derivative", "critical cone", "metric subregularity" ], "literature": [], "source": { "title": "Convergence of Augmented Lagrangian Methods for Composite Optimization Problems", "authors": [ "Nguyen Thi Van Hang", "Ebrahim Sarabi" ], "doi": "10.1287/moor.2023.0324", "url": "https://doi.org/10.1287/moor.2023.0324", "volume": "51", "issue": "1", "pages": "591-620" }, "number": 299 }, { "title": "Lower envelope of bounded monomials in higher dimension", "topic": "Mixed-integer & global optimization", "series": "B", "publicationDate": "2025-03-17", "sourceLocation": "Section 4", "sourceQuote": "The lower envelope for Wᵢⱼ is an open problem for n > 2.", "problemStatement": "The convex envelope of a bounded monomial is the tightest convex relaxation of that monomial over its domain and is central to global optimization formulations. The source paper determines these envelopes on two-variable cones X ∩ Wᵢⱼ, but its lower-envelope construction does not extend to ambient dimension n > 2. Derive the missing lower envelope in higher dimension, or characterize why the two-dimensional description cannot be extended.", "status": "resolved", "statusEvidence": "Yang and Zhang’s June 2026 paper proves that the missing higher-dimensional lower envelope is flat at the lower monomial bound and gives the resulting convexification.", "literature": [ { "title": "Flat lower envelopes solve bounded monomial convexification on two-variable cones", "url": "https://doi.org/10.36922/IJOCTA026220100", "relation": "Resolution published online 29 June 2026" } ], "keywords": [ "convex envelope", "bounded monomial", "factorable programming", "global optimization" ], "source": { "title": "Convex envelopes of bounded monomials on two-variable cones", "authors": [ "Pietro Belotti" ], "doi": "10.1007/s10107-025-02212-5", "url": "https://doi.org/10.1007/s10107-025-02212-5", "preprintUrl": "https://arxiv.org/abs/2308.12650" }, "id": "mp-2025-006", "number": 300, "recordType": "reviewed", "sourceClaim": "explicit_open_problem", "statusCheckedThrough": "2026-07-13", "verificationNote": "This status is tied to the cited primary-literature result and the precise scope stated above.", "journal": "Mathematical Programming", "publicationYear": 2025 }, { "title": "Improve feasibility rates for primal constrained VI methods", "topic": "Convex & nonlinear optimization", "series": "B", "publicationDate": "2025-03-20", "sourceLocation": "Section 7, p. 23", "sourceQuote": "It remains unknown whether such guarantees can be improved.", "problemStatement": "A primal first-order method for a functionally constrained variational inequality must reduce both the variational-inequality gap and violation of the nonlinear constraints without access to optimal multipliers. Under strong monotonicity, the source analysis gives a constraint-violation rate slower than O(1/T). Improve that rate to O(1/T) or better while retaining a purely primal algorithm and no multiplier bound.", "keywords": [ "variational inequality", "functional constraints", "feasibility rate", "first-order method" ], "source": { "title": "Primal methods for variational inequality problems with functional constraints", "authors": [ "Liang Zhang", "Niao He", "Michael Muehlebach" ], "doi": "10.1007/s10107-025-02206-3", "url": "https://doi.org/10.1007/s10107-025-02206-3" }, "id": "mp-2025-030", "number": 301, "recordType": "reviewed", "sourceClaim": "explicit_open_problem", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "statusEvidence": "Targeted title, exact-statement, and forward-literature searches located no later paper claiming a complete resolution.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "literature": [], "journal": "Mathematical Programming", "publicationYear": 2025 }, { "id": "focm-2025-tetrahedral-refinement-closure-estimate", "journal": "Foundations of Computational Mathematics", "publicationYear": 2025, "title": "Tetrahedral Refinement Closure Estimate", "topic": "Adaptive finite element methods", "publicationDate": "2025-03-24", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conclusion, discussion of Theorem 8", "sourceQuote": "It remains open whether the standard tetrahedral refinement strategies satisfy the required closure estimate.", "problemStatement": "For marked tetrahedra M_ell and successive triangulations T_ell, prove the closure estimate #T_L-#T_0 <= C sum_(ell=0)^(L-1) #M_ell with C independent of the marking history. The unresolved strategies are the Bansch rule and the five-type Arnold-Mukherjee-Pouly refinement; the Schon alternative has an estimate whose constant depends on the number of initial tetrahedra. Establish a dependence on the initial mesh strong enough for the adaptive optimality theorem, rather than a generic claim about every tetrahedral refinement rule.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "tetrahedral refinement", "mesh closure", "adaptive FEM", "complexity" ], "literature": [], "source": { "title": "Adaptive Mesh Refinement for Arbitrary Initial Triangulations", "authors": [ "Lars Diening", "Lukas Gehring", "Johannes Storn" ], "doi": "10.1007/s10208-025-09698-7", "url": "https://doi.org/10.1007/s10208-025-09698-7", "volume": "26", "issue": "3", "pages": "1193-1218" }, "number": 302 }, { "id": "mor-2025-W4408830023_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2025, "title": "Achieve doubly-exponential competition complexity using dynamic prices in combinatorial auctions", "topic": "Online Posted Pricing", "publicationDate": "2025-03-25", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 5, Section 1.5 (Extensions)", "sourceQuote": "whether a double-exponentially fast approach is possible with dynamic prices.", "problemStatement": "An online combinatorial auction sells finitely many items to independent-distribution buyers with submodular or XOS valuations. A fully dynamic posted-price policy may update prices on unsold items after every purchase; its (1−ε)-competition complexity is the number k of independent buyer blocks needed to attain expected welfare at least (1−ε) times the one-block offline optimum. Determine whether this complexity is Θ(log log(1/ε)).", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "competition complexity", "prophet inequalities", "combinatorial auctions", "dynamic posted pricing", "XOS valuations", "submodular valuations" ], "literature": [], "source": { "title": "The Competition Complexity of Prophet Inequalities", "authors": [ "Johannes Brustle", "José Correa", "Paul Dütting", "Tomer Ezra", "Michal Feldman", "Victor Verdugo" ], "doi": "10.1287/moor.2024.0684", "url": "https://doi.org/10.1287/moor.2024.0684", "volume": "51", "issue": "1", "pages": "641-665" }, "number": 303 }, { "id": "mor-2025-W4408928070_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2025, "title": "Remove zero-gradient error requirement in partial relative inexact Bregman proximal gradient", "topic": "Inexact Proximal Gradient", "publicationDate": "2025-03-28", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 9, Section 3 (discussion following Algorithm 1, paragraph comparing (AbSC) and (ReSC)).", "sourceQuote": "it would be interesting to explore whether the requirement ‖Δₖ‖=0 can be removed in our methods", "problemStatement": "Each inexact Bregman proximal-gradient step approximately minimizes a relative-smoothness local model. Its partial-relative stopping rule uses Δₖ to record the first-order residual of that subproblem, yet the convergence proof requires Δₖ=0. Give a verifiable nonzero bound on Δₖ that preserves descent, stationarity, and convergence.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "bregman proximal gradient", "relative smoothness", "inexact methods", "relative stopping criterion", "subproblem accuracy", "first-order methods" ], "literature": [], "source": { "title": "Inexact Bregman Proximal Gradient Method and Its Inertial Variant with Absolute and Partial Relative Stopping Criteria", "authors": [ "Lei Yang", "Kim-Chuan Toh" ], "doi": "10.1287/moor.2023.0328", "url": "https://doi.org/10.1287/moor.2023.0328", "volume": "51", "issue": "1", "pages": "686-714" }, "number": 304 }, { "id": "mor-2025-W3080982993_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2025, "title": "Characterize when constant expected unvisited vertices requires duration at least hitting time", "topic": "Random Walks On Growinggraphs", "publicationDate": "2025-04-04", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 5 (end of Section 1.2, discussion after Theorem 1.3) and Page 23, Section 5 Concluding Remarks.", "sourceQuote": "A natural question remains unsettled whether E[U] = O(1) requires d(i) = Ω(t_(hit)(i)) for any RWoGG (d,(G(i))_(i≥ 1),(P^((i)))_(i≥ 1)).", "problemStatement": "A random walk evolves on a graph that gains one connected vertex at each stage; d(i) is the time spent on the i-vertex graph, h(i) its worst-case hitting time, and U(n) the number of vertices still unvisited by stage n. Known sufficient conditions compare d(i) with h(i). Decide whether bounded expected unvisited count forces d(i)=Ω(h(i)), or construct a growing-graph walk violating this necessity.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "random walks", "dynamic graphs", "hitting time", "cover time", "growing networks", "unvisited vertices" ], "literature": [], "source": { "title": "How Many Vertices Does a Random Walk Miss in a Network with a Moderately Increasing Number of Vertices?", "authors": [ "Shuji Kijima", "Nobutaka Shimizu", "Takeharu Shiraga" ], "doi": "10.1287/moor.2023.0060", "url": "https://doi.org/10.1287/moor.2023.0060", "volume": "51", "issue": "1", "pages": "783-805" }, "number": 305 }, { "id": "siopt-2025-061", "journal": "SIAM Journal on Optimization", "publicationYear": 2025, "title": "Aubin strong regularity general C² cones", "topic": "Variational analysis", "publicationDate": "2025-04-09", "recordType": "reviewed", "status": "resolved", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conclusion; cached preprint lines 1598-1602", "sourceQuote": "The equivalence between the Aubin property and the strong regularity for general optimization problems, particularly those with non-polyhedral C2-cone reducible constraints, remains an open question.", "problemStatement": "The source proves that the Aubin property of the canonically perturbed KKT solution map is equivalent to strong regularity for nonlinear second-order-cone programs. It asks whether the equivalence extends to general optimization systems with nonpolyhedral C2-cone-reducible constraints. A later preprint proves the equivalence for generalized equations over C2-cone-reducible sets, so this problem is marked resolved.", "statusEvidence": "Proves equivalence of the Aubin property and strong regularity for generalized equations over C2-cone-reducible sets.", "verificationNote": "This status is tied to the cited primary-literature result and the precise scope stated above.", "keywords": [ "Aubin property", "strong regularity", "C2-cone reducible", "generalized equations" ], "literature": [ { "title": "The Aubin Property for Generalized Equations over C2-cone Reducible Sets", "url": "https://arxiv.org/abs/2509.14194", "relation": "Proves equivalence of the Aubin property and strong regularity for generalized equations over C2-cone-reducible sets." } ], "source": { "title": "Aubin Property and Strong Regularity Are Equivalent for Nonlinear Second-Order Cone Programming", "authors": [ "Liang Chen", "Ruoning Chen", "Defeng Sun", "Junyuan Zhu" ], "doi": "10.1137/24m1670676", "url": "https://doi.org/10.1137/24m1670676", "preprintUrl": "https://arxiv.org/abs/2406.13798", "volume": "35", "issue": "2", "pages": "712-738" }, "number": 306 }, { "id": "mor-2025-W4409316647_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2025, "title": "Construct Hamiltonian policy-iteration paths for multi-action Markov decision processes", "topic": "Markov Decision Processes", "publicationDate": "2025-04-10", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 6 (Summary and outlook), page 31.", "sourceQuote": "It remains unknown if such a Hamiltonian path through the set of policies exists for any MDP with k ≥ 3 actions.", "problemStatement": "For a finite discounted Markov decision process, the policy-iteration directed acyclic graph has one vertex for each deterministic stationary policy and an edge for each valid policy-improvement step. With n states and k actions per state, it has kⁿ vertices. For every n≥1 and k≥3, determine whether some such process has a Hamiltonian directed path visiting every policy exactly once.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "markov decision processes", "policy iteration", "policy improvement digraph", "hamiltonian path", "worst-case complexity", "multi-action MDPs" ], "literature": [], "source": { "title": "Upper Bounds for All and Max-Gain Policy Iteration Algorithms on Deterministic MDPs", "authors": [ "Ritesh Goenka", "Eashan Gupta", "Sushil Khyalia", "Shivaram Kalyanakrishnan" ], "doi": "10.1287/moor.2023.0317", "url": "https://doi.org/10.1287/moor.2023.0317", "volume": "51", "issue": "1", "pages": "806-828" }, "number": 307 }, { "id": "mor-2025-W4409528654_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2025, "title": "Characterize the exact worst-case approximation ratio for i.i.d. k-selection prophet inequalities", "topic": "Prophet Inequalities", "publicationDate": "2025-04-17", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 6 (Final Remarks), page 28 (arXiv v2 pagination).", "sourceQuote": "Finding an analytical formula for the approximation ratio of (k, n)-SPI remains a problem, but our findings offer a potential avenue toward this goal.", "problemStatement": "In the IID k-selection prophet problem, independent values arrive sequentially and an online rule may accept at most k; γ_n,k is the best worst-case ratio to the expected sum of the top k values. The paper derives bounds from a coupled nonlinear differential-equation system but no closed formula for general k. Give an analytical characterization of γ_n,k and its worst-case or large-n limit.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "prophet inequality", "optimal stopping", "k-selection", "approximation ratio", "worst-case analysis", "differential equations" ], "literature": [], "source": { "title": "Splitting Guarantees for Prophet Inequalities via Nonlinear Systems", "authors": [ "Johannes Brustle", "Sebastian Perez-Salazar", "Victor Verdugo" ], "doi": "10.1287/moor.2024.0413", "url": "https://doi.org/10.1287/moor.2024.0413", "volume": "51", "issue": "2", "pages": "877-904" }, "number": 308 }, { "title": "Acceleration using only currently violated nonlinear constraints", "topic": "Convex & nonlinear optimization", "series": "A", "publicationDate": "2025-04-21", "sourceLocation": "Introduction, p. 2", "sourceQuote": "We conjecture that the same rates can be achieved asymptotically for I = Iₓ.", "problemStatement": "The discrete velocity-projection method accelerates smooth optimization subject to nonlinear inequalities by projecting a momentum-like direction onto linearized feasible velocities. Proven accelerated linear rates use all constraints in the projection, whereas implementations may include only constraints violated at the current iterate, denoted I = Iₓ. Prove that this active-violation variant achieves the same rates asymptotically, or state conditions under which it does.", "keywords": [ "nonlinear constraints", "acceleration", "active set", "velocity projection" ], "source": { "title": "Accelerated first-order optimization under nonlinear constraints", "authors": [ "Michael Muehlebach", "Michael I. Jordan" ], "doi": "10.1007/s10107-025-02224-1", "url": "https://doi.org/10.1007/s10107-025-02224-1" }, "id": "mp-2025-031", "number": 309, "recordType": "reviewed", "sourceClaim": "explicit_open_problem", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "statusEvidence": "Targeted title, exact-statement, and forward-literature searches located no later paper claiming a complete resolution.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "literature": [], "journal": "Mathematical Programming", "publicationYear": 2025 }, { "id": "focm-2025-multivariate-rational-minimal-telescoper", "journal": "Foundations of Computational Mathematics", "publicationYear": 2025, "title": "Multivariate Rational Minimal Telescoper", "topic": "Symbolic computation", "publicationDate": "2025-04-24", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conclusion", "sourceQuote": "Computing a minimal telescoper for a multivariate rational function remains open.", "problemStatement": "Design a complete algorithm that computes a telescoper of minimal order for a multivariate rational function. The method should certify minimality rather than merely output some annihilating operator. A telescoper is a univariate differential operator whose action on the integrand becomes a sum of total derivatives, enabling elimination of the integration variables.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "creative telescoping", "rational functions", "minimal operator", "algorithms" ], "literature": [], "source": { "title": "Symbolic Summation of Multivariate Rational Functions", "authors": [ "Shaoshi Chen", "Lixin Du", "Hanqian Fang" ], "doi": "10.1007/s10208-025-09710-0", "url": "https://doi.org/10.1007/s10208-025-09710-0", "preprintUrl": "https://arxiv.org/abs/2204.06968", "volume": "26", "issue": "3", "pages": "1635-1697" }, "number": 310 }, { "id": "focm-2025-rational-integrability-more-than-two-variables", "journal": "Foundations of Computational Mathematics", "publicationYear": 2025, "title": "Rational Integrability More Than Two Variables", "topic": "Symbolic computation", "publicationDate": "2025-04-24", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conclusion", "sourceQuote": "In more than two variables, there is no complete algorithm for rational integrability.", "problemStatement": "Develop a terminating complete algorithm deciding rational integrability in more than two variables and constructing an antiderivative decomposition when one exists. Current algorithms cover only restricted cases or two variables.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "rational integration", "multivariate functions", "decision algorithm", "Hermite reduction" ], "literature": [], "source": { "title": "Symbolic Summation of Multivariate Rational Functions", "authors": [ "Shaoshi Chen", "Lixin Du", "Hanqian Fang" ], "doi": "10.1007/s10208-025-09710-0", "url": "https://doi.org/10.1007/s10208-025-09710-0", "preprintUrl": "https://arxiv.org/abs/2204.06968", "volume": "26", "issue": "3", "pages": "1635-1697" }, "number": 311 }, { "id": "focm-2025-summability-remainder-minimal-bounds", "journal": "Foundations of Computational Mathematics", "publicationYear": 2025, "title": "Summability Remainder Minimal Bounds", "topic": "Symbolic computation", "publicationDate": "2025-04-24", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 5, discussion of remainder bounds", "sourceQuote": "Finding minimal degree and arithmetic-size bounds for the summability remainder remains open.", "problemStatement": "Determine sharp minimal degree and coefficient-size bounds for the remainder in rational summability reduction. The bounds should improve the general estimates produced by the paper's constructive algorithm. Summability reduction decomposes a rational term into an exact discrete difference plus a canonical obstruction; the open issue is the smallest possible algebraic complexity of that obstruction.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "rational summability", "degree bounds", "arithmetic complexity", "remainder" ], "literature": [], "source": { "title": "Symbolic Summation of Multivariate Rational Functions", "authors": [ "Shaoshi Chen", "Lixin Du", "Hanqian Fang" ], "doi": "10.1007/s10208-025-09710-0", "url": "https://doi.org/10.1007/s10208-025-09710-0", "preprintUrl": "https://arxiv.org/abs/2204.06968", "volume": "26", "issue": "3", "pages": "1635-1697" }, "number": 312 }, { "id": "mor-2025-W4409800077_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2025, "title": "Global convergence theory for Anderson mixing methods for Euclidean fixed-point problems", "topic": "Anderson Acceleration", "publicationDate": "2025-04-25", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 4 (Regularied RAM and Global Convergence Analysis), page 10; also reiterated in Section 5.1 (Details Concerning Numerical Implementations), page 14.", "sourceQuote": "The global convergence of Anderson mixing remains a problem, even in the Euclidean setting.", "problemStatement": "Anderson mixing accelerates a fixed-point iteration by choosing each new iterate from a linear combination of recent residuals. Local convergence is understood under regularity assumptions, but unrestricted coefficients can destroy global stability. Develop a global convergence theory for Euclidean fixed-point problems, specifying safeguards or assumptions that guarantee convergence and rates.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "Anderson mixing", "Anderson acceleration", "fixed-point iteration", "global convergence", "nonlinear equations" ], "literature": [], "source": { "title": "Riemannian Anderson Mixing Methods for Minimizing C² Functions on Riemannian Manifolds", "authors": [ "Zanyu Li", "Chenglong Bao" ], "doi": "10.1287/moor.2023.0284", "url": "https://doi.org/10.1287/moor.2023.0284", "volume": "51", "issue": "2", "pages": "905-937" }, "number": 313 }, { "id": "siopt-2025-047", "journal": "SIAM Journal on Optimization", "publicationYear": 2025, "title": "Signomial maximal s free scaling", "topic": "Global optimization", "publicationDate": "2025-04-28", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 7.1; extracted preprint lines 1204-1210", "sourceQuote": "Which gamma yields DCC constraints that result in maximal S-free sets", "problemStatement": "A global power transformation with scaling vector gamma converts signomial constraints into difference-of-convex constraints used to generate intersection cuts. Cut strength depends on whether the resulting convex set is maximal S-free. Characterize scalings gamma that yield maximal S-free sets and determine whether, as conjectured, no such gamma exists for general mixed signomial systems.", "statusEvidence": "No forward-citing primary paper or exact-query result explicitly claiming a proof or counterexample was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "signomial programming", "intersection cuts", "S-free sets", "power transformation" ], "literature": [], "source": { "title": "Cutting Planes for Signomial Programming", "authors": [ "Liding Xu", "Claudia D’Ambrosio", "Leo Liberti", "Sonia Haddad-Vanier" ], "doi": "10.1137/23m1599537", "url": "https://doi.org/10.1137/23m1599537", "preprintUrl": "https://arxiv.org/abs/2212.02857", "volume": "35", "issue": "2", "pages": "899-926" }, "number": 314 }, { "id": "siopt-2025-051", "journal": "SIAM Journal on Optimization", "publicationYear": 2025, "title": "Lipschitz stability derivative structure", "topic": "Sensitivity analysis", "publicationDate": "2025-04-28", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conclusion; extracted preprint lines 1602-1606", "sourceQuote": "whether S is directional differentiable or possesses the conservative Jacobian is an open question", "problemStatement": "The paper characterizes Lipschitz stability of solution mappings S for convex composite problems with group-Lasso and nuclear-norm regularizers. Lipschitz continuity alone does not provide a usable derivative for sensitivity algorithms. Determine whether S is directionally differentiable or admits a conservative Jacobian in these models, and characterize that derivative if it exists.", "statusEvidence": "No forward-citing primary paper or exact-query result explicitly claiming a proof or counterexample was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "Lipschitz stability", "directional differentiability", "conservative Jacobian", "group Lasso" ], "literature": [], "source": { "title": "Geometric Characterizations of Lipschitz Stability for Convex Optimization Problems", "authors": [ "Tran T. A. Nghia" ], "doi": "10.1137/24m1637532", "url": "https://doi.org/10.1137/24m1637532", "preprintUrl": "https://arxiv.org/abs/2402.05215", "volume": "35", "issue": "2", "pages": "927-958" }, "number": 315 }, { "id": "siopt-2025-059", "journal": "SIAM Journal on Optimization", "publicationYear": 2025, "title": "Time varying distributed optimal complexity", "topic": "Distributed optimization", "publicationDate": "2025-04-28", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conclusion; extracted preprint lines 1242-1249", "sourceQuote": "optimal communication costs and convergence rates for distributed algorithms with potentially disconnected time-varying graphs have yet to be established.", "problemStatement": "The paper constructs graph sequences with finite-time consensus and embeds them in gradient tracking, allowing individual communication graphs to be disconnected. It proves convergence for specific constructions but does not establish information-theoretic optima. Determine the best achievable communication complexity and optimization rate over potentially disconnected time-varying networks.", "statusEvidence": "No forward-citing primary paper or exact-query result explicitly claiming a proof or counterexample was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "finite-time consensus", "time-varying graph", "communication complexity", "gradient tracking" ], "literature": [], "source": { "title": "On Graphs with Finite-Time Consensus and Their Use in Gradient Tracking", "authors": [ "Edward Duc Hien Nguyen", "Xin Jiang", "Bicheng Ying", "César A. Uribe" ], "doi": "10.1137/24m1661455", "url": "https://doi.org/10.1137/24m1661455", "preprintUrl": "https://arxiv.org/abs/2311.01317", "volume": "35", "issue": "2", "pages": "872-898" }, "number": 316 }, { "id": "mor-2025-W4409922425_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2025, "title": "Analytically prove equality of asymptotic and worst-case IID k-unit prophet ratios", "topic": "Prophet Inequalities", "publicationDate": "2025-04-29", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 6, Introduction, item 2 under “1.2. New Results”.", "sourceQuote": "our numerical observation that lim_(n→∞) gamma_(k,n) = inf_(n≥ 1) gamma_(k,n), which would resolve the open question of k-unit IID prophet inequality.", "problemStatement": "In the IID k-unit prophet inequality, an online rule accepts at most k values and γ_k,n is its optimal worst-case ratio to the offline top-k benchmark. Numerical evidence suggests that the asymptotic ratio as n grows equals the infimum over all finite horizons. Prove that equality analytically for every k > 1, or provide a counterexample.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "iid prophet inequality", "multi-unit selection", "k-uniform matroid", "tight guarantee", "asymptotic horizon", "worst-case horizon" ], "literature": [], "source": { "title": "Tightness Without Counterexamples: A New Approach and New Results for Prophet Inequalities", "authors": [ "Jiashuo Jiang", "Will Ma", "Jiawei Zhang" ], "doi": "10.1287/moor.2023.0221", "url": "https://doi.org/10.1287/moor.2023.0221", "volume": "51", "issue": "2", "pages": "956-987" }, "number": 317 }, { "id": "mor-2025-W4410154284_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2025, "title": "Define and justify competitive fairness under lower congestion bounds", "topic": "Congested Assignment", "publicationDate": "2025-05-07", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 7 (Concluding comments), page 29 (paper pagination shown as 28 in the provided text dump), under the heading 'two open questions', item 1", "sourceQuote": "how to account for the situation where a lower bound s_a forces the agents to populate a post a that they unanimously loath?", "problemStatement": "Agents choose posts, and congestion determines their utilities; lower bounds may require a minimum number of agents at designated posts. Without lower bounds, a competitive assignment lets no agent improve by moving unilaterally to another post at its resulting congestion. Formulate an appropriate competitive-fairness notion with lower congestion bounds and prove its existence and incentive properties.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "congested assignment", "fairness", "envy freeness", "lower bounds", "competitive equilibrium", "congestion constraints" ], "literature": [], "source": { "title": "Fair Congested Assignment", "authors": [ "Anna Bogomolnaia", "Hervé Moulin" ], "doi": "10.1287/moor.2024.0581", "url": "https://doi.org/10.1287/moor.2024.0581", "volume": "51", "issue": "2", "pages": "1061-1079" }, "number": 318 }, { "id": "mor-2025-W4410155359_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2025, "title": "Krivine-Stengle Positivstellensatz analog for universal-quantifier-defined polynomial positivity sets", "topic": "Positivstellensatz Certificates", "publicationDate": "2025-05-07", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 7 (Conclusions and Discussions), page 26.", "sourceQuote": "Is there an analog of the Krivine-Stengle Positivstellensatz for such sets K?", "problemStatement": "The feasible set K consists of points x for which polynomial inequalities g_j(x,y) ≥ 0 hold for every y in a closed set Q; K need not be semialgebraic. Classical Krivine–Stengle Positivstellensatz results certify polynomial nonnegativity and infeasibility for quantifier-free semialgebraic sets. Develop an analogous algebraic certificate for nonnegativity on K, including a certificate for K = ∅, without imposing an Archimedean compactness assumption.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "positivstellensatz", "krivine-stengle", "universal quantifiers", "preorderings", "semi-infinite constraints", "sum of squares" ], "literature": [], "source": { "title": "Positivstellensätze and Moment Problems with Universal Quantifiers", "authors": [ "Xiaomeng Hu", "Igor Klep", "Jiawang Nie" ], "doi": "10.1287/moor.2024.0402", "url": "https://doi.org/10.1287/moor.2024.0402", "volume": "51", "issue": "2", "pages": "1037-1060" }, "number": 319 }, { "id": "mor-2025-W4410233669_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2025, "title": "Convergence of exact Nash equilibria to mean field equilibria in ergodic Markovian games", "topic": "Mean Field Games", "publicationDate": "2025-05-09", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 4.1, Remark 4.6 (page 14).", "sourceQuote": "This paper does not address the convergence of exact Nash equilibria to mean field equilibria.", "problemStatement": "Consider a symmetric continuous-time n-player stochastic game on a finite state space [d]=1,…,d. Establish a rigorous convergence result for exact Nash equilibria of the finite-state, infinite-horizon ergodic n-player game with Markovian (closed-loop) strategies to the corresponding mean field equilibrium as n→∞.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "mean field games", "ergodic control", "Markovian Nash equilibrium", "master equation", "finite-state Markov chains", "mean field limit" ], "literature": [], "source": { "title": "Asymptotic Nash Equilibria of Finite-State Ergodic Markovian Mean Field Games", "authors": [ "Asaf Cohen", "Ethan Zell" ], "doi": "10.1287/moor.2024.0478", "url": "https://doi.org/10.1287/moor.2024.0478", "volume": "51", "issue": "2", "pages": "1139-1173" }, "number": 320 }, { "id": "mor-2025-W4399151428_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2025, "title": "Establish sequential convergence of Lloyd’s algorithms without globally subanalytic densities", "topic": "Optimal Quantization", "publicationDate": "2025-05-09", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 3, Introduction (paragraph ending with 'remains open')", "sourceQuote": "Beyond existing results, a general sequential convergence guaranty for Lloyd’s methods in the semi-discrete setting remains open.", "problemStatement": "For a compactly supported, absolutely continuous measure μ on ℝᵈ, standard Lloyd iteration replaces each center by the μ-barycenter of its Voronoi cell; uniform Lloyd iteration instead uses equal-mass Laguerre cells. Global subanalyticity of the density supplies a Kurdyka–Łojasiewicz inequality and whole-sequence convergence. Find broader assumptions on μ ensuring that every iterate sequence converges to a single limit for either map.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "lloyds algorithm", "quantization", "semi-discrete optimal transport", "sequential convergence", "kurdyka-lojasiewicz", "o-minimal structures" ], "literature": [], "source": { "title": "On the Sequential Convergence of Lloyd’s Algorithms", "authors": [ "Leo Portales", "Elsa Cazelles", "Edouard Pauwels" ], "doi": "10.1287/moor.2024.0550", "url": "https://doi.org/10.1287/moor.2024.0550", "preprintUrl": "https://arxiv.org/pdf/2405.20744", "volume": "51", "issue": "2", "pages": "1120-1138" }, "number": 321 }, { "title": "A 3/2-approximation for Prize-Collecting TSP", "topic": "Combinatorial optimization", "series": "B", "publicationDate": "2025-05-12", "sourceLocation": "Introduction, p. 4", "sourceQuote": "It remains open whether there is an efficient algorithm for PCTSP that matches … the 3/2-approximation for TSP.", "problemStatement": "Metric Prize-Collecting TSP lets a tour omit vertices by paying penalties, combining tour construction with a selection decision. Its best known approximation ratio remains above the 3/2 guarantee available for ordinary metric TSP. Find a polynomial-time 3/2-approximation for Prize-Collecting TSP, or give an approximation-preserving reduction that transfers a 3/2 TSP algorithm.", "statusEvidence": "No 3/2-approximation was located. An August 2025 paper still identifies the source paper’s ratio, slightly below 1.6, as the best known guarantee.", "literature": [ { "title": "Prize-Collecting TSP and Related Problems", "url": "https://doi.org/10.4230/LIPIcs.MFCS.2025.7", "relation": "Later 2025 status context" } ], "keywords": [ "prize-collecting TSP", "approximation", "LP rounding", "metric routing" ], "source": { "title": "A better-than-1.6-approximation for prize-collecting TSP", "authors": [ "Jannis Blauth", "Nathan Klein", "Martin Nägele" ], "doi": "10.1007/s10107-025-02221-4", "url": "https://doi.org/10.1007/s10107-025-02221-4" }, "id": "mp-2025-029", "number": 322, "recordType": "reviewed", "sourceClaim": "explicit_open_problem", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "journal": "Mathematical Programming", "publicationYear": 2025 }, { "id": "mor-2025-W4410395976_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2025, "title": "Prove polynomial-time convergence of Scarf’s algorithm for any ordinal matrix", "topic": "Stable Matching", "publicationDate": "2025-05-15", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 8 (Conclusions and Future Work), page 35.", "sourceQuote": "Second, it would be interesting to understand whether Scarf’s algorithm converges in polynomial time on the bipartite matching polytope for any ordinal matrix C.", "problemStatement": "Scarf's algorithm alternates cardinal pivots in a bipartite matching polytope with ordinal pivots determined by an integer preference matrix, terminating when the two bases coincide at a dominating basis. Polynomial convergence is known for a particular consistent preference matrix, but not for an arbitrary one. Determine whether every arbitrary matrix admits a polynomial pivot sequence under a suitable degeneracy rule, or exhibit a superpolynomial instance.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "Scarf's algorithm", "pivoting algorithms", "bipartite matching polytope", "degeneracy", "polynomial-time convergence", "ordinal matrices" ], "literature": [], "source": { "title": "Scarf’s Algorithm and Stable Marriages", "authors": [ "Yuri Faenza", "Chengyue He", "Jay Sethuraman" ], "doi": "10.1287/moor.2023.0055", "url": "https://doi.org/10.1287/moor.2023.0055", "volume": "51", "issue": "2", "pages": "1174-1204" }, "number": 323 }, { "title": "Matrix-LASSO landscape at intermediate ranks", "topic": "Matrix & semidefinite optimization", "series": "A", "publicationDate": "2025-05-24", "sourceLocation": "Section 4, rank-constrained guarantees", "sourceQuote": "What the situation is for intermediate values of r is still an open question.", "problemStatement": "Matrix LASSO minimizes a least-squares loss plus a nuclear-norm penalty and is often solved through a factorization whose rank is fixed in advance. Restricted-isometry assumptions give benign landscapes at sufficiently large factor ranks, while full rank removes the nonconvex obstruction trivially. Characterize the critical points and global-optimality guarantees at intermediate ranks between the proved RIP regime and the full-rank regime.", "keywords": [ "matrix LASSO", "rank constraint", "restricted isometry", "overparameterization" ], "source": { "title": "Low solution rank of the matrix LASSO under RIP with consequences for rank-constrained algorithms", "authors": [ "Andrew D. McRae" ], "doi": "10.1007/s10107-025-02236-x", "url": "https://doi.org/10.1007/s10107-025-02236-x" }, "id": "mp-2025-007", "number": 324, "recordType": "reviewed", "sourceClaim": "explicit_open_problem", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "statusEvidence": "Targeted title, exact-statement, and forward-literature searches located no later paper claiming a complete resolution.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "literature": [], "journal": "Mathematical Programming", "publicationYear": 2025 }, { "id": "focm-2025-cayley-octad-compatible-block-decomposition", "journal": "Foundations of Computational Mathematics", "publicationYear": 2025, "title": "Cayley Octad Compatible Block Decomposition", "topic": "Arithmetic geometry", "publicationDate": "2025-06-02", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conjecture 5.4", "sourceQuote": "We conjecture that every Cayley octad valuation datum admits a compatible block decomposition.", "problemStatement": "Prove existence of the compatible building-block decomposition used to associate an octad picture to arbitrary normalized Cayley-octad valuation data. The paper proves uniqueness conditional on existence. A Cayley octad is the eight-point base locus of a net of quadrics attached to a plane quartic, and compatible blocks encode clusters of valuation degenerations used to predict stable reduction.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "Cayley octads", "valuation data", "block decomposition", "plane quartics" ], "literature": [], "source": { "title": "Reduction of Plane Quartics and Cayley Octads", "authors": [ "Raymond van Bommel", "Jordan Docking", "Vladimir Dokchitser", "Reynald Lercier", "Elisa Lorenzo García" ], "doi": "10.1007/s10208-025-09704-y", "url": "https://doi.org/10.1007/s10208-025-09704-y", "volume": "26", "issue": "3", "pages": "1425-1496" }, "number": 325 }, { "id": "focm-2025-octad-picture-sp6-equivariance", "journal": "Foundations of Computational Mathematics", "publicationYear": 2025, "title": "Octad Picture Sp(6,2) Equivariance", "topic": "Arithmetic geometry", "publicationDate": "2025-06-02", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conjecture 6.13", "sourceQuote": "We conjecture that the octad-picture map is Sp(6,2)-equivariant.", "problemStatement": "Prove equivariance of the octad-picture construction under the natural Sp(6,2) action relating the 36 Cayley octads of a plane quartic. This would establish independence of the predicted stable type beyond single building-block degenerations.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "Cayley octads", "Sp(6,2)", "equivariance", "stable reduction" ], "literature": [], "source": { "title": "Reduction of Plane Quartics and Cayley Octads", "authors": [ "Raymond van Bommel", "Jordan Docking", "Vladimir Dokchitser", "Reynald Lercier", "Elisa Lorenzo García" ], "doi": "10.1007/s10208-025-09704-y", "url": "https://doi.org/10.1007/s10208-025-09704-y", "volume": "26", "issue": "3", "pages": "1425-1496" }, "number": 326 }, { "id": "focm-2025-octad-picture-stable-reduction", "journal": "Foundations of Computational Mathematics", "publicationYear": 2025, "title": "Octad Picture Stable Reduction", "topic": "Arithmetic geometry", "publicationDate": "2025-06-02", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conjecture 5.12", "sourceQuote": "The stable reduction type, including hyperellipticity, is determined by the octad picture of any Cayley octad.", "problemStatement": "Prove that the unlabeled octad picture of any Cayley octad determines the stable-reduction dual graph of a smooth plane quartic and whether the special fiber is hyperelliptic. The claim is over nonarchimedean local fields of residue characteristic not equal to two.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "plane quartics", "stable reduction", "Cayley octads", "hyperellipticity" ], "literature": [], "source": { "title": "Reduction of Plane Quartics and Cayley Octads", "authors": [ "Raymond van Bommel", "Jordan Docking", "Vladimir Dokchitser", "Reynald Lercier", "Elisa Lorenzo García" ], "doi": "10.1007/s10208-025-09704-y", "url": "https://doi.org/10.1007/s10208-025-09704-y", "volume": "26", "issue": "3", "pages": "1425-1496" }, "number": 327 }, { "title": "Strong regularity for general C²-cone-reducible constraints", "topic": "Convex & nonlinear optimization", "series": "A", "publicationDate": "2025-06-02", "sourceLocation": "Remark 4.4, p. 30", "sourceQuote": "It is still unknown if the strong regularity of the KKT system … is equivalent to the constraint nondegeneracy.", "problemStatement": "For nonlinear programs over a C²-cone-reducible set, strong regularity of the KKT generalized equation gives local uniqueness and Lipschitz stability of primal–dual solutions. In nonlinear semidefinite programming it can be characterized by constraint nondegeneracy together with an appropriate second-order sufficient condition. Determine whether the same equivalence holds for every C²-cone-reducible constraint set, or identify the additional geometry required.", "keywords": [ "strong regularity", "KKT system", "cone reducibility", "Aubin property" ], "source": { "title": "Characterizations of the Aubin property of the solution mapping for nonlinear semidefinite programming", "authors": [ "Liang Chen", "Ruoning Chen", "Defeng Sun", "Liping Zhang" ], "doi": "10.1007/s10107-025-02231-2", "url": "https://doi.org/10.1007/s10107-025-02231-2", "preprintUrl": "https://arxiv.org/abs/2408.08232" }, "id": "mp-2025-032", "number": 328, "recordType": "reviewed", "sourceClaim": "explicit_open_problem", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "statusEvidence": "Targeted title, exact-statement, and forward-literature searches located no later paper claiming a complete resolution.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "literature": [], "journal": "Mathematical Programming", "publicationYear": 2025 }, { "id": "mor-2025-W4411023035_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2025, "title": "Achieve asymptotic three-quarters approximation for impartial selection in single-nomination graphs", "topic": "Impartial Selection Mechanisms", "publicationDate": "2025-06-04", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 14, Section 5 (end of the section introduction, immediately after stating Theorem 5.1 and Corollary 5.2).", "sourceQuote": "the existence of an impartial mechanism that is close to 3/4-optimal when n is large. Whether such a mechanism exists is an intriguing open question", "problemStatement": "In a single-nomination graph every vertex nominates exactly one other vertex, and a randomized mechanism selects one vertex. Impartiality means a vertex cannot change its own selection probability by changing its outgoing nomination; performance is selected indegree divided by maximum indegree. Determine whether impartial mechanisms can achieve an asymptotic worst-case ratio of 3/4, or prove a smaller upper bound.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "impartial selection", "single-nomination model", "randomized mechanisms", "approximation guarantee", "asymptotic optimality", "directed graphs" ], "literature": [], "source": { "title": "Improved Bounds for Single-Nomination Impartial Selection", "authors": [ "Javier Cembrano", "Felix Fischer", "Max Klimm" ], "doi": "10.1287/moor.2024.0431", "url": "https://doi.org/10.1287/moor.2024.0431", "volume": "51", "issue": "2", "pages": "1486-1513" }, "number": 329 }, { "title": "The strongest practical encoding of a disjunctive program", "topic": "Mixed-integer & global optimization", "series": "B", "publicationDate": "2025-06-09", "sourceLocation": "Introduction", "sourceQuote": "The best encoding of a disjunctive program … as an MIP is an open question.", "problemStatement": "A disjunctive constraint requires at least one member of a family of convex systems to hold and is commonly encoded as a mixed-integer formulation. P-split formulations interpolate between compact big-M models and strong but larger convex-hull formulations by partitioning the constraint functions. Determine how to choose the partition and encoding so that relaxation strength, formulation size, and computational performance are jointly optimized.", "keywords": [ "P-split", "disjunctive programming", "mixed-integer programming", "extended formulation" ], "source": { "title": "P-split formulations: a class of intermediate formulations between big-M and convex hull for disjunctive constraints", "authors": [ "Jan Kronqvist", "Ruth Misener", "Calvin Tsay" ], "doi": "10.1007/s10107-025-02232-1", "url": "https://doi.org/10.1007/s10107-025-02232-1", "preprintUrl": "https://arxiv.org/abs/2202.05198" }, "id": "mp-2025-008", "number": 330, "recordType": "reviewed", "sourceClaim": "explicit_open_problem", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "statusEvidence": "Targeted title, exact-statement, and forward-literature searches located no later paper claiming a complete resolution.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "literature": [], "journal": "Mathematical Programming", "publicationYear": 2025 }, { "id": "mor-2025-W4411234993_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2025, "title": "Achieve tight oracle complexity for composite nonconvex optimization with affine constraints", "topic": "First Order Convex Optimization", "publicationDate": "2025-06-12", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 4 (Conclusion and Open Questions), page 25.", "sourceQuote": "First, for Algorithm Class 1 on solving Problem Class 1, can the lower bound O(kappa([Ā;A])L_fΔ F_0ε^(-2)) be achieved, i.e., is it tight?", "problemStatement": "Minimize f₀(x)+g(x) subject to Ax+b=0, where f₀ is smooth and possibly nonconvex and g is convex and proximable. A direct oracle returns ∇f₀, products by A and Aᵀ, and the proximal map of g; its lower bound is Ω(κLΔε⁻²), where κ is the spectral condition number of A stacked with the linear map inside g and Δ is the initial objective gap. Design a direct method matching this bound up to logarithmic factors, since the known splitting upper bound changes the oracle model.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "first-order methods", "oracle complexity", "affine constraints", "composite nonconvex optimization", "proximal operator", "lower complexity bounds" ], "literature": [], "source": { "title": "Lower Complexity Bounds of First-Order Methods for Affinely Constrained Composite Nonconvex Problems", "authors": [ "Wei Liu", "Qihang Lin", "Yangyang Xu" ], "doi": "10.1287/moor.2023.0377", "url": "https://doi.org/10.1287/moor.2023.0377", "volume": "51", "issue": "2", "pages": "1659-1682" }, "number": 331 }, { "id": "mor-2025-W4411449955_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2025, "title": "Convergence guarantees for numerical algorithms for time-inconsistent equilibrium transport problems", "topic": "Bicausal Optimal Transport", "publicationDate": "2025-06-19", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 2.1.1, Remark 2.1 (page 7 of 56 in the provided parsed text)", "sourceQuote": "Convergence of numerical algorithms for the transport problem with time-inconsistent costs remains a problem.", "problemStatement": "Bicausal transport plans couple two stochastic paths without using future information in either direction. With a time-inconsistent cost, an equilibrium plan is defined by optimality against every one-step deviation while later kernels remain fixed. Design a numerical method for these equilibrium transport problems and prove convergence of its iterates under verifiable assumptions.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "bicausal optimal transport", "time inconsistency", "equilibrium transport", "numerical algorithms", "convergence analysis", "f-divergence regularization" ], "literature": [], "source": { "title": "Equilibrium Transport with Time-Inconsistent Costs", "authors": [ "Erhan Bayraktar", "Bingyan Han" ], "doi": "10.1287/moor.2023.0323", "url": "https://doi.org/10.1287/moor.2023.0323" }, "number": 332 }, { "title": "Reinhardt’s maximum-perimeter problem for powers of two", "topic": "Mixed-integer & global optimization", "series": "B", "publicationDate": "2025-06-21", "sourceLocation": "Introduction", "sourceQuote": "The perimeter problem is still open for the case when n > 8 is a power of two.", "problemStatement": "Reinhardt’s perimeter problem asks for the largest perimeter of a convex n-gon whose Euclidean diameter is at most one. The optimum is understood for many values of n, but the cases in which n > 8 is a power of two remain unresolved. Determine the optimal perimeter in those cases and characterize every polygon attaining it.", "keywords": [ "small polygons", "maximum perimeter", "Reinhardt problem", "global optimization" ], "source": { "title": "A zonogon approach for computing small convex polygons of maximum perimeter", "authors": [ "Bernd Mulansky", "Andreas Potschka" ], "doi": "10.1007/s10107-025-02244-x", "url": "https://doi.org/10.1007/s10107-025-02244-x" }, "id": "mp-2025-009", "number": 333, "recordType": "reviewed", "sourceClaim": "explicit_open_problem", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "statusEvidence": "Targeted title, exact-statement, and forward-literature searches located no later paper claiming a complete resolution.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "literature": [], "journal": "Mathematical Programming", "publicationYear": 2025 }, { "id": "mor-2025-W4411674410_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2025, "title": "Label-setting criteria for stochastic shortest path games with two players", "topic": "Stochastic Shortest Path Games", "publicationDate": "2025-06-26", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 6 (Conclusions), page 33.", "sourceQuote": "deterministic games on graphs have been developed in the last ten years [7, 11], but the case of general stochastic games on graphs remains open.", "problemStatement": "A stochastic shortest-path game has an absorbing target and two players choosing actions that affect transition costs. Label-setting algorithms finalize states in increasing value order when a monotone-causality condition prevents later states from lowering an earlier label. Characterize checkable conditions under which such a label-setting method is valid for two-player stochastic shortest-path games.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "stochastic shortest path", "zero-sum games", "label-setting algorithms", "dijkstra-like methods", "monotone causality", "dynamic programming" ], "literature": [], "source": { "title": "Monotone Causality in Opportunistically Stochastic Shortest Path Problems", "authors": [ "Mallory E. Gaspard", "Alexander Vladimirsky" ], "doi": "10.1287/moor.2023.0362", "url": "https://doi.org/10.1287/moor.2023.0362" }, "number": 334 }, { "id": "mor-2025-W4411801764_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2025, "title": "Well-posedness theory for coupled multidimensional quadratic BSDEs in portfolio equilibria", "topic": "Quadratic Bsdes", "publicationDate": "2025-06-30", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 15, Section 6 (Remark 6.2(a))", "sourceQuote": "Two-dimensional QBSDEs are technically much more challenging than the one-dimensional ones, and unless other assumptions are imposed, the well-posedness remains open", "problemStatement": "The portfolio-equilibrium model leads to a coupled multidimensional backward stochastic differential equation whose generator has quadratic growth in the control variable. Standard scalar and diagonally structured quadratic-BSDE results do not cover this fully coupled system. Give existence, uniqueness, and stability conditions strong enough to establish the proposed equilibrium.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "quadratic BSDE", "multidimensional BSDE", "well-posedness", "BMO martingales", "coupled systems", "portfolio equilibrium" ], "literature": [], "source": { "title": "Dynamic Portfolio Selection for Nonlinear Law-Dependent Preferences", "authors": [ "Zongxia Liang", "Jianming Xia", "Fengyi Yuan" ], "doi": "10.1287/moor.2023.0345", "url": "https://doi.org/10.1287/moor.2023.0345" }, "number": 335 }, { "id": "mor-2025-W4411787576_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2025, "title": "Prove global convergence to the physical invariant state for hydrodynamic equations", "topic": "Mean Field Queueing", "publicationDate": "2025-06-30", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 1.1 (Introduction), page 3.", "sourceQuote": "show that the solution to the hydrodynamic equations converges to the invariant state characterized here for a large class of initial conditions.", "problemStatement": "The SQ(d) load-balancing model routes each arrival to the shortest of d sampled queues, and its many-server limit is a measure-valued hydrodynamic system tracking queue lengths and service ages. In regimes where the paper identifies a unique physical invariant state, stationarity does not by itself imply attraction. Prove that solutions from a suitably broad class of initial states converge globally to that invariant state.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "load balancing", "hydrodynamic limit", "invariant state", "mean-field limit", "measure-valued equations", "convergence to equilibrium" ], "literature": [], "source": { "title": "Invariant States of Hydrodynamic Limits of Randomized Load-Balancing Networks", "authors": [ "Pooja Agarwal", "Kavita Ramanan" ], "doi": "10.1287/moor.2020.0282", "url": "https://doi.org/10.1287/moor.2020.0282" }, "number": 336 }, { "title": "Sinkhorn rates under a Poincaré inequality", "topic": "Optimal transport", "series": "A", "publicationDate": "2025-07-01", "sourceLocation": "Section 3, before Remark 3.2", "sourceQuote": "Assuming that μ satisfies a Poincaré inequality should be a sufficient condition for our results to hold.", "problemStatement": "Continuous Sinkhorn iteration alternately rescales a reference kernel so that its coupling matches two prescribed probability marginals. The source paper obtains sharper exponential convergence rates under functional-inequality assumptions stronger than a basic Poincaré inequality. Prove comparable rates when the marginal μ is assumed only to satisfy an appropriate Poincaré inequality, or identify an additional condition that is necessary.", "keywords": [ "Sinkhorn algorithm", "entropic optimal transport", "Poincaré inequality", "exponential convergence" ], "source": { "title": "Sharper exponential convergence rates for Sinkhorn’s algorithm in continuous settings", "authors": [ "Lénaïc Chizat", "Alex Delalande", "Tomas Vaškevičius" ], "doi": "10.1007/s10107-025-02242-z", "url": "https://doi.org/10.1007/s10107-025-02242-z", "preprintUrl": "https://arxiv.org/abs/2407.01202" }, "id": "mp-2025-010", "number": 337, "recordType": "reviewed", "sourceClaim": "explicit_open_problem", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "statusEvidence": "Targeted title, exact-statement, and forward-literature searches located no later paper claiming a complete resolution.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "literature": [], "journal": "Mathematical Programming", "publicationYear": 2025 }, { "id": "mor-2025-W4411879499_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2025, "title": "Degree bound for Lasserre relaxation under regular sequence assumptions without infinity condition", "topic": "Polynomial Optimization", "publicationDate": "2025-07-01", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 8 (Conclusion and perspectives), page 21 (in the provided text, the paragraph beginning 'However, Assumption 1 is stronger ...')", "sourceQuote": "It would be interesting to investigate the degree bound under the regular sequence condition only.", "problemStatement": "Lasserre's sum-of-squares hierarchy is exact once f-f* has a bounded-degree representation in the equality ideal plus the inequality quadratic module. The paper obtains an explicit degree bound under a no-solutions-at-infinity condition, which implies an H-basis property. Derive an effective degree bound assuming only that the homogenized equality polynomials form a regular sequence, even when solutions at infinity exist.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "Lasserre hierarchy", "sum of squares", "effective degree bounds", "regular sequence", "H-basis", "finite variety", "moment-SOS relaxations" ], "literature": [], "source": { "title": "Exactness and Effective Degree Bound of Lasserre’s Relaxation for Polynomial Optimization over Finite Variety", "authors": [ "Zheng Hua", "Zheng Qu" ], "doi": "10.1287/moor.2024.0483", "url": "https://doi.org/10.1287/moor.2024.0483" }, "number": 338 }, { "title": "Convergence of PDQP’s practical adaptive rules", "topic": "Convex & nonlinear optimization", "series": "A", "publicationDate": "2025-07-02", "sourceLocation": "Section 6, p. 27", "sourceQuote": "Currently, we do not have a proof of convergence for the adaptive step size rule.", "problemStatement": "PDQP is a first-order primal–dual method for large-scale convex quadratic programs whose practical implementation adapts its step size and restarts based on observed progress. The paper proves optimal complexity for a controlled theoretical variant, but not for the adaptive rules used by the solver. Establish global convergence, and ideally a rate, for PDQP with both its practical adaptive step-size and adaptive-restart mechanisms.", "statusEvidence": "A 2026 analysis proves adaptive restart for a different nonlinear-conic primal–dual method under metric subregularity; it does not establish convergence of PDQP’s exact rules.", "literature": [ { "title": "Restarted Accelerated Primal-Dual Algorithms with Adaptive Stepsizes for Nonlinear Conic Constrained Convex Optimization", "url": "https://arxiv.org/abs/2605.29291", "relation": "Related 2026 analysis for a different algorithm" } ], "keywords": [ "quadratic programming", "PDQP", "adaptive restart", "step size" ], "source": { "title": "A Practical and Optimal First-Order Method for Large-Scale Convex Quadratic Programming", "authors": [ "Haihao Lu", "Jinwen Yang" ], "doi": "10.1007/s10107-025-02241-0", "url": "https://doi.org/10.1007/s10107-025-02241-0", "preprintUrl": "https://arxiv.org/abs/2311.07710" }, "id": "mp-2025-011", "number": 339, "recordType": "reviewed", "sourceClaim": "explicit_open_problem", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "journal": "Mathematical Programming", "publicationYear": 2025 }, { "id": "mor-2025-W4411957271_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2025, "title": "Prove tangential derivative convergence of NSDP central path as barrier vanishes", "topic": "Nonlinear Semidefinite Programming", "publicationDate": "2025-07-02", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 4 (Concluding remarks and future work), page 19.", "sourceQuote": "We have the following conjecture: lim_(μ→ 0) ẋ(μ) = ξ^(*), where ξ^(*) was defined in Proposition 5.", "problemStatement": "A nonlinear semidefinite program has a smooth primal-dual central path indexed by a barrier parameter μ under strict complementarity and second-order regularity. The paper proves that (x(μ)-x*)/μ converges to a vector ξ* characterized by the linearized barrier KKT system. Determine whether the path derivative dx(μ)/dμ also converges to ξ* as μ decreases to zero.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "nonlinear semidefinite programming", "central path", "primal-dual interior point methods", "log-det barrier", "analytic center", "tangential direction", "strict complementarity" ], "literature": [], "source": { "title": "Analysis of the Primal-Dual Central Path for Nonlinear Semidefinite Optimization Without the Nondegeneracy Condition", "authors": [ "Takayuki Okuno" ], "doi": "10.1287/moor.2022.0298", "url": "https://doi.org/10.1287/moor.2022.0298" }, "number": 340 }, { "id": "mor-2025-W4411959281_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2025, "title": "Characterize necessity of the I-property for isolated calmness of KKT mapping", "topic": "Variational Inequalities", "publicationDate": "2025-07-02", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 7 (Conclusions), page 27", "sourceQuote": "is the I-property also necessary for the isolated calmness of S_(KKT)?", "problemStatement": "In a canonically perturbed Nash equilibrium problem, isolated calmness means that every nearby KKT solution stays within a constant times the perturbation size of a reference KKT pair. The I-property says that the only critical-cone direction y for which every player's Lagrangian Hessian has zero quadratic interaction with all players' directions is y=0; the cone imposes equality for active positive-multiplier constraints and nonpositivity for active zero-multiplier inequalities. Determine whether isolated calmness necessarily implies this I-property.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "nash equilibrium problems", "kkt solution mapping", "isolated calmness", "variational analysis", "second-order conditions", "stability" ], "literature": [], "source": { "title": "Stability for Nash Equilibrium Problems", "authors": [ "Ruoyu Diao", "Yu-Hong Dai", "Liwei Zhang" ], "doi": "10.1287/moor.2024.0609", "url": "https://doi.org/10.1287/moor.2024.0609" }, "number": 341 }, { "id": "focm-2025-expected-characteristic-polynomial-differential-operator", "journal": "Foundations of Computational Mathematics", "publicationYear": 2025, "title": "Expected Characteristic Polynomial Differential Operator", "topic": "Random matrix theory", "publicationDate": "2025-07-14", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Introduction, discussion of P_k(x;A,B)", "sourceQuote": "it is unknown whether such a differential-operator expression exists for general B", "problemStatement": "For A in R^(n x d), B in R^(n x d_B), and Q_S=I-B_:S(B_:S)^dagger, define P_k(x;A,B)=sum_(|S|=k) det((B_:S)^T B_:S) det(xI-A^T Q_S A). Find an expression for this polynomial using only a univariate differential operator in x for general B. Such formulas are known when B=A through a flip operator and d^k/dx^k, and when B=I through x^(d-n+k)(d^k/dx^k)det(xI-AA^T), but not when B differs from both A and I.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "expected characteristic polynomial", "differential operator", "matrix pencils", "mixed determinants" ], "literature": [], "source": { "title": "Interlacing Polynomial Method for Matrix Approximation via Generalized Column and Row Selection", "authors": [ "Jian-Feng Cai", "Zhiqiang Xu", "Zili Xu" ], "doi": "10.1007/s10208-025-09719-5", "url": "https://doi.org/10.1007/s10208-025-09719-5", "preprintUrl": "https://arxiv.org/abs/2312.01715", "volume": "26", "issue": "3", "pages": "1759-1808" }, "number": 342 }, { "id": "focm-2025-higher-order-commuting-feec-elements", "journal": "Foundations of Computational Mathematics", "publicationYear": 2025, "title": "Higher Order Commuting FEEC Elements", "topic": "Finite element exterior calculus", "publicationDate": "2025-07-14", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conclusion", "sourceQuote": "The construction for higher order r > 1 in dimensions n > 2 remains open.", "problemStatement": "For the differential-form Hamiltonian complex, construct local finite-element spaces in dimension n>2 and polynomial order r>1 with neighboring spaces P_(r-1) Lambda^(k-1) and P_(r-1) Lambda^(k+1), and a central k-form space containing P_r Lambda^k plus suitable harmonic polynomial forms. Build a projection that commutes with both the exterior derivative d and its codifferential delta. The paper's two-dimensional construction does not extend directly, and even the three-dimensional higher-order case remains open.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "commuting projections", "higher-order elements", "differential forms", "finite elements" ], "literature": [], "source": { "title": "Multisymplecticity in Finite Element Exterior Calculus", "authors": [ "Ari Stern", "Enrico Zampa" ], "doi": "10.1007/s10208-025-09720-y", "url": "https://doi.org/10.1007/s10208-025-09720-y", "preprintUrl": "https://arxiv.org/abs/2312.03657", "volume": "26", "issue": "3", "pages": "1809-1858" }, "number": 343 }, { "title": "Complexity of sortation on out-trees", "topic": "Combinatorial optimization", "series": "B", "publicationDate": "2025-07-17", "sourceLocation": "Section 6, p. 19", "sourceQuote": "It remains open whether or not out-tree instances are NP-hard.", "problemStatement": "The sortation planning problem selects facilities at which parcel flows are consolidated in a directed logistics network, with min-degree-SPP minimizing the required sortation degree. General instances are NP-hard, whereas several restricted network families admit faster combinatorial algorithms. Decide whether min-degree-SPP remains NP-hard when the underlying directed network is an out-tree, or give a polynomial-time algorithm for that class.", "keywords": [ "sortation", "out-trees", "NP-hardness", "network design" ], "source": { "title": "Fast Combinatorial Algorithms for Efficient Sortation", "authors": [ "Madison Van Dyk", "Kim Klause", "Jochen Koenemann", "Nicole Megow" ], "doi": "10.1007/s10107-025-02239-8", "url": "https://doi.org/10.1007/s10107-025-02239-8", "preprintUrl": "https://arxiv.org/abs/2311.05094" }, "id": "mp-2025-012", "number": 344, "recordType": "reviewed", "sourceClaim": "explicit_open_problem", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "statusEvidence": "Targeted title, exact-statement, and forward-literature searches located no later paper claiming a complete resolution.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "literature": [], "journal": "Mathematical Programming", "publicationYear": 2025 }, { "id": "mor-2025-W4413077228_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2025, "title": "Prove tight worst-case characterization of gradient descent averaging benefit conditions", "topic": "First Order Convex Optimization", "publicationDate": "2025-07-17", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 3.2.1 (Discussion of Conditions (C1) and (C2)), page 12.", "sourceQuote": "numerical results suggest that the converses of Theorem 3.1 and Theorem 3.2 are true, i.e., (C1)/(C2) hold if and only if averaging is not beneficial.", "problemStatement": "Gradient descent on an L-smooth convex function uses fixed normalized steps hᵢ/L from an initial point within distance D of a minimizer, and an averaging scheme reports a convex combination of all iterates through the final iterate. The benchmark last-iterate objective gap is LD²/(4Σhᵢ+2), while the gradient-norm benchmark is LD/(Σhᵢ+1). Whenever a benchmark exactly equals the corresponding worst-case last-iterate value, no nontrivial average improves it. Prove the converses: if either equality fails, some nondegenerate average has a strictly smaller worst-case criterion.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "gradient descent", "iterate averaging", "worst-case analysis", "performance estimation problem", "smooth convex optimization", "tight bounds" ], "literature": [], "source": { "title": "On Averaging and Extrapolation for Gradient Descent", "authors": [ "Alan Luner", "Benjamin Grimmer" ], "doi": "10.1287/moor.2024.0449", "url": "https://doi.org/10.1287/moor.2024.0449" }, "number": 345 }, { "id": "mor-2025-W4412612512_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2025, "title": "Prove or disprove saddle-escaping for subgradient methods in robust matrix sensing", "topic": "Nonsmooth Optimization Dynamics", "publicationDate": "2025-07-23", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 8 (Discussion and Future Directions), page 39 (arXiv PDF numbering)", "sourceQuote": "mere randomness in the initial point may be insufficient for the sub-gradient method to escape saddle points in robust low-rank matrix recovery", "problemStatement": "Robust low-rank matrix recovery minimizes an ℓ₁ residual over a fixed-rank factorization, producing a nonsmooth nonconvex objective. Existing subgradient theory reaches approximate stationarity but does not show that the dynamics avoid strict saddles. Determine whether deterministic subgradient descent escapes the strict saddles associated with true solutions, or whether such saddles can attract iterates.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "nonsmooth optimization", "subgradient method", "strict saddle points", "robust matrix sensing", "low-rank recovery", "saddle escaping" ], "literature": [], "source": { "title": "Can Learning Be Explained by Local Optimality in Robust Low-Rank Matrix Recovery?", "authors": [ "Jianhao Ma", "Salar Fattahi" ], "doi": "10.1287/moor.2023.0371", "url": "https://doi.org/10.1287/moor.2023.0371" }, "number": 346 }, { "id": "mor-2025-W4412630514_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2025, "title": "Establish strong duality for distorted optimal transport beyond affine costs", "topic": "Distorted Optimal Transport", "publicationDate": "2025-07-24", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 9 (Concluding remarks), page 33 (PDF page 33/37), second open-question paragraph.", "sourceQuote": "Similarly, strong duality is only established for convex h and affine cost, and we wonder whether it also holds in more general situations.", "problemStatement": "A distorted expectation reweights outcome probabilities through a distortion function h and induces an optimal-transport problem with prescribed marginals and cost c. The paper proves weak duality generally and strong duality for affine costs. Characterize non-affine costs and distortions for which the full Kantorovich-type strong-duality identity still holds.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "optimal transport", "duality", "distorted expectation", "Choquet integral", "risk measures", "Kantorovich problem" ], "literature": [], "source": { "title": "Distorted Optimal Transport", "authors": [ "Haiyan Liu", "Bin Wang", "Ruodu Wang", "Sheng Chao Zhuang" ], "doi": "10.1287/moor.2024.0591", "url": "https://doi.org/10.1287/moor.2024.0591" }, "number": 347 }, { "id": "focm-2025-realization-space-connected-components", "journal": "Foundations of Computational Mathematics", "publicationYear": 2025, "title": "Realization Space Connected Components", "topic": "Computational geometry", "publicationDate": "2025-08-07", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Remark 26", "sourceQuote": "Determining the number of real connected components of the lambda-compatible realization space remains open.", "problemStatement": "For a graph G=(V,E) and edge-length data lambda, let the planar realization space consist of maps p:V->R^2 satisfying ||p(v)-p(w)||^2=lambda_(vw) on every edge, modulo translations and rotations. Determine the number of connected components of this real semialgebraic space, characterize them combinatorially, and decide when it is connected. The answer cannot be inferred from the paper's complex irreducible-component count because the real count changes on Zariski-dense sets of lambda.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "realization spaces", "connected components", "planar geometry", "semialgebraic sets" ], "literature": [], "source": { "title": "Irreducible Components of Sets of Points in the Plane that Satisfy Distance Conditions", "authors": [ "Niels Lubbes", "Mehdi Makhul", "Josef Schicho", "Audie Warren" ], "doi": "10.1007/s10208-025-09725-7", "url": "https://doi.org/10.1007/s10208-025-09725-7" }, "number": 348 }, { "id": "focm-2025-realization-space-decomposition-higher-dimensions", "journal": "Foundations of Computational Mathematics", "publicationYear": 2025, "title": "Realization Space Decomposition Higher Dimensions", "topic": "Computational geometry", "publicationDate": "2025-08-07", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 6", "sourceQuote": "The generalization of our component decomposition theorem to dimensions n at least 3 remains an open problem.", "problemStatement": "The planar theorem decomposes the complex realization space using maximal tight subgraphs and counts components through the resulting factorization. Extend this theorem to realizations p:V->R^m for m>=3 by identifying the correct maximal-rigid pieces and a valid component formula. A direct substitution of maximal rigid for maximal tight is insufficient because higher-dimensional graphs need not admit an edge-disjoint maximal-rigid decomposition.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "realization spaces", "higher dimensions", "component decomposition", "semialgebraic geometry" ], "literature": [], "source": { "title": "Irreducible Components of Sets of Points in the Plane that Satisfy Distance Conditions", "authors": [ "Niels Lubbes", "Mehdi Makhul", "Josef Schicho", "Audie Warren" ], "doi": "10.1007/s10208-025-09725-7", "url": "https://doi.org/10.1007/s10208-025-09725-7" }, "number": 349 }, { "id": "focm-2025-policy-optimization-average-cost", "journal": "Foundations of Computational Mathematics", "publicationYear": 2025, "title": "Policy Optimization Average Cost", "topic": "Reinforcement learning theory", "publicationDate": "2025-08-11", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Remark 2.9", "sourceQuote": "An interesting question is whether an analogous convergence result holds for the average-cost criterion.", "problemStatement": "Extend the paper's convergence theorem from discounted or entropy-regularized objectives to the average-cost criterion. This requires new control of the optimization landscape and long-run state distribution. Average cost removes geometric discounting and depends on the invariant state distribution, so the contraction and occupancy estimates used for the discounted result no longer apply directly.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "average cost", "policy gradient", "Markov decision processes", "convergence" ], "literature": [], "source": { "title": "A Fisher–Rao Gradient Flow for Entropy-Regularised Markov Decision Processes in Polish Spaces", "authors": [ "Bekzhan Kerimkulov", "James-Michael Leahy", "David Siska", "Lukasz Szpruch", "Yufei Zhang" ], "doi": "10.1007/s10208-025-09729-3", "url": "https://doi.org/10.1007/s10208-025-09729-3" }, "number": 350 }, { "id": "focm-2025-policy-optimization-function-approximation", "journal": "Foundations of Computational Mathematics", "publicationYear": 2025, "title": "Policy Optimization Function Approximation", "topic": "Reinforcement learning theory", "publicationDate": "2025-08-11", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Introduction", "sourceQuote": "A convergence analysis with function approximation for general continuous state and action spaces remains open.", "problemStatement": "Consider the infinite-horizon discounted Markov decision process on general state and action spaces with transition kernel P, bounded cost c, discount gamma<1, and entropy penalty tau times the relative entropy of a policy against a reference policy. The paper proves global exponential convergence of the exact Fisher-Rao policy flow over all stochastic kernels and stability to some inexact gradients. Prove convergence for implementable function-approximated policies or values in continuous state and action spaces, with quantitative control of representation and gradient errors.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "policy optimization", "function approximation", "continuous MDP", "convergence" ], "literature": [], "source": { "title": "A Fisher–Rao Gradient Flow for Entropy-Regularised Markov Decision Processes in Polish Spaces", "authors": [ "Bekzhan Kerimkulov", "James-Michael Leahy", "David Siska", "Lukasz Szpruch", "Yufei Zhang" ], "doi": "10.1007/s10208-025-09729-3", "url": "https://doi.org/10.1007/s10208-025-09729-3" }, "number": 351 }, { "id": "focm-2025-wfr-mixture-flow-convergence-rates", "journal": "Foundations of Computational Mathematics", "publicationYear": 2025, "title": "WFR Mixture Flow Convergence Rates", "topic": "Optimization over probability measures", "publicationDate": "2025-08-12", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conclusion", "sourceQuote": "We leave it as an open question to obtain convergence rates for this flow.", "problemStatement": "Prove quantitative convergence rates for the Wasserstein-Fisher-Rao gradient flow over mixture distributions developed in the paper. The rate should specify dependence on geometry, regularity, and initialization rather than only asymptotic convergence.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "Wasserstein-Fisher-Rao", "mixture models", "gradient flow", "convergence rate" ], "literature": [], "source": { "title": "Algorithms for Mean-Field Variational Inference Via Polyhedral Optimization in the Wasserstein Space", "authors": [ "Yiheng Jiang", "Sinho Chewi", "Aram-Alexandre Pooladian" ], "doi": "10.1007/s10208-025-09721-x", "url": "https://doi.org/10.1007/s10208-025-09721-x", "preprintUrl": "https://arxiv.org/abs/2312.02849", "volume": "26", "issue": "3", "pages": "1859-1910" }, "number": 352 }, { "id": "siopt-2025-044", "journal": "SIAM Journal on Optimization", "publicationYear": 2025, "title": "Jacobi newton step bounded away zero", "topic": "Equilibrium computation", "publicationDate": "2025-08-13", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conclusion; extracted preprint lines 1796-1801", "sourceQuote": "to prove whether or not it is possible to guarantee the boundedness away from zero of tk", "problemStatement": "The Jacobi-type Newton method for Nash equilibrium problems has global descent and convergence when accepted step sizes t_k stay uniformly positive. The paper verifies that property only under additional model conditions. Find general assumptions ensuring inf_k t_k>0, or construct a natural problem on which the line search forces t_k to zero.", "statusEvidence": "No forward-citing primary paper or exact-query result explicitly claiming a proof or counterexample was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "Nash equilibrium", "Newton method", "line search", "step size" ], "literature": [], "source": { "title": "A Jacobi-Type Newton Method for Nash Equilibrium Problems with Descent Guarantees", "authors": [ "L. F. Bueno", "G. Haeser", "O. Kolossoski" ], "doi": "10.1137/23m1575639", "url": "https://doi.org/10.1137/23m1575639", "preprintUrl": "https://arxiv.org/abs/2209.11571", "volume": "35", "issue": "3", "pages": "1761-1791" }, "number": 353 }, { "title": "A stronger pseudopolynomial state space for matroid interdiction", "topic": "Combinatorial optimization", "series": "A", "publicationDate": "2025-08-19", "sourceLocation": "Section 2.3, p. 13", "sourceQuote": "The existence of a pseudopolynomial sized T … with worst-case performance of o(m)OPT is an open question.", "problemStatement": "Matroid-basis interdiction removes elements within a budget so as to maximize the minimum weight of a remaining basis, and a dynamic program can represent partial interdiction outcomes by a finite state space T. Existing pseudopolynomial constructions may have a worst-case upper bound proportional to m times the optimum, where m is the ground-set size. Construct a pseudopolynomial-size state space with worst-case bound o(m)OPT, or prove that no such strengthening is possible.", "keywords": [ "interdiction", "matroid bases", "dynamic programming", "pseudopolynomial" ], "source": { "title": "Interdiction of minimum spanning trees and other matroid bases", "authors": [ "Noah Weninger", "Ricardo Fukasawa" ], "doi": "10.1007/s10107-025-02273-6", "url": "https://doi.org/10.1007/s10107-025-02273-6", "preprintUrl": "https://arxiv.org/abs/2407.14906" }, "id": "mp-2025-013", "number": 354, "recordType": "reviewed", "sourceClaim": "explicit_open_problem", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "statusEvidence": "Targeted title, exact-statement, and forward-literature searches located no later paper claiming a complete resolution.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "literature": [], "journal": "Mathematical Programming", "publicationYear": 2025 }, { "title": "Guarantees for a tractable copositive-cone replacement", "topic": "Mixed-integer & global optimization", "series": "B", "publicationDate": "2025-08-25", "sourceLocation": "Section 5, p. 36", "sourceQuote": "An interesting open question is to provide some theoretical guarantee on such modification.", "problemStatement": "A copositive reformulation can express sensitivity bounds for mixed-binary quadratic programs exactly, but optimizing over the copositive cone is intractable. The proposed computable surrogate replaces that cone by the sum S₊ + Sₚ of the positive-semidefinite and entrywise-nonnegative cones. Quantify the resulting loss through an approximation ratio, additive error, or other rigorous guarantee for the derived sensitivity bounds.", "keywords": [ "mixed-binary quadratic", "copositive optimization", "sensitivity", "tractable relaxation" ], "source": { "title": "Sensitivity analysis for mixed binary quadratic programming", "authors": [ "Diego Cifuentes", "Santanu S. Dey", "Jingye Xu" ], "doi": "10.1007/s10107-025-02265-6", "url": "https://doi.org/10.1007/s10107-025-02265-6", "preprintUrl": "https://arxiv.org/abs/2312.06714" }, "id": "mp-2025-014", "number": 355, "recordType": "reviewed", "sourceClaim": "explicit_open_problem", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "statusEvidence": "Targeted title, exact-statement, and forward-literature searches located no later paper claiming a complete resolution.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "literature": [], "journal": "Mathematical Programming", "publicationYear": 2025 }, { "id": "mor-2025-W4413635748_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2025, "title": "Extend uniform subdifferential convergence rates to max-of-smooth weakly convex models", "topic": "Stochastic Nonsmooth Optimization", "publicationDate": "2025-08-26", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 24, Section 7 (Discussion), future directions list", "sourceQuote": "Can Theorem 5 be extended to other weakly-convex models, such as max-of-smooth functions [56, 57]?", "problemStatement": "Uniform subdifferential convergence asks whether empirical weakly convex objectives have subgradients close to those of their population objective throughout a region. The paper proves high-probability rates for a one-dimensional convex-composite model using a VC-dimension bound. Extend comparable rates to max-of-smooth weakly convex models and identify the correct dimension and sample-size dependence.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "weakly convex functions", "subdifferentials", "uniform convergence", "hausdorff distance", "max of smooth", "sample average approximation", "finite sample rates" ], "literature": [], "source": { "title": "On the Uniform Convergence of Subdifferentials in Stochastic Optimization and Learning", "authors": [ "Feng Ruan" ], "doi": "10.1287/moor.2024.0533", "url": "https://doi.org/10.1287/moor.2024.0533" }, "number": 356 }, { "id": "mor-2025-W4413848299_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2025, "title": "Determine tight PTAS running times for q-norm load balancing with intermediate machines", "topic": "Parallel Machine Scheduling", "publicationDate": "2025-08-29", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 2, Introduction (Contribution Overview); also Page 46, Conclusion (Section H).", "sourceQuote": "Another interesting problem is to show subexponential lower bound or develop FPTAS for P||sum_i C_i^q where m = Θ((1/ε)^θ) for θ ∈ (1/2, 1].", "problemStatement": "The problem schedules jobs on m identical machines to minimize a fixed q-norm of the load vector using a polynomial-time approximation scheme. Nearly tight exponential dependence on 1/ε is known when m is proportional to 1/ε, while polynomial dependence is possible for much larger m. Determine the optimal running time when m scales as (1/ε) to a power θ with 1/2<θ≤1, matching algorithms with ETH-based lower bounds.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "scheduling", "ptas", "load balancing", "ellq norm", "fine-grained complexity", "ETH lower bounds" ], "literature": [], "source": { "title": "Tight Running Times for Minimum ℓq-Norm Load Balancing: Beyond Exponential Dependencies on 1/ϵ", "authors": [ "Lin Chen", "Liangde Tao", "José Verschae" ], "doi": "10.1287/moor.2023.0231", "url": "https://doi.org/10.1287/moor.2023.0231" }, "number": 357 }, { "title": "Uniform high-probability price of adaptivity", "topic": "Stochastic optimization", "series": "A", "publicationDate": "2025-09-03", "sourceLocation": "Section 6, p. 20", "sourceQuote": "Characterizing the uniform-high-probability PoA is an open problem.", "problemStatement": "The price of adaptivity compares the performance of an adaptive stochastic convex-optimization procedure with that of a nonadaptive one when the distance to an optimum is uncertain by a factor ρ. The source paper obtains expected and confidence-dependent bounds, but not a bound that holds simultaneously for every confidence level. Determine the tight dependence on ρ for this uniform high-probability price of adaptivity.", "keywords": [ "stochastic convex optimization", "parameter-free methods", "price of adaptivity", "oracle complexity" ], "source": { "title": "The Price of Adaptivity in Stochastic Convex Optimization", "authors": [ "Yair Carmon", "Oliver Hinder" ], "doi": "10.1007/s10107-025-02268-3", "url": "https://doi.org/10.1007/s10107-025-02268-3", "preprintUrl": "https://arxiv.org/abs/2402.10898" }, "id": "mp-2025-015", "number": 358, "recordType": "reviewed", "sourceClaim": "explicit_open_problem", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "statusEvidence": "Targeted title, exact-statement, and forward-literature searches located no later paper claiming a complete resolution.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "literature": [], "journal": "Mathematical Programming", "publicationYear": 2025 }, { "id": "mor-2025-W4414091988_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2025, "title": "Prove existence of a simpler all-norms approximation clustering for correlation clustering", "topic": "Correlation Clustering", "publicationDate": "2025-09-10", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 17, Section 5 (Conclusion).", "sourceQuote": "One question is whether there is a simpler existential (not necessarily algorithmic) proof that there exists an O(1)-approximation for the all-norm objective for correlation clustering.", "problemStatement": "In correlation clustering, positive edges prefer endpoints in one cluster and negative edges prefer endpoints in different clusters; each vertex has a disagreement count. The all-norms objective seeks one clustering whose disagreement vector is within a universal constant of the optimum in every ℓp norm, 1≤p≤∞. Give a simpler existential proof that such a simultaneous constant-factor clustering always exists, without relying on the full algorithmic construction.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "correlation clustering", "all-norms objective", "simultaneous approximation", "lp norms", "disagreement vector" ], "literature": [], "source": { "title": "Fast Combinatorial Algorithms for Simultaneously Approximating All ℓp-Norms in Correlation Clustering", "authors": [ "Sami Davies", "Benjamin Moseley", "Heather Newman" ], "doi": "10.1287/moor.2024.0567", "url": "https://doi.org/10.1287/moor.2024.0567" }, "number": 359 }, { "title": "Trajectory predictions with sample reuse", "topic": "Stochastic optimization", "series": "B", "publicationDate": "2025-09-17", "sourceLocation": "Concluding discussion, p. 34", "sourceQuote": "Understand the effect of sample re-use and develop trajectory predictions in this setting.", "problemStatement": "Trajectory prediction aims to describe the entire iteration path of stochastic prox-linear algorithms for matrix sensing, rather than only their final convergence rate. The source analysis assumes fresh independent minibatches at successive iterations, while practical implementations often reuse one finite dataset. Establish deterministic trajectory predictions and convergence guarantees under sample reuse, explicitly accounting for the dependence created across iterations.", "keywords": [ "matrix sensing", "prox-linear", "sample reuse", "trajectory prediction" ], "source": { "title": "Hyperparameter tuning via trajectory predictions: stochastic prox-linear methods in matrix sensing", "authors": [ "Mengqi Lou", "Kabir Aladin Verchand", "Ashwin Pananjady" ], "doi": "10.1007/s10107-025-02279-0", "url": "https://doi.org/10.1007/s10107-025-02279-0", "preprintUrl": "https://arxiv.org/abs/2402.01599" }, "id": "mp-2025-016", "number": 360, "recordType": "reviewed", "sourceClaim": "explicit_open_problem", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "statusEvidence": "Targeted title, exact-statement, and forward-literature searches located no later paper claiming a complete resolution.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "literature": [], "journal": "Mathematical Programming", "publicationYear": 2025 }, { "id": "siopt-2025-063", "journal": "SIAM Journal on Optimization", "publicationYear": 2025, "title": "Sharp RIP rank benign low rank landscape", "topic": "Low-rank optimization", "publicationDate": "2025-09-19", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Introduction; extracted preprint lines 359-365", "sourceQuote": "We conjecture that k >= r + r-star is both necessary and sufficient for a globally benign landscape", "problemStatement": "The paper studies factorized recovery of a rank-r-star positive-semidefinite matrix with an overparameterized rank-r factor under rank-k restricted isometry. It proves minimax-optimal recovery and benignity when k is at least r+r-star. Show this rank order is also necessary by constructing spurious second-order points below it, or improve the sufficient threshold.", "statusEvidence": "No forward-citing primary paper or exact-query result explicitly claiming a proof or counterexample was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "matrix recovery", "RIP", "benign landscape", "overparameterization" ], "literature": [], "source": { "title": "Sharp Global Guarantees for Nonconvex Low-Rank Recovery in the Noisy Overparameterized Regime", "authors": [ "Richard Y. Zhang" ], "doi": "10.1137/24m1697980", "url": "https://doi.org/10.1137/24m1697980", "preprintUrl": "https://arxiv.org/abs/2104.10790", "volume": "35", "issue": "3", "pages": "2128-2154" }, "number": 361 }, { "title": "Diamond-type quadratic-form optimal transport", "topic": "Optimal transport", "series": "A", "publicationDate": "2025-09-22", "sourceLocation": "Appendix D(iv)(a), p. 35", "sourceQuote": "We conjecture that some ‘diamond-type’ coupling is the minimizer of the corresponding QOT problem.", "problemStatement": "Quadratic-form optimal transport minimizes ∬ c(x,y,x′,y′) dπ(x,y)dπ(x′,y′) over couplings π with fixed one-dimensional marginals. When both marginals are symmetric and c = φ((x−x′)²)φ((y−y′)²), with φ completely monotone and φ′(u) + 2uφ″(u) ≤ 0 on the relevant squared-distance range, the diamond coupling is optimal and is unique for nonconstant φ. Remove the marginal-symmetry assumption: define the conjectured coupling assembled from four comonotone and antitone pieces and prove it minimizes the same objective, or construct a counterexample.", "keywords": [ "quadratic-form transport", "diamond coupling", "dependence optimization", "marginal symmetry" ], "source": { "title": "Quadratic-form optimal transport", "authors": [ "Ruodu Wang", "Zhenyuan Zhang" ], "doi": "10.1007/s10107-025-02282-5", "url": "https://doi.org/10.1007/s10107-025-02282-5", "preprintUrl": "https://arxiv.org/abs/2501.04658" }, "id": "mp-2025-017", "number": 362, "recordType": "reviewed", "sourceClaim": "explicit_open_problem", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "statusEvidence": "Targeted title, exact-statement, and forward-literature searches located no later paper claiming a complete resolution.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "literature": [], "journal": "Mathematical Programming", "publicationYear": 2025 }, { "id": "mor-2025-W4414408152_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2025, "title": "Resolve smoothed complexity of pure Nash equilibria in unrestricted congestion games", "topic": "Smoothed Complexity Congestion Games", "publicationDate": "2025-09-22", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 6 (Conclusion and problems), page 33.", "sourceQuote": "network congestion games (see Theorem 4.2 and Theorem 4.8, respectively). However, outside of these restricted instances of the problem, the complexity remains unresolved.", "problemStatement": "A congestion game is an exact potential game in which each player's cost is the sum of latency functions on the resources she selects. Better-response dynamics repeatedly applies a cost-improving unilateral move until reaching a pure Nash equilibrium. Determine its smoothed complexity for unrestricted congestion games under independently perturbed general, polynomial, or step-function latency models, including whether the expected number of moves is polynomial.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "smoothed analysis", "congestion games", "pure nash equilibrium", "better-response dynamics", "PLS", "local search" ], "literature": [], "source": { "title": "On the Smoothed Complexity of Combinatorial Local Search", "authors": [ "Yiannis Giannakopoulos", "Alexander Grosz", "Themistoklis Melissourgos" ], "doi": "10.1287/moor.2024.0610", "url": "https://doi.org/10.1287/moor.2024.0610" }, "number": 363 }, { "id": "mor-2025-W4414449781_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2025, "title": "Existence of equilibrium with rich firm using control and poor firm using strategy", "topic": "Stochastic Differential Game", "publicationDate": "2025-09-23", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 23, Remark 4.9 (end of Section 4, immediately after Theorem 4.1 proof).", "sourceQuote": "whether an equilibrium exists in which Player 1 uses a strategy and Player 2 uses a control (both either pure or randomised) is open", "problemStatement": "The stochastic game models two firms of unequal wealth that choose investment or extraction behavior using either ordinary controls or nonanticipative strategies. The paper establishes an equilibrium for one assignment of these roles. Determine whether an equilibrium also exists when the richer firm uses a control and the poorer firm uses a strategy.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "stochastic games", "singular control", "nash equilibrium", "feedback strategies", "dividend distribution", "free boundary problems" ], "literature": [], "source": { "title": "Nash Equilibria for Dividend Distribution with Competition", "authors": [ "Tiziano De Angelis", "Fabien Gensbittel", "Stéphane Villeneuve" ], "doi": "10.1287/moor.2023.0374", "url": "https://doi.org/10.1287/moor.2023.0374", "preprintUrl": "https://publications.ut-capitole.fr/id/eprint/51609/1/wp_tse_1495.pdf" }, "number": 364 }, { "id": "mor-2025-W4414427837_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2025, "title": "Deterministic polynomial-time algorithm for exact matching in bipartite graphs", "topic": "Exact Matching", "publicationDate": "2025-09-23", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 6 (Introduction, discussion before Theorem 4) and page 22 (Section 5.1, statement of Theorem 4 and surrounding text).", "sourceQuote": "the exact matching problem [46] (see Section 5.1 for the definition), for which the existence of a deterministic efficient algorithm remains a longstanding open question.", "problemStatement": "Every edge of a balanced bipartite graph is colored red or blue. Given an integer k, exact matching asks whether a perfect matching contains exactly k red edges. Design a deterministic polynomial-time decision and construction algorithm, or establish a barrier to derandomization.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "exact matching", "bipartite perfect matching", "deterministic polynomial time", "colored edges", "reductions" ], "literature": [], "source": { "title": "Popular Maximum-Utility Matchings with Matroid Constraints", "authors": [ "Gergely Csáji", "Tamás Király", "Kenjiro Takazawa", "Yu Yokoi" ], "doi": "10.1287/moor.2024.0633", "url": "https://doi.org/10.1287/moor.2024.0633" }, "number": 365 }, { "id": "focm-2025-random-feature-bound-linear-n-dependence", "journal": "Foundations of Computational Mathematics", "publicationYear": 2025, "title": "Sharp Ambient-Dimension Dependence for Regular-Set Covering Numbers", "topic": "Approximation theory and learning", "publicationDate": "2025-09-26", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conjecture 2.8 discussion", "sourceQuote": "We conjecture that an N dependence is sufficient.", "problemStatement": "A set V in R^N is (K,n)-regular when almost every affine slice of codimension at most n has at most K path components and almost every slice of larger codimension is empty. For V in the unit ball, prove Conjecture 2.8 by constructing a regular set with covering number N(V,epsilon)=Omega(N^n) at fixed epsilon and K, or refute it. This would decide whether linear ambient-dimension dependence inside the base of the covering-number bound is necessary, improving the proof's N^(3/2) slicing factor.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "random features", "sample complexity", "dimension dependence", "concentration" ], "literature": [], "source": { "title": "Covering Number of Real Algebraic Varieties and Beyond: Improved Bounds and Applications", "authors": [ "Yifan Zhang", "Joe Kileel" ], "doi": "10.1007/s10208-025-09735-5", "url": "https://doi.org/10.1007/s10208-025-09735-5", "preprintUrl": "https://arxiv.org/abs/2311.05116" }, "number": 366 }, { "id": "mor-2025-W4414541699_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2025, "title": "Polynomial-time approximation of justified envy-free Pareto-optimal lotteries in two-sided markets", "topic": "Fair Matching", "publicationDate": "2025-09-26", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 23, Section 4 (Conclusion)", "sourceQuote": "Is there a polynomial time algorithm to find α-approximately JEF+PO lotteries in two-sided markets, for any constant α?", "problemStatement": "In a two-sided cardinal-utility matching market, justified envy compares an agent with someone who receives a partner preferred by both sides. The paper obtains lotteries with efficiency or fairness properties separately, but not their desired combination. Design a polynomial-time algorithm returning a Pareto-optimal lottery with a universal constant-factor approximation to justified envy-freeness.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "two-sided matching", "cardinal utilities", "justified envy-freeness", "pareto optimality", "polynomial-time algorithms", "approximation" ], "literature": [], "source": { "title": "Cardinal-Utility Matching Markets: The Quest for Envy-Freeness, Pareto-Optimality, and Efficient Computability", "authors": [ "Thorben Tröbst", "Vijay V. Vazirani" ], "doi": "10.1287/moor.2024.0770", "url": "https://doi.org/10.1287/moor.2024.0770" }, "number": 367 }, { "id": "focm-2025-trotter-ground-state-lower-bound", "journal": "Foundations of Computational Mathematics", "publicationYear": 2025, "title": "Trotter Ground State Lower Bound", "topic": "Mathematical quantum mechanics", "publicationDate": "2025-09-29", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Introduction, discussion after Corollary 2.23", "sourceQuote": "Proving an analytical lower bound, however, remains an open problem.", "problemStatement": "Prove an analytical lower bound matching the observed n^-1/4 ground-state convergence rate for the Trotter product formula studied in the paper. This would show the available upper estimate is essentially sharp. The Trotter product alternates semigroups generated by split Hamiltonian terms; the paper proves an upper rate near n^-1/4 for rough ground states but has only numerical evidence of sharpness.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "Trotter product formula", "ground state", "convergence rate", "lower bound" ], "literature": [], "source": { "title": "Convergence Rates for the Trotter Splitting for Unbounded Operators", "authors": [ "Simon Becker", "Niklas Galke", "Lauritz van Luijk", "Robert Salzmann" ], "doi": "10.1007/s10208-025-09730-w", "url": "https://doi.org/10.1007/s10208-025-09730-w" }, "number": 368 }, { "id": "mor-2025-W4414783466_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2025, "title": "Improve Wasserstein convergence rate for vector martingale central limit theorem", "topic": "Martingale Central Limit Theorem", "publicationDate": "2025-10-03", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 5 (Conclusions), page 19", "sourceQuote": "if the O(log n/sqrt n) rate of convergence in Theorem 1 can be improved to O(1/sqrt n)", "problemStatement": "The paper proves a multivariate martingale central-limit bound in Wasserstein distance that typically scales as log(n)/√n. The logarithm arises from accumulating conditional covariance and moment errors. Give verifiable additional assumptions under which the rate improves to 1/√n while retaining vector-valued martingale differences and a nondegenerate Gaussian limit.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "martingale central limit theorem", "Wasserstein distance", "Stein's method", "rate of convergence", "multivariate normal approximation", "log factor removal" ], "literature": [], "source": { "title": "Rates of Convergence in the Central Limit Theorem for Markov Chains, with an Application to TD Learning", "authors": [ "R. Srikant" ], "doi": "10.1287/moor.2024.0444", "url": "https://doi.org/10.1287/moor.2024.0444" }, "number": 369 }, { "title": "Unconditional expectation complexity for stochastic Bregman proximal gradient", "topic": "Stochastic optimization", "series": "A", "publicationDate": "2025-10-06", "sourceLocation": "Discussion after Theorem 3.8, p. 17", "sourceQuote": "Whether one can … obtain the standard in expectation complexity bound.", "problemStatement": "Stochastic Bregman proximal-gradient methods use a geometry-generating kernel to handle relatively smooth composite objectives. The source paper’s complexity theorem is conditioned on a high-probability event controlling the stochastic error, so it does not directly give the standard unconditional expected complexity. Derive a comparable expectation bound without conditioning, or show that further moment or kernel assumptions are necessary.", "keywords": [ "Bregman proximal gradient", "variance reduction", "expectation complexity", "nonconvex optimization" ], "source": { "title": "Stochastic Bregman Proximal Gradient Method Revisited: Kernel Conditioning and Painless Variance Reduction", "authors": [ "Junyu Zhang" ], "doi": "10.1007/s10107-025-02285-2", "url": "https://doi.org/10.1007/s10107-025-02285-2", "preprintUrl": "https://arxiv.org/abs/2401.03155" }, "id": "mp-2025-018", "number": 370, "recordType": "reviewed", "sourceClaim": "explicit_open_problem", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "statusEvidence": "Targeted title, exact-statement, and forward-literature searches located no later paper claiming a complete resolution.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "literature": [], "journal": "Mathematical Programming", "publicationYear": 2025 }, { "id": "mor-2025-W4415178980_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2025, "title": "Achieve quota, population monotonicity, and ex-ante proportionality without varying house size", "topic": "Randomized Apportionment", "publicationDate": "2025-10-14", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 6 (Discussion), page 23.", "sourceQuote": "proportionality guarantees if we allow to violate the house size (Section 4.2). Whether a similar result holds without violating the house size is open.", "problemStatement": "An apportionment instance specifies state populations and a fixed house size H. Existing randomized methods satisfy quota, population monotonicity, and ex ante proportionality by allowing the realized total seat count to differ from H while equaling H in expectation. Achieve all three properties while assigning exactly H seats in every realization.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "apportionment", "population monotonicity", "quota compliance", "randomized methods", "ex-ante proportionality", "house size constraint" ], "literature": [], "source": { "title": "New Combinatorial Insights for Monotone Apportionment", "authors": [ "Javier Cembrano", "José Correa", "Ulrike Schmidt-Kraepelin", "Alexandros Tsigonias-Dimitriadis", "Victor Verdugo" ], "doi": "10.1287/moor.2024.0817", "url": "https://doi.org/10.1287/moor.2024.0817" }, "number": 371 }, { "id": "siopt-2025-067", "journal": "SIAM Journal on Optimization", "publicationYear": 2025, "title": "Euclidean distance degrees manifold families", "topic": "Algebraic optimization", "publicationDate": "2025-10-22", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conclusion, two open problems; extracted preprint lines 696-703", "sourceQuote": "Determine the Euclidean distance degree for the symplectic Grassmannian", "problemStatement": "The Euclidean distance degree of an algebraic manifold counts the complex critical points of squared distance to generic data and predicts the algebraic complexity of projection. The paper computes this invariant for several matrix manifolds. Determine it for the embedded symplectic Grassmannian and for a general Schubert variety, first constructing a suitable quadratic model in the latter case.", "statusEvidence": "No forward-citing primary paper or exact-query result explicitly claiming a proof or counterexample was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "Euclidean distance degree", "symplectic Grassmannian", "Schubert variety", "manifold projection" ], "literature": [], "source": { "title": "Euclidean Distance Degree in Manifold Optimization", "authors": [ "Zehua Lai", "Lek-Heng Lim", "Ke Ye" ], "doi": "10.1137/25m1735032", "url": "https://doi.org/10.1137/25m1735032", "preprintUrl": "https://arxiv.org/abs/2502.10336", "volume": "35", "issue": "4", "pages": "2402-2422" }, "number": 372 }, { "id": "siopt-2025-060", "journal": "SIAM Journal on Optimization", "publicationYear": 2025, "title": "Graph derived forward backward unification", "topic": "Operator splitting", "publicationDate": "2025-10-23", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 4.3; cached preprint lines 1343-1348", "sourceQuote": "it remains as an open question for future investigation to expand our framework to fully cover the framework in [14].", "problemStatement": "Consider 0∈Σ_(i=1)^nA_i(x)+Σ_(j=1)^mB_j(x) on a Hilbert space, with maximally monotone A_i evaluated by backward resolvent steps and cocoercive B_j evaluated by forward steps. The new framework uses a directed graph and two subgraphs to construct frugal minimal-lifting splittings, while the Morin–Banert–Giselsson framework represents all frugal minimal-lifting splittings directly. For the biparallel and parallel graph instance, the latter framework corresponds to μ=2/(n−1) and relaxation θ=2, but the new framework with step λ and cocoercivity constant β requires θ≤2−(n−1)λ/(2β)<2 and therefore cannot recover it. Construct an extension covering both parameter regimes and determine how the algebraic connectivity of the chosen subgraph controls convergence and practical speed.", "statusEvidence": "No forward-citing primary paper or exact-query result explicitly claiming a proof or counterexample was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "forward-backward splitting", "graph algorithms", "algebraic connectivity", "unification" ], "literature": [], "source": { "title": "Forward-Backward Algorithms Devised by Graphs", "authors": [ "Francisco J. Aragón-Artacho", "Rubén Campoy", "César López-Pastor" ], "doi": "10.1137/24m166721x", "url": "https://doi.org/10.1137/24m166721x", "preprintUrl": "https://arxiv.org/abs/2406.03309", "volume": "35", "issue": "4", "pages": "2423-2451" }, "number": 373 }, { "title": "Improve the lower bound for the max-entropy TSP algorithm", "topic": "Combinatorial optimization", "series": "B", "publicationDate": "2025-10-27", "sourceLocation": "Section 5, p. 14", "sourceQuote": "Whether one can obtain a lower bound … larger than 11/8.", "problemStatement": "The maximum-entropy algorithm for metric TSP samples spanning trees from an entropy-maximizing distribution guided by a Subtour-LP solution and then repairs parity. The source paper constructs instances with asymptotic approximation ratio 11/8, establishing a lower bound on the algorithm’s guarantee. Find a family with limiting ratio strictly greater than 11/8, or prove that 11/8 is the worst possible ratio.", "keywords": [ "TSP", "maximum entropy", "lower bound", "randomized rounding" ], "source": { "title": "A lower bound for the max entropy algorithm for TSP", "authors": [ "Billy Jin", "Nathan Klein", "David P. Williamson" ], "doi": "10.1007/s10107-025-02289-y", "url": "https://doi.org/10.1007/s10107-025-02289-y", "preprintUrl": "https://arxiv.org/abs/2311.01950" }, "id": "mp-2025-020", "number": 374, "recordType": "reviewed", "sourceClaim": "explicit_open_problem", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "statusEvidence": "Targeted title, exact-statement, and forward-literature searches located no later paper claiming a complete resolution.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "literature": [], "journal": "Mathematical Programming", "publicationYear": 2025 }, { "title": "Heavy-ball acceleration in one dimension", "topic": "Convex & nonlinear optimization", "series": "B", "publicationDate": "2025-10-27", "sourceLocation": "Section 7, p. 23", "sourceQuote": "The potential acceleration of (HB) in a one-dimensional space is thus left open.", "problemStatement": "The fixed-parameter heavy-ball method augments gradient descent with momentum and is known not to achieve uniform acceleration over broad multidimensional smooth strongly convex classes. Those lower bounds do not settle the one-dimensional class of L-smooth, μ-strongly convex functions. Determine whether a fixed choice of heavy-ball parameters can attain an accelerated worst-case rate in one dimension.", "keywords": [ "heavy-ball", "acceleration", "one dimension", "performance estimation" ], "source": { "title": "Provable non-accelerations of the heavy-ball method", "authors": [ "Baptiste Goujaud", "Adrien Taylor", "Aymeric Dieuleveut" ], "doi": "10.1007/s10107-025-02269-2", "url": "https://doi.org/10.1007/s10107-025-02269-2", "preprintUrl": "https://arxiv.org/abs/2307.11291" }, "id": "mp-2025-019", "number": 375, "recordType": "reviewed", "sourceClaim": "explicit_open_problem", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "statusEvidence": "Targeted title, exact-statement, and forward-literature searches located no later paper claiming a complete resolution.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "literature": [], "journal": "Mathematical Programming", "publicationYear": 2025 }, { "title": "PTAS versus constant-additive hardness for robust Stackelberg equilibria", "topic": "Game theory & matching", "series": "A", "publicationDate": "2025-11-12", "sourceLocation": "Section 4.2, p. 16", "sourceQuote": "To understand whether a PTAS exists for δ-RSE.", "problemStatement": "A δ-robust Stackelberg equilibrium requires the leader’s commitment to remain effective when followers may choose any response whose utility is within δ of optimal. The source paper develops approximation results but leaves a gap between them and its hardness bounds. Decide whether the approximate δ-robust equilibrium problem has a polynomial-time approximation scheme, or prove hardness for some constant additive error.", "keywords": [ "Stackelberg equilibrium", "robust equilibrium", "PTAS", "bilevel optimization" ], "source": { "title": "Robust Stackelberg Equilibria", "authors": [ "Jiarui Gan", "Minbiao Han", "Jibang Wu", "Haifeng Xu" ], "doi": "10.1007/s10107-025-02291-4", "url": "https://doi.org/10.1007/s10107-025-02291-4", "preprintUrl": "https://arxiv.org/abs/2304.14990" }, "id": "mp-2025-021", "number": 376, "recordType": "reviewed", "sourceClaim": "explicit_open_problem", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "statusEvidence": "Targeted title, exact-statement, and forward-literature searches located no later paper claiming a complete resolution.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "literature": [], "journal": "Mathematical Programming", "publicationYear": 2025 }, { "id": "mor-2025-W4416295153_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2025, "title": "Fixed-parameter tractability of knapsack facet recognition by distinct coefficients parameter", "topic": "Knapsack Polytope Facets", "publicationDate": "2025-11-17", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 16, Section 5 (Concluding Remarks)", "sourceQuote": "whether the knapsack facet recognition problem is fixed-parameter tractable.", "problemStatement": "KNAPSACK_FACETS asks whether a valid inequality defines a facet of a 0/1 knapsack polytope. An XP algorithm is known when K is the number of distinct positive coefficients in the inequality. Determine whether the problem is fixed-parameter tractable in K, meaning solvable in f(K) times a polynomial in the input size.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "knapsack polytope", "facet recognition", "parameterized complexity", "fixed-parameter tractability", "Dp-completeness", "polyhedral combinatorics" ], "literature": [], "source": { "title": "The Complexity of Recognizing Facets for the Knapsack Polytope", "authors": [ "Rui Chen", "Haoran Zhu" ], "doi": "10.1287/moor.2024.0481", "url": "https://doi.org/10.1287/moor.2024.0481" }, "number": 377 }, { "title": "Characterization of uniformly robust norms", "topic": "Robust & network optimization", "series": "A", "publicationDate": "2025-11-24", "sourceLocation": "Section 9, Conjecture 9.1, p. 31", "sourceQuote": "A norm is uniformly robust if and only if it is polyhedral.", "problemStatement": "A Fermat–Weber estimator minimizes the sum of distances to observed points, and its breakdown point measures the fraction of adversarial observations it can tolerate. A norm is called uniformly robust when these estimators retain a uniform positive robustness guarantee across all data configurations. Prove or refute that a finite-dimensional norm is uniformly robust exactly when its unit ball is a polytope.", "keywords": [ "Fermat–Weber", "breakdown point", "polyhedral norm", "robust location" ], "source": { "title": "Breakdown points of Fermat–Weber problems under gauge distances", "authors": [ "Andrei Comăneci", "Frank Plastria" ], "doi": "10.1007/s10107-025-02302-4", "url": "https://doi.org/10.1007/s10107-025-02302-4", "preprintUrl": "https://arxiv.org/abs/2306.13424" }, "id": "mp-2025-022", "number": 378, "recordType": "reviewed", "sourceClaim": "explicit_open_problem", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "statusEvidence": "Targeted title, exact-statement, and forward-literature searches located no later paper claiming a complete resolution.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "literature": [], "journal": "Mathematical Programming", "publicationYear": 2025 }, { "id": "mor-2025-W4416664990_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2025, "title": "Settle complexity of min-cost popular maximum matchings in many-to-one settings", "topic": "Popular Matchings", "publicationDate": "2025-11-25", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 4 (Conclusions and problems), page 20", "sourceQuote": "Settling the computational complexity of the min-cost popular maximum matching problem in a many-to-one instance with strict and incomplete preferences is an interesting problem.", "problemStatement": "A popular maximum matching in a hospitals/residents instance defeats or ties every other maximum-cardinality matching in a pairwise election based on preferences. Costs are attached to acceptable resident-hospital edges. Determine the complexity of finding a minimum-cost popular maximum matching when strict preference lists may be incomplete.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "popular matching", "hospital residents", "maximum matching", "computational complexity", "minimum cost", "incomplete preferences" ], "literature": [], "source": { "title": "Min-Cost Popular Matchings in a Hospitals/Residents Instance with Complete Preferences", "authors": [ "Telikepalli Kavitha", "Kazuhisa Makino" ], "doi": "10.1287/moor.2024.0634", "url": "https://doi.org/10.1287/moor.2024.0634" }, "number": 379 }, { "id": "mor-2025-W4416665164_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2025, "title": "Efficiently update density-based decompositions under single-element insertions and deletions", "topic": "Dynamic Matroid Decomposition", "publicationDate": "2025-11-25", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 6, Introduction (end of the discussion on density-based decomposition and principal partitions).", "sourceQuote": "whether updating a density-based decomposition when performing these kinds of operations can be done more efficiently, without re-computing the whole decomposition each time.", "problemStatement": "A matroid's density decomposition partitions a changing ground set into layers determined by ranks and maximum-density subsets. Recomputing it after every inserted or deleted element is expensive. Design a dynamic data structure that maintains the decomposition under single-element updates with provably better update time.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "matroid intersection", "density-based decomposition", "principal partitions", "dynamic algorithms", "submodular minimization" ], "literature": [], "source": { "title": "Robust Sparsification for Matroid Intersection with Applications", "authors": [ "Chien-Chung Huang", "François Sellier" ], "doi": "10.1287/moor.2024.0562", "url": "https://doi.org/10.1287/moor.2024.0562" }, "number": 380 }, { "id": "focm-2025-christoffel-sampling-sparse-regression", "journal": "Foundations of Computational Mathematics", "publicationYear": 2025, "title": "Christoffel Sampling Sparse Regression", "topic": "Approximation theory and sampling", "publicationDate": "2025-12-01", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 8, open questions", "sourceQuote": "it is unknown whether Christoffel sampling yields near-optimal sample complexity for sparse regression", "problemStatement": "Determine whether Christoffel sampling achieves near-optimal sample complexity for sparse regression in standard nonlinear approximation settings. The result should compare rigorously with leverage-score or coherence-optimal designs.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "Christoffel sampling", "sparse regression", "sample complexity", "nonlinear approximation" ], "literature": [], "source": { "title": "Optimal sampling for least-squares approximation", "authors": [ "Ben Adcock" ], "doi": "10.1007/s10208-025-09738-2", "url": "https://doi.org/10.1007/s10208-025-09738-2", "preprintUrl": "https://arxiv.org/abs/2409.02342", "volume": "25", "issue": "6", "pages": "1975-2034" }, "number": 381 }, { "id": "focm-2025-canny-emiris-dilation-free-resultant", "journal": "Foundations of Computational Mathematics", "publicationYear": 2025, "title": "Canny Emiris Dilation Free Resultant", "topic": "Computational algebraic geometry", "publicationDate": "2025-12-01", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conclusion", "sourceQuote": "Whether this dilation could be avoided is left as an open problem.", "problemStatement": "Remove the up-to-(n+1) dilation of the Canny-Emiris support set in the paper's general tropical resultant relation. Prove the undilated statement or construct a counterexample showing dilation is necessary. The Canny-Emiris set is a sparse resultant support selected from mixed subdivisions, and the proved tropical relation becomes valid only after enlarging this set by the stated dilation.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "tropical resultants", "Canny-Emiris", "support sets", "dilation" ], "literature": [], "source": { "title": "The Nullstellensatz and Positivstellensatz for Sparse Tropical Polynomial Systems", "authors": [ "Marianne Akian", "Antoine Béreau", "Stéphane Gaubert" ], "doi": "10.1007/s10208-025-09708-8", "url": "https://doi.org/10.1007/s10208-025-09708-8", "preprintUrl": "https://arxiv.org/abs/2312.05859", "volume": "26", "issue": "3", "pages": "1587-1633" }, "number": 382 }, { "title": "Constant-factor approximation under fixed non-uniform fault width", "topic": "Robust & network optimization", "series": "A", "publicationDate": "2025-12-01", "sourceLocation": "Section 6, p. 27", "sourceQuote": "Is there a constant factor approximation for fixed width/connectivity?", "problemStatement": "Bulk-Robust survivable network design and flexible network design require connectivity to persist under specified nonuniform failure scenarios. Approximation factors generally deteriorate with the scenario width or required connectivity, even when that parameter is fixed. Give a polynomial-time constant-factor approximation for every fixed width/connectivity, or prove that no such approximation exists under a standard complexity assumption.", "keywords": [ "survivable network design", "non-uniform faults", "approximation", "fixed width" ], "source": { "title": "Approximation algorithms for network design in non-uniform fault models", "authors": [ "Chandra Chekuri", "Rhea Jain" ], "doi": "10.1007/s10107-025-02298-x", "url": "https://doi.org/10.1007/s10107-025-02298-x", "preprintUrl": "https://arxiv.org/abs/2403.15547" }, "id": "mp-2025-023", "number": 383, "recordType": "reviewed", "sourceClaim": "explicit_open_problem", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "statusEvidence": "Targeted title, exact-statement, and forward-literature searches located no later paper claiming a complete resolution.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "literature": [], "journal": "Mathematical Programming", "publicationYear": 2025 }, { "id": "mor-2025-W4417166251_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2025, "title": "Solvability conditions for asymmetric indefinite Riccati equation in mean-field game feedback", "topic": "Indefinite Riccati Equations", "publicationDate": "2025-12-09", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 14, Remark 4.13(ii) (following Theorem 4.12, Section 4.3 \"Feedback representation\").", "sourceQuote": "Noticing that Riccati equation (35) is an asymmetric indefinite equation with complex structure, its solvability is still a problem.", "problemStatement": "A partially observed linear-quadratic mean-field game with common noise yields an asymmetric, indefinite Riccati equation for a matrix Σ. The equation contains the inverse of an effective control-weight matrix that itself depends on Σ. Give verifiable coefficient conditions guaranteeing a unique solution on the full horizon and uniform invertibility of that effective matrix, so the decentralized feedback law is well defined.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "indefinite riccati equation", "mean-field game", "linear-quadratic control", "partial observation", "common noise", "feedback equilibrium" ], "literature": [], "source": { "title": "Indefinite Linear-Quadratic Partially Observed Mean-Field Game", "authors": [ "Tian Chen", "Tianyang Nie", "Zhen Wu" ], "doi": "10.1287/moor.2024.0748", "url": "https://doi.org/10.1287/moor.2024.0748" }, "number": 384 }, { "id": "mor-2025-W7114802374_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2025, "title": "Designing fully polynomial approximation schemes for stochastic probing and stopping problems", "topic": "Stochastic Combinatorial Optimization", "publicationDate": "2025-12-11", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 7 (Future Directions), page 34.", "sourceQuote": "An immediate open question for future research is whether one or more of these problems admit fully polynomial time approximation schemes (FPTAS).", "problemStatement": "The source gives efficient approximation schemes for Free-Order Prophets, where one chooses an inspection order and stopping rule; Pandora's Box with Commitment, which adds inspection costs and irrevocable acceptance; and adaptive and nonadaptive ProbeMax, which probes feasible subsets to maximize the largest revealed value. A fully polynomial-time approximation scheme must return a (1−ε)-approximation in time polynomial in the input size and 1/ε. These are four separate open questions: determine for each problem whether such a scheme exists, using techniques that avoid the strongly NP-hard multidimensional Santa Claus subroutine, or prove that an FPTAS is impossible.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "FPTAS", "stochastic optimization", "probing problems", "optimal stopping", "EPTAS", "complexity" ], "literature": [], "source": { "title": "Efficient Approximation Schemes for Stochastic Probing and Selection-Stopping Problems", "authors": [ "Danny Segev", "Sahil Singla" ], "doi": "10.1287/moor.2023.0242", "url": "https://doi.org/10.1287/moor.2023.0242" }, "number": 385 }, { "id": "mor-2025-W4417331274_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2025, "title": "Characterize the optimal equilibrium stopping region under general aggregation preferences", "topic": "Time Inconsistent Optimal Stopping", "publicationDate": "2025-12-15", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 28 (PDF page label 28/38), Section 5.1 \"Geometric Brownian motion\", Remark 5.3 (end of remark).", "sourceQuote": "find a general characterization of the optimal equilibrium if it exists", "problemStatement": "The stopping payoff aggregates exponentially discounted values across a distribution of discount rates through a nonlinear preference function. The paper characterizes optimal equilibrium stopping regions only under a structural monotonicity condition on that aggregator. Remove that condition by giving general existence and characterization results for an optimal equilibrium stopping region, or identify a counterexample.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "time-inconsistent stopping", "equilibrium stopping", "optimal equilibrium", "smooth ambiguity preference", "heterogeneous discount rates", "diffusion processes" ], "literature": [], "source": { "title": "On Time-Consistent Equilibrium Stopping Under Aggregation of Diverse Discount Rates", "authors": [ "Shuoqing Deng", "Xiang Yu", "Jiacheng Zhang" ], "doi": "10.1287/moor.2024.0752", "url": "https://doi.org/10.1287/moor.2024.0752" }, "number": 386 }, { "title": "Dimension-independent guarantees for matrix fixed-point projection", "topic": "Matrix & semidefinite optimization", "series": "A", "publicationDate": "2025-12-16", "sourceLocation": "Section 9, p. 19", "sourceQuote": "Of particular interest is whether this approach can yield theoretical dimension-independent guarantees.", "problemStatement": "Given a positive-definite matrix, the method projects it in Bures distance onto the positive-definite matrices invariant under a prescribed symmetry by repeating a fixed-point update; the same update is a unit-step Riemannian descent and includes Bures–Wasserstein barycenters as a special case. Exponential convergence is proved, while dimension-independent guarantees are known for related Riemannian descent only with a smaller, nonunit step. Prove a convergence rate for the unit-step fixed-point iteration whose constants do not grow with matrix dimension, or exhibit a family showing that such a rate is impossible.", "keywords": [ "matrix projection", "fixed point", "Bures–Wasserstein", "quantum information" ], "source": { "title": "A fixed-point algorithm for matrix projections with applications in quantum information", "authors": [ "Shrigyan Brahmachari", "Roberto Rubboli", "Marco Tomamichel" ], "doi": "10.1007/s10107-025-02318-w", "url": "https://doi.org/10.1007/s10107-025-02318-w", "preprintUrl": "https://arxiv.org/abs/2312.14615" }, "id": "mp-2025-026", "number": 387, "recordType": "reviewed", "sourceClaim": "explicit_open_problem", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "statusEvidence": "Targeted title, exact-statement, and forward-literature searches located no later paper claiming a complete resolution.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "literature": [], "journal": "Mathematical Programming", "publicationYear": 2025 }, { "title": "Efficiently enforce route-level TSP optimality", "topic": "Combinatorial optimization", "series": "B", "publicationDate": "2025-12-16", "sourceLocation": "Concluding discussion, p. 31", "sourceQuote": "Efficiently ensuring TSP-optimality of selected routes in the CVRP remains an interesting open problem.", "problemStatement": "In capacitated vehicle routing with a nonmonotone route-fairness objective, a selected customer set may be served in an order that is not a shortest traveling-salesman tour. Standard pricing and branching machinery does not efficiently enforce TSP optimality separately within every chosen route. Develop an exact formulation or branch-price-and-cut mechanism that imposes route-level TSP optimality without destroying computational tractability.", "status": "partial_progress", "statusEvidence": "The same authors’ April 2026 paper adds TSP-optimality cuts to branch-price-and-cut and reports strong computational performance, but does not provide a general complexity-theoretic resolution.", "literature": [ { "title": "Enforcing TSP-Optimality in Fair Vehicle Routing by Cutting Planes", "url": "https://arxiv.org/abs/2604.23748", "relation": "Direct 2026 partial progress" } ], "keywords": [ "vehicle routing", "TSP-optimality", "branch-and-price", "fairness" ], "source": { "title": "Efficient branching rules for optimizing range and order-based objective functions", "authors": [ "Bart van Rossum", "Rui Chen", "Andrea Lodi" ], "doi": "10.1007/s10107-025-02306-0", "url": "https://doi.org/10.1007/s10107-025-02306-0", "preprintUrl": "https://arxiv.org/abs/2311.03885" }, "id": "mp-2025-025", "number": 388, "recordType": "reviewed", "sourceClaim": "explicit_open_problem", "statusCheckedThrough": "2026-07-13", "verificationNote": "This status is tied to the cited primary-literature result and the precise scope stated above.", "journal": "Mathematical Programming", "publicationYear": 2025 }, { "title": "Upper boundary of caterpillar and even-tree profiles", "topic": "Combinatorial optimization", "series": "B", "publicationDate": "2025-12-16", "sourceLocation": "Section 5.2, Conjecture 1, p. 21", "sourceQuote": "DCat∞ₚ,₁₋ₚ lies on the upper boundary … for all k ≥ 4 and p ∈ [0,1].", "problemStatement": "Tree-density profiles record the simultaneous limiting densities of selected finite trees inside sequences of increasingly large trees. For the joint profile of the k-vertex caterpillar Catₖ and the six-vertex even tree E₆, infinite double caterpillars DCat∞ₚ,₁₋ₚ are conjectured to trace the upper boundary. Prove this boundary characterization for every k ≥ 4 and p ∈ [0,1], or identify a tree limit lying above it.", "keywords": [ "tree limits", "caterpillars", "flag algebras", "inducibility" ], "source": { "title": "Getting to the root of the problem: sums of squares for limits of trees", "authors": [ "Daniel Brosch", "Diane Puges" ], "doi": "10.1007/s10107-025-02305-1", "url": "https://doi.org/10.1007/s10107-025-02305-1", "preprintUrl": "https://arxiv.org/abs/2404.12838" }, "id": "mp-2025-024", "number": 389, "recordType": "reviewed", "sourceClaim": "explicit_open_problem", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "statusEvidence": "Targeted title, exact-statement, and forward-literature searches located no later paper claiming a complete resolution.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "literature": [], "journal": "Mathematical Programming", "publicationYear": 2025 }, { "id": "mor-2025-W4417437095_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2025, "title": "Characterize Nash product rule under strategyproofness and core fairness conditions", "topic": "Mechanism Design", "publicationDate": "2025-12-17", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 17, Section 6.2 (Characterization) and Page 19, Section 8 (Conclusion)", "sourceQuote": "Is continuity required for the characterization of the Nash product rule?", "problemStatement": "Outcomes are probability allocations q over m alternatives, and agent i with ideal allocation pᵢ has Leontief utility minqⱼ/pᵢⱼ:pᵢⱼ>0. The Nash rule selects the unique allocation maximizing the product of agents' utilities. Core fair share forbids a coalition from using its population share of resources to make every member strictly better off regardless of how the remainder is allocated. Determine whether every continuous, individually strategyproof mechanism satisfying this fairness axiom must equal the Nash rule, making group-strategyproofness unnecessary.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "participatory budgeting", "Leontief utilities", "Nash welfare", "strategyproofness", "group-strategyproofness", "core fair share", "axiomatic characterization" ], "literature": [], "source": { "title": "Optimal Budget Aggregation with Star-Shaped Preference Domains", "authors": [ "Felix Brandt", "Matthias Greger", "Erel Segal-Halevi", "Warut Suksompong" ], "doi": "10.1287/moor.2024.0723", "url": "https://doi.org/10.1287/moor.2024.0723" }, "number": 390 }, { "id": "mor-2026-W7118180028_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2026, "title": "Compute Lipschitz modulus of regularized least-squares solution mapping under stability", "topic": "Variational Analysis", "publicationDate": "2026-01-05", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 19 (Conclusion), paragraph beginning 'One of the open questions we plan to investigate in the future...'; also reiterated on page 20.", "sourceQuote": "the computation of the Lipschitz modulus of the solution mapping when Lipschitz stability occurs.", "problemStatement": "Tikhonov-regularized least squares has a solution mapping from the data vector to the unique minimizer, and the paper characterizes when this mapping is locally Lipschitz. The exact Lipschitz modulus controls sensitivity but normally requires coderivative information beyond the first-order test. Derive a computable formula or sharp bounds for that modulus under the stability conditions.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "lipschitz modulus", "solution mapping", "regularized least squares", "sensitivity analysis", "fenchel conjugate", "coderivative" ], "literature": [], "source": { "title": "Lipschitz Stability of Least-Squares Problems Regularized by Functions with C2-Cone Reducible Conjugates", "authors": [ "Ying Cui", "Tim Hoheisel", "Tran T. A. Nghia", "Defeng Sun" ], "doi": "10.1287/moor.2024.0692", "url": "https://doi.org/10.1287/moor.2024.0692" }, "number": 391 }, { "id": "mp-2026-effective-stochastic-ipm-directions", "journal": "Mathematical Programming", "publicationYear": 2026, "title": "Effective Directions for Stochastic Interior Points", "topic": "Stochastic constrained optimization", "publicationDate": "2026-01-08", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conclusion, first open question", "sourceQuote": "a computationally effective strategy for computing search directions that adhere to our theoretical guarantees is not straightforward to design.", "problemStatement": "The method minimizes a stochastic objective subject to smooth inequalities and affine equations Ax=b while keeping every iterate feasible. Design a computable direction d in Null(A) whose norm is comparable to the projected stochastic gradient, makes a uniformly obtuse angle with it, and lies sufficiently inside the polar cone of gradients of nearly active inequalities. In the stochastic version d must also solve the projected positive-semidefinite linear system with the uniform bounds used in the convergence proof.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact problem wording, title and authors across publisher pages, arXiv, and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "interior-point method", "search direction", "stochastic gradient", "nonlinear constraints" ], "literature": [], "source": { "title": "A single-loop stochastic feasible interior-point algorithm for nonlinear inequality-constrained optimization", "authors": [ "Frank E. Curtis", "Xin Jiang", "Qi Wang" ], "doi": "10.1007/s10107-025-02320-2", "url": "https://doi.org/10.1007/s10107-025-02320-2" }, "number": 392 }, { "id": "mp-2026-single-loop-infeasible-stochastic-ipm", "journal": "Mathematical Programming", "publicationYear": 2026, "title": "Single Loop Infeasible Stochastic Interior Points", "topic": "Stochastic constrained optimization", "publicationDate": "2026-01-08", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conclusion, second open question", "sourceQuote": "whether it is possible to design a single-loop infeasible interior-point framework that possesses theoretical convergence guarantees", "problemStatement": "Develop a single-loop infeasible interior-point algorithm for nonlinear inequality-constrained stochastic optimization with convergence guarantees at least as strong as the source paper's feasible method. The feasible analysis depends on a prescribed barrier sequence and feasibility of every iterate. The challenge is to control slack-variable directions and neighborhood positivity when the original-variable iterate may violate the constraints.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact problem wording, title and authors across publisher pages, arXiv, and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "infeasible interior-point method", "single-loop algorithm", "stochastic optimization", "slack variables" ], "literature": [], "source": { "title": "A single-loop stochastic feasible interior-point algorithm for nonlinear inequality-constrained optimization", "authors": [ "Frank E. Curtis", "Xin Jiang", "Qi Wang" ], "doi": "10.1007/s10107-025-02320-2", "url": "https://doi.org/10.1007/s10107-025-02320-2" }, "number": 393 }, { "id": "mp-2026-barriers-general-trace-functions", "journal": "Mathematical Programming", "publicationYear": 2026, "title": "Barriers for General Trace Functions", "topic": "Convex optimization and barriers", "publicationDate": "2026-01-08", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conclusion, final paragraph", "sourceQuote": "It remains an open question how to construct self-concordant barriers for the remaining scenarios", "problemStatement": "For positive-definite matrices X and Y, define Ψp,q,s(X,Y)=tr[(Y^(q/2)X^pY^(q/2))^s]. Construct self-concordant barriers for its jointly convex epigraph when −1≤q≤p≤0 and s>−1/(p+q), and when −1≤q≤0, 1≤p<2, (p,q)≠(1,−1), and s≥1/(p+q). These are the convexity regimes left outside the paper's compatibility constructions.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact problem wording, title and authors across publisher pages, arXiv, and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "trace functions", "self-concordant barrier", "matrix convexity", "compatibility" ], "literature": [], "source": { "title": "Operator convexity along lines, self-concordance, and sandwiched Rényi entropies", "authors": [ "Kerry He", "James Saunderson", "Hamza Fawzi" ], "doi": "10.1007/s10107-025-02314-0", "url": "https://doi.org/10.1007/s10107-025-02314-0" }, "number": 394 }, { "id": "mp-2026-matrix-compatibility-renyi-alpha", "journal": "Mathematical Programming", "publicationYear": 2026, "title": "Matrix Compatibility for Rényi Trace Functions", "topic": "Convex optimization and barriers", "publicationDate": "2026-01-08", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conclusion, Conjecture 1", "sourceQuote": "Based on numerical experiments, we believe this compatibility result extends to the situation", "problemStatement": "Prove that −Ψα is compatible on pairs of positive semidefinite n-by-n matrices with compatibility constant (2α−1)/3 for every α≥2 and every positive integer n. The scalar version is proved and its constant is tight, while numerical experiments motivate the matrix conjecture. Establishing it would yield self-concordant barriers beyond α=2, although with parameters that grow with α.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact problem wording, title and authors across publisher pages, arXiv, and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "compatibility", "positive semidefinite matrices", "Rényi entropy", "self-concordance" ], "literature": [], "source": { "title": "Operator convexity along lines, self-concordance, and sandwiched Rényi entropies", "authors": [ "Kerry He", "James Saunderson", "Hamza Fawzi" ], "doi": "10.1007/s10107-025-02314-0", "url": "https://doi.org/10.1007/s10107-025-02314-0" }, "number": 395 }, { "id": "mp-2026-optimal-renyi-barrier-alpha-above-two", "journal": "Mathematical Programming", "publicationYear": 2026, "title": "Optimal Rényi Barriers Beyond Alpha Two", "topic": "Convex optimization and barriers", "publicationDate": "2026-01-08", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conclusion, first open question", "sourceQuote": "with optimal barrier parameter 1+2n remains an open question.", "problemStatement": "Construct a self-concordant barrier with the optimal parameter 1+2n for the epigraph of the sandwiched Rényi trace function Ψα when α>2. The paper proves optimal barriers for 1≤α≤2, but operator convexity along lines fails beyond that range. A solution therefore needs a different compatibility, lifting, or direct barrier argument that does not worsen the parameter with α.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact problem wording, title and authors across publisher pages, arXiv, and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "self-concordant barrier", "sandwiched Rényi entropy", "interior-point methods", "operator convexity" ], "literature": [], "source": { "title": "Operator convexity along lines, self-concordance, and sandwiched Rényi entropies", "authors": [ "Kerry He", "James Saunderson", "Hamza Fawzi" ], "doi": "10.1007/s10107-025-02314-0", "url": "https://doi.org/10.1007/s10107-025-02314-0" }, "number": 396 }, { "id": "mor-2026-W7119488906_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2026, "title": "Scaling limit of dynamic risk measures with Wasserstein-type one-step penalties", "topic": "Dynamic Risk Measures", "publicationDate": "2026-01-08", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 7, Section 2.1, Remark 2.4.", "sourceQuote": "We consider the scaling limit of (2.1) for our choice alpha_1(·)=± W_1(·,P)^p as an interesting open question which we do not tackle here.", "problemStatement": "The paper studies nested distributional risk penalties for an N-period sampled martingale and identifies the continuous-time scaling of the associated adapted Wasserstein divergence. It does not identify the limit of the robust pricing or hedging functional built from that penalty. Find the correct normalization and continuous-time limit of the multi-period predictable-hedging problem when each one-step penalty is a power of Wasserstein-1 distance.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "dynamic risk measures", "scaling limits", "Wasserstein distance", "time-consistent divergences", "martingales", "convex duality" ], "literature": [], "source": { "title": "Specific Wasserstein Divergence Between Continuous Martingales", "authors": [ "Julio Backhoff-Veraguas", "Xin Zhang" ], "doi": "10.1287/moor.2025.0896", "url": "https://doi.org/10.1287/moor.2025.0896" }, "number": 397 }, { "id": "mp-2026-xfel-invariant-distribution-characterization", "journal": "Mathematical Programming", "publicationYear": 2026, "title": "Invariant Distributions for X-FEL Reconstruction", "topic": "Stochastic imaging and inverse problems", "publicationDate": "2026-01-12", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 4.1, invariant-distribution discussion", "sourceQuote": "It is an open question to determine exactly when and in what way this holds.", "problemStatement": "In the X-FEL model, x parametrizes an electron density and each photon-count image has an unknown orientation in SO(3); the objective is the empirical negative log likelihood whose component likelihoods average the forward model over orientations. Characterize when invariant laws of the randomized component-map iteration are supported only on global minimizers of that objective. Also determine when those minimizers, or the invariant laws concentrated on them, recover the true electron density up to the model's symmetries.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact problem wording, title and authors across publisher pages, arXiv, and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "X-FEL imaging", "invariant distribution", "Markov operator", "global optimum" ], "literature": [], "source": { "title": "Stochastic algorithms for large-scale composite optimization: the case of likelihood maximization for X-FEL imaging", "authors": [ "D. Russell Luke", "Steffen Schultze", "Helmut Grubmüller" ], "doi": "10.1007/s10107-025-02319-9", "url": "https://doi.org/10.1007/s10107-025-02319-9" }, "number": 398 }, { "id": "mp-2026-necessity-metric-subregularity-random-iterations", "journal": "Mathematical Programming", "publicationYear": 2026, "title": "Necessity of Metric Subregularity for Random Iterations", "topic": "Stochastic fixed-point methods", "publicationDate": "2026-01-12", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 3.2, after Proposition 1", "sourceQuote": "It is an open problem whether metric subregularity is necessary in the present setting", "problemStatement": "Let G be compact, let Ti:G→G be continuous random self-maps sampled independently, and let P be the induced Markov operator on probability measures with finite second moment. Determine whether quantitative Wasserstein convergence to inv(P) requires a metric-subregularity bound that controls distance to inv(P) by a nonlinear gauge of the paper's Markov transport discrepancy Ψ(μ). The unresolved necessity direction is for maps that are pointwise almost α-firmly nonexpansive in expectation, possibly with a positive violation parameter.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact problem wording, title and authors across publisher pages, arXiv, and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "metric subregularity", "random function iteration", "stochastic feasibility", "convergence" ], "literature": [], "source": { "title": "Stochastic algorithms for large-scale composite optimization: the case of likelihood maximization for X-FEL imaging", "authors": [ "D. Russell Luke", "Steffen Schultze", "Helmut Grubmüller" ], "doi": "10.1007/s10107-025-02319-9", "url": "https://doi.org/10.1007/s10107-025-02319-9" }, "number": 399 }, { "id": "mp-2026-xfel-scattering-events-orientation-weights", "journal": "Mathematical Programming", "publicationYear": 2026, "title": "Scattering Events Needed for Informative Orientations", "topic": "Stochastic imaging and inverse problems", "publicationDate": "2026-01-12", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Concluding remarks, X-FEL questions", "sourceQuote": "What is not clear is how many scattering events in each image are required", "problemStatement": "Quantify how many scattering events per X-FEL image are required for the posterior weights on molecular orientations to differ significantly from uniform weights. The reconstruction model can use a candidate solution to assign orientation probabilities before evaluating the rotational integral. The open task is a statistical or information-theoretic sample threshold tied to the physical forward model and a stated notion of significant nonuniformity.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact problem wording, title and authors across publisher pages, arXiv, and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "X-FEL imaging", "orientation inference", "sample complexity", "scattering events" ], "literature": [], "source": { "title": "Stochastic algorithms for large-scale composite optimization: the case of likelihood maximization for X-FEL imaging", "authors": [ "D. Russell Luke", "Steffen Schultze", "Helmut Grubmüller" ], "doi": "10.1007/s10107-025-02319-9", "url": "https://doi.org/10.1007/s10107-025-02319-9" }, "number": 400 }, { "id": "mp-2026-powell-exponential-trust-region", "journal": "Mathematical Programming", "publicationYear": 2026, "title": "Powell's Exponential Trust Region Conjecture", "topic": "Complexity of optimization algorithms", "publicationDate": "2026-01-26", "recordType": "reviewed", "status": "resolved", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Abstract, final contribution sentence", "sourceQuote": "a first step towards addressing a conjecture of Powell [38] that trust-region methods may require an exponential number of iterations in such a case.", "problemStatement": "Determine whether a standard trust-region method can require exponentially many iterations when its model Hessian approximations are allowed to grow without a uniform bound. The source establishes sharp polynomial deterioration when growth is controlled by a power p<1, leaving Powell's linear-growth case outside its theorem. A contemporaneous independent article appeared online one week earlier and proves a sharp exponential bound when p=1, so the current status is resolved.", "statusEvidence": "Establishes a sharp exponential evaluation-complexity bound when model Hessians grow linearly with the iteration count, confirming Powell's conjectured behavior.", "verificationNote": "This status is tied to the cited primary-literature result and the precise scope stated above.", "keywords": [ "trust-region method", "unbounded Hessian", "evaluation complexity", "Powell conjecture" ], "literature": [ { "title": "Complexity of trust-region methods in the presence of unbounded Hessian approximations", "url": "https://doi.org/10.1007/s10107-025-02287-0", "relation": "Establishes a sharp exponential evaluation-complexity bound when model Hessians grow linearly with the iteration count, confirming Powell's conjectured behavior." } ], "source": { "title": "Complexity of trust-region methods with potentially unbounded Hessian approximations for smooth and nonsmooth optimization", "authors": [ "Geoffroy Leconte", "Dominique Orban" ], "doi": "10.1007/s10107-025-02304-2", "url": "https://doi.org/10.1007/s10107-025-02304-2" }, "number": 401 }, { "id": "mp-2026-polynomial-simplex-pivot-rule", "journal": "Mathematical Programming", "publicationYear": 2026, "title": "Polynomial Simplex Pivot Rule", "topic": "Linear optimization and complexity", "publicationDate": "2026-01-28", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Abstract, opening paragraph", "sourceQuote": "The existence of a pivot rule for the simplex method that guarantees a polynomial run-time is a longstanding, fundamental open problem", "problemStatement": "Determine whether there is a pivot rule for the simplex method whose number of pivots is polynomially bounded for every linear program. The paper constructs exponential lower bounds covering every projection choice in broad shadow-rule variants and every norm in steepest-edge variants. Those exclusions sharpen the boundary but do not rule out a different pivot rule or provide a polynomial one.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact problem wording, title and authors across publisher pages, arXiv, and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "simplex method", "pivot rule", "polynomial time", "linear programming" ], "literature": [], "source": { "title": "Exponential Lower Bounds for Many Pivot Rules for the Simplex Method", "authors": [ "Alexander E. Black" ], "doi": "10.1007/s10107-026-02325-5", "url": "https://doi.org/10.1007/s10107-026-02325-5" }, "number": 402 }, { "id": "mor-2026-W7125931785_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2026, "title": "Dynamic regret bounds for distributionally robust stochastic model predictive control policies", "topic": "Distributionally Robust Mpc", "publicationDate": "2026-01-28", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 7, Remark 5; and page 13, Section 4 (Conclusions and Future Work), third bullet.", "sourceQuote": "Nevertheless, the dynamic regret analysis regarding the distributional robustness property remains an open question.", "problemStatement": "Stochastic model-predictive control repeatedly solves a finite look-ahead problem and applies its first control, producing dynamic regret relative to the full-horizon policy. A distributionally robust version uses an ambiguity set around a nominal disturbance law but evaluates performance under the unknown true law. Derive finite-time dynamic-regret bounds that expose horizon, ambiguity radius, and model-mismatch dependence.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "stochastic model predictive control", "dynamic regret", "distributional robustness", "wasserstein distance", "stochastic control", "multistage stochastic programming" ], "literature": [], "source": { "title": "Near-Optimal Performance of Stochastic Model Predictive Control", "authors": [ "Sungho Shin", "Sen Na", "Mihai Anitescu" ], "doi": "10.1287/moor.2023.0159", "url": "https://doi.org/10.1287/moor.2023.0159" }, "number": 403 }, { "id": "mor-2026-W7126221168_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2026, "title": "Approximate symmetric convex bodies by intersections of co-centered ellipsoids in high dimensions", "topic": "Convex Body Approximation", "publicationDate": "2026-01-30", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 7 (Future Research Directions), page 31 (PDF pagination in the provided text)", "sourceQuote": "how well can one approximate an n-dimensional symmetric convex body with a finite intersection of m (co-centered) ellipsoids", "problemStatement": "A generalized ellipsoid represents a symmetric convex set through a polynomial matrix inequality; the paper shows that intersections of m co-centered ellipsoids yield bounded-degree generalized ellipsoids. It also approximates arbitrary symmetric convex bodies indirectly through polytopes. Determine dimension-dependent approximation guarantees using intersections of co-centered ellipsoids directly, including the number of ellipsoids needed for a prescribed error.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "convex geometry", "symmetric convex bodies", "ellipsoid intersections", "geometric approximation", "John ellipsoid", "semidefinite representations" ], "literature": [], "source": { "title": "Generalized Ellipsoids", "authors": [ "Amir Ali Ahmadi", "Abraar Chaudhry", "Cemil Dibek" ], "doi": "10.1287/moor.2024.0643", "url": "https://doi.org/10.1287/moor.2024.0643" }, "number": 404 }, { "id": "focm-2026-critical-mass-structure-preserving-discretization", "journal": "Foundations of Computational Mathematics", "publicationYear": 2026, "title": "Convergent Explicit-Formula Scheme for Cubic Szegő", "topic": "Geometric numerical integration", "publicationDate": "2026-02-02", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Introduction, numerical approximation discussion", "sourceQuote": "Constructing a convergent structure-preserving approximation remains an interesting open problem.", "problemStatement": "For the cubic Szegő equation i u_t=Pi(|u|^2u) on the torus, analyze the paper's exact-in-time, Fourier-truncated explicit-formula approximation and prove that it converges as the truncation grows. The analogous argument works for the Benjamin-Ono and Calogero-Sutherland equations, but for Szegő the missing derivative operator prevents the norm-equivalence estimate used in those proofs. This is a convergence problem for a specified structure-preserving scheme, not a critical-mass existence problem.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "cubic Szego equation", "structure preservation", "critical dynamics", "numerical approximation" ], "literature": [], "source": { "title": "Spectrally Accurate Fully Discrete Schemes for Some Nonlocal and Nonlinear Integrable PDEs via Explicit Formulas", "authors": [ "Yvonne Alama Bronsard", "Xi Chen", "Matthieu Dolbeault" ], "doi": "10.1007/s10208-026-09740-2", "url": "https://doi.org/10.1007/s10208-026-09740-2", "preprintUrl": "https://arxiv.org/abs/2412.13480" }, "number": 405 }, { "id": "focm-2026-critical-mass-cubic-szego-global-existence", "journal": "Foundations of Computational Mathematics", "publicationYear": 2026, "title": "Critical-Mass Focusing Calogero–Sutherland Global Existence", "topic": "Nonlinear partial differential equations", "publicationDate": "2026-02-02", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Introduction", "sourceQuote": "global existence for initial data with critical or supercritical mass remains a compelling open problem", "problemStatement": "For the focusing Calogero-Sutherland derivative nonlinear Schrodinger equation i u_t+u_xx+(2/i)u partial_x Pi(|u|^2)=0 on the torus, the paper proves global well-posedness when ||u_0||^2_L2<1. Determine global existence versus finite-time breakdown for initial mass at least one, including the critical value. This problem concerns the focusing Calogero-Sutherland model, not the cubic Szegő equation treated separately in the same article.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "cubic Szego equation", "critical mass", "global existence", "blow-up" ], "literature": [], "source": { "title": "Spectrally Accurate Fully Discrete Schemes for Some Nonlocal and Nonlinear Integrable PDEs via Explicit Formulas", "authors": [ "Yvonne Alama Bronsard", "Xi Chen", "Matthieu Dolbeault" ], "doi": "10.1007/s10208-026-09740-2", "url": "https://doi.org/10.1007/s10208-026-09740-2", "preprintUrl": "https://arxiv.org/abs/2412.13480" }, "number": 406 }, { "id": "siopt-2026-062", "journal": "SIAM Journal on Optimization", "publicationYear": 2026, "title": "Phase synchronization SDP tightness threshold", "topic": "Semidefinite optimization", "publicationDate": "2026-02-02", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 4.3; cached preprint lines 1200-1212", "sourceQuote": "prove that the SDP fails to be tight when σ exceeds a certain threshold, or to establish a lower bound on the discrepancy n⁻²‖ẐSDP−ẑMLE(ẑMLE)ᴴ‖²_F.", "problemStatement": "In the phase-synchronization model Y=z*(z*)ᴴ+σW for n unit-modulus phases, W has independent standard complex-Gaussian off-diagonal entries and σ is the noise level. Let ẑMLE maximize ⟨Y,zzᴴ⟩ and let ẐSDP solve its positive-semidefinite relaxation; define the normalized discrepancy d_n=n⁻²‖ẐSDP−ẑMLE(ẑMLE)ᴴ‖²_F, so tightness means d_n=0. Earlier experiments give two different heuristic scales—the figure suggests tightness for σ≤√n/3, while its text suggests σ=√n/polylog(n)—whereas the rigorous sufficient scale is σ≲√(n/log n) and the paper bounds E[d_n] by C exp(−n/(8σ²))+2n⁻¹⁰ in the complementary high-noise regime. Prove SDP failure above a threshold or a matching lower bound for d_n; if the latter upper bound is sharp, σ≲√(n/log n) is necessary and sufficient.", "statusEvidence": "No forward-citing primary paper or exact-query result explicitly claiming a proof or counterexample was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "phase synchronization", "SDP tightness", "Burer-Monteiro", "noise threshold" ], "literature": [], "source": { "title": "Tightness of SDP and Burer–Monteiro Factorization for Phase Synchronization in a High-Noise Regime", "authors": [ "Anderson Ye Zhang" ], "doi": "10.1137/24m1689429", "url": "https://doi.org/10.1137/24m1689429", "preprintUrl": "https://arxiv.org/abs/2510.01522", "volume": "36", "issue": "1", "pages": "60-89" }, "number": 407 }, { "id": "mor-2026-W7127637708_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2026, "title": "Existence of non-deterministic equilibrium portfolio strategies under rank-dependent utility", "topic": "Time Inconsistent Portfolio Choice", "publicationDate": "2026-02-04", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 29, Section 5 (Concluding Remarks).", "sourceQuote": "We even do not know whether there is a non-deterministic equilibrium strategy.", "problemStatement": "A sophisticated equilibrium strategy (SES) is a time-consistent equilibrium policy for a continuous-time portfolio problem with rank-dependent utility. Known constructions in the paper are deterministic functions of time. Determine whether a genuinely state- or randomness-dependent SES exists in an incomplete Brownian market, or prove that every SES must be deterministic.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "rank-dependent utility", "time inconsistency", "equilibrium strategy", "incomplete markets", "stochastic control", "portfolio selection" ], "literature": [], "source": { "title": "Time-Consistent Portfolio Selection for Rank-Dependent Utilities in a Constrained Market", "authors": [ "Jiaqin Wei", "Jianming Xia", "Qian Zhao" ], "doi": "10.1287/moor.2024.0720", "url": "https://doi.org/10.1287/moor.2024.0720" }, "number": 408 }, { "id": "mor-2026-W7128591806_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2026, "title": "Determine the price of strategyproofness for social welfare with three resources", "topic": "Fair Division Mechanisms", "publicationDate": "2026-02-11", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 5 (Price of Strategyproofness), page 30 (discussion after Theorem 7) and page 33 (statement preceding Lemma 16).", "sourceQuote": "We leave the gap between 2 and 3 as an open question.", "problemStatement": "For three divisible resources and Leontief utilities, the price of strategyproofness compares the best social welfare attainable under sharing incentives, envy-freeness, Pareto optimality, and strategyproofness with the best fair allocation. The known lower and upper bounds place this worst-case ratio between 2 and 3. Compute its exact value by matching a mechanism with a lower-bound instance, or tighten either side of the interval.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "multi-resource allocation", "dominant resource fairness", "strategyproofness", "envy freeness", "share incentive", "Leontief preferences", "price of anarchy" ], "literature": [], "source": { "title": "Fair and Efficient Multi-resource Allocation for Cloud Computing: Beyond Dominant Resource Fairness", "authors": [ "Xiaohui Bei", "Zihao Li", "Junjie Luo" ], "doi": "10.1287/moor.2024.0714", "url": "https://doi.org/10.1287/moor.2024.0714" }, "number": 409 }, { "id": "mp-2026-exact-complexity-polytope-diameter", "journal": "Mathematical Programming", "publicationYear": 2026, "title": "Exact Complexity of Polytope Diameter", "topic": "Polyhedral complexity", "publicationDate": "2026-02-19", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Abstract, opening paragraph", "sourceQuote": "determining the precise complexity of computing the diameter of an input polytope remains a long-standing open problem.", "problemStatement": "Classify the exact computational complexity of computing the graph diameter of a polytope supplied by an inequality description. The paper proves strong NP-hardness even for bipartite perfect-matching polytopes and also establishes hardness for monotone and circuit variants. The remaining task is a matching upper classification or a sharper completeness result for the general diameter problem under a precise encoding model.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact problem wording, title and authors across publisher pages, arXiv, and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "polytope diameter", "NP-hardness", "perfect matching polytope", "complexity classification" ], "literature": [], "source": { "title": "Complexity of polytope diameters via perfect matchings", "authors": [ "Christian Nöbel", "Raphael Steiner" ], "doi": "10.1007/s10107-026-02324-6", "url": "https://doi.org/10.1007/s10107-026-02324-6" }, "number": 410 }, { "id": "siopt-2026-066", "journal": "SIAM Journal on Optimization", "publicationYear": 2026, "title": "Dual quantitative de finetti on sphere", "topic": "Polynomial optimization", "publicationDate": "2026-02-20", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 3; extracted preprint lines 664-671", "sourceQuote": "A natural question is whether there is an analog on the dual side of the quantitative result", "problemStatement": "Nonnegative polynomials on the sphere are dual to linear functionals admitting representing measures. A quantitative primal theorem approximates nonnegative forms by lower-degree SOS certificates. Prove a dual theorem that approximates a functional nonnegative on spherical sums of squares by a lower-degree functional with an actual representing measure, with explicit error rates.", "statusEvidence": "No forward-citing primary paper or exact-query result explicitly claiming a proof or counterexample was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "de Finetti theorem", "representing measure", "sphere", "sum of squares" ], "literature": [], "source": { "title": "Moment-Sos and Spectral Hierarchies for Polynomial Optimization on the Sphere and Quantum de Finetti Theorems", "authors": [ "Alexander Taveira Blomenhofer", "Monique Laurent" ], "doi": "10.1137/24m1717750", "url": "https://doi.org/10.1137/24m1717750", "preprintUrl": "https://arxiv.org/abs/2412.13191", "volume": "36", "issue": "1", "pages": "204-232" }, "number": 411 }, { "id": "siopt-2026-065", "journal": "SIAM Journal on Optimization", "publicationYear": 2026, "title": "Sphere SOS and spectral exact rates", "topic": "Polynomial optimization", "publicationDate": "2026-02-20", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conclusion; extracted preprint lines 1495-1502", "sourceQuote": "it is also an open question to settle what is the exact convergence rate for the sos lower bounds", "problemStatement": "For polynomial optimization on the sphere, moment-SOS and spectral hierarchies give lower bounds at relaxation order r. The paper proves an O(1/r^2) upper bound for SOS, and for the spectral hierarchy an O(1/r) upper bound with an Omega(1/r^2) example. Determine the exact worst-case rates for both hierarchies by closing these upper-lower gaps.", "statusEvidence": "No forward-citing primary paper or exact-query result explicitly claiming a proof or counterexample was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "sphere optimization", "moment-SOS", "spectral hierarchy", "convergence rate" ], "literature": [], "source": { "title": "Moment-Sos and Spectral Hierarchies for Polynomial Optimization on the Sphere and Quantum de Finetti Theorems", "authors": [ "Alexander Taveira Blomenhofer", "Monique Laurent" ], "doi": "10.1137/24m1717750", "url": "https://doi.org/10.1137/24m1717750", "preprintUrl": "https://arxiv.org/abs/2412.13191", "volume": "36", "issue": "1", "pages": "204-232" }, "number": 412 }, { "id": "focm-2026-rescaled-navier-stokes-steady-state-limit", "journal": "Foundations of Computational Mathematics", "publicationYear": 2026, "title": "Rescaled Navier Stokes Steady State Limit", "topic": "Fluid dynamics", "publicationDate": "2026-02-25", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 6, numerical blow-up discussion", "sourceQuote": "We conjecture that the dynamically rescaled solution converges to a steady state.", "problemStatement": "In the paper's generalized axisymmetric Navier-Stokes model, the viscosity and effective dimension depend dynamically on the solution, and a rescaling around the maximum vorticity produces profiles Psi, V, and Omega in similarity variables. Prove that these dynamically rescaled profiles converge to a steady state as rescaled time tends to infinity, or identify a nonsteady limiting behavior. Such convergence would validate the numerically observed self-similar singularity for this generalized model; it is not a claimed construction for the fixed physical three-dimensional Navier-Stokes equation.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "Navier-Stokes", "dynamic rescaling", "self-similar blow-up", "steady state" ], "literature": [], "source": { "title": "Nearly Self-similar Blowup of Generalized Axisymmetric Navier–Stokes Equations", "authors": [ "Thomas Y. Hou" ], "doi": "10.1007/s10208-026-09748-8", "url": "https://doi.org/10.1007/s10208-026-09748-8", "preprintUrl": "https://arxiv.org/abs/2405.10916" }, "number": 413 }, { "id": "mp-2026-compromise-regularizer-dro-equivalence", "journal": "Mathematical Programming", "publicationYear": 2026, "title": "Compromise Regularization and DRO", "topic": "Stochastic programming", "publicationDate": "2026-03-02", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conclusion, third enumerated question", "sourceQuote": "whether it is possible to connect our decision-based regularizer to a class of Distributionally Robust Optimization", "problemStatement": "Determine whether the decision-based quadratic regularizer used in compromise stochastic programs is exactly or approximately equivalent to a class of distributionally robust optimization models. Known results link divergence-based ambiguity to variance regularization and transport ambiguity to norm regularization. The task is to identify an ambiguity set and assumptions that recover the compromise objective, or prove that no such general representation exists.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact problem wording, title and authors across publisher pages, arXiv, and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "distributionally robust optimization", "regularization", "ambiguity set", "compromise decision" ], "literature": [], "source": { "title": "A Reliability Theory of Compromise Decisions for Large-Scale Stochastic Programs", "authors": [ "Shuotao Diao", "Suvrajeet Sen" ], "doi": "10.1007/s10107-026-02336-2", "url": "https://doi.org/10.1007/s10107-026-02336-2" }, "number": 414 }, { "id": "mp-2026-nonconvex-compromise-decision-formulation", "journal": "Mathematical Programming", "publicationYear": 2026, "title": "Nonconvex Compromise Decision Formulations", "topic": "Stochastic programming", "publicationDate": "2026-03-02", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conclusion, fourth enumerated question", "sourceQuote": "how one might reformulate compromise decision problems for nonconvex SP problems and problems with constraint sets", "problemStatement": "Formulate and analyze compromise-decision models for nonconvex stochastic programs and for feasible sets defined by nonlinear equalities or inequalities. The paper's reliability theory relies on convexity and sampled objective aggregation, so it does not directly cover neural-network-type landscapes or nonlinear constraints. A solution should specify an appropriate pessimistic-distance notion and prove reliability guarantees despite nonunique local solutions.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact problem wording, title and authors across publisher pages, arXiv, and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "nonconvex stochastic programming", "nonlinear constraints", "compromise decision", "reliability" ], "literature": [], "source": { "title": "A Reliability Theory of Compromise Decisions for Large-Scale Stochastic Programs", "authors": [ "Shuotao Diao", "Suvrajeet Sen" ], "doi": "10.1007/s10107-026-02336-2", "url": "https://doi.org/10.1007/s10107-026-02336-2" }, "number": 415 }, { "id": "mp-2026-optimal-replication-sample-allocation", "journal": "Mathematical Programming", "publicationYear": 2026, "title": "Optimal Replication and Sample Allocation", "topic": "Stochastic programming", "publicationDate": "2026-03-02", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conclusion, second enumerated question", "sourceQuote": "design an optimal strategy of choosing (m, n) for a given total sample size N.", "problemStatement": "Choose the number m of independent replications and the per-replication sample size n optimally when the total sampling budget N=mn is fixed in the compromise-decision framework. Existing expectation and variance bounds exhibit different dependences on m and n, but the source does not turn them into an optimal design. The task is to specify an objective reliability criterion and derive an allocation rule with finite-sample guarantees.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact problem wording, title and authors across publisher pages, arXiv, and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "sample allocation", "replication", "compromise decision", "finite-sample reliability" ], "literature": [], "source": { "title": "A Reliability Theory of Compromise Decisions for Large-Scale Stochastic Programs", "authors": [ "Shuotao Diao", "Suvrajeet Sen" ], "doi": "10.1007/s10107-026-02336-2", "url": "https://doi.org/10.1007/s10107-026-02336-2" }, "number": 416 }, { "id": "mp-2026-tight-reliability-bounds-compromise-decisions", "journal": "Mathematical Programming", "publicationYear": 2026, "title": "Tighter Reliability Bounds for Compromise Decisions", "topic": "Stochastic programming", "publicationDate": "2026-03-02", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conclusion, first enumerated question", "sourceQuote": "One open question is how to tighten the upper bounds on the reliability of the compromise decisions", "problemStatement": "A compromise decision aggregates m independently replicated sample-average objectives, each built from n observations, and compares its ε-optimal solution set A with the true ε-optimal set B. Tighten bounds on the expectation and variance of the directed pessimistic distance Δ(A,B)=sup_(a∈A) inf_(b∈B)‖a−b‖. The specific targets are the variance estimate and the SAA or cutting-plane reliability bounds for two-stage stochastic quadratic-quadratic programs, including any dependence on m and n.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact problem wording, title and authors across publisher pages, arXiv, and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "compromise decision", "reliability bound", "sample-average approximation", "pessimistic distance" ], "literature": [], "source": { "title": "A Reliability Theory of Compromise Decisions for Large-Scale Stochastic Programs", "authors": [ "Shuotao Diao", "Suvrajeet Sen" ], "doi": "10.1007/s10107-026-02336-2", "url": "https://doi.org/10.1007/s10107-026-02336-2" }, "number": 417 }, { "id": "mp-2026-stability-empirical-portfolio-ratios", "journal": "Mathematical Programming", "publicationYear": 2026, "title": "Distributional Stability of Portfolio Ratios", "topic": "Statistical robustness", "publicationDate": "2026-03-09", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Concluding remarks, second open issue", "sourceQuote": "we are unable to assert distributional stability of the empirical estimators of various ratios of interest in the context of portfolio theory", "problemStatement": "Prove quantitative distributional stability for empirical estimators of the Sharpe, Omega, and Rachev portfolio ratios under contaminated sampling distributions. The source's transfer principle applies to globally Lipschitz transformations of empirical means or covariance matrices, whereas these ratio maps can be non-Lipschitz near small denominators. The task is to find useful domain conditions, regularization, or weaker metrics that yield finite-sample stability bounds.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact problem wording, title and authors across publisher pages, arXiv, and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "Sharpe ratio", "Omega ratio", "Rachev ratio", "distributional stability" ], "literature": [], "source": { "title": "Distributional stability of sparse inverse covariance matrix estimators", "authors": [ "Renjie Chen", "Huifu Xu", "Henryk Zähle" ], "doi": "10.1007/s10107-026-02339-z", "url": "https://doi.org/10.1007/s10107-026-02339-z" }, "number": 418 }, { "id": "mp-2026-stability-alternative-precision-penalties", "journal": "Mathematical Programming", "publicationYear": 2026, "title": "Stability Under Alternative Precision Penalties", "topic": "Statistical robustness", "publicationDate": "2026-03-09", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Concluding remarks, first open issue", "sourceQuote": "we are unable to assert Lipschitz continuity of the minimizer of the optimization problem underlying the estimator", "problemStatement": "Establish quantitative distributional stability for sparse inverse-covariance estimators when the entrywise ℓ1 penalty is replaced by another sparsity-inducing norm, such as the off-diagonal ℓ1 norm discussed in the paper. The existing proof depends on global Lipschitz continuity of the optimization minimizer as a function of the empirical covariance. The task is to prove an adequate continuity bound or derive stability by a different route.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact problem wording, title and authors across publisher pages, arXiv, and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "precision matrix", "distributional stability", "sparsity penalty", "Lipschitz continuity" ], "literature": [], "source": { "title": "Distributional stability of sparse inverse covariance matrix estimators", "authors": [ "Renjie Chen", "Huifu Xu", "Henryk Zähle" ], "doi": "10.1007/s10107-026-02339-z", "url": "https://doi.org/10.1007/s10107-026-02339-z" }, "number": 419 }, { "id": "mp-2026-subset-sum-n-plus-w", "journal": "Mathematical Programming", "publicationYear": 2026, "title": "Linear n Plus w Subset Sum", "topic": "Algorithms and complexity", "publicationDate": "2026-03-10", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Abstract, stated open question", "sourceQuote": "can Subset Sum be solved in O(n + w) time?", "problemStatement": "Determine whether Subset Sum on n positive integers, with largest item value w, can be decided in O(n+w) time in the standard word-RAM model. Bringmann's earlier bound depends on the target t, while the source improves the pseudopolynomial running time to soft-O(n+√(wt)). The precise task is to remove the remaining dependence on t and the polylogarithmic overhead, or prove a conditional lower bound ruling that out.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact problem wording, title and authors across publisher pages, arXiv, and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "Subset Sum", "pseudopolynomial algorithm", "word-RAM", "running time" ], "literature": [], "source": { "title": "An Improved Pseudopolynomial Time Algorithm for Subset Sum", "authors": [ "Lin Chen", "Jiayi Lian", "Yuchen Mao", "Guochuan Zhang" ], "doi": "10.1007/s10107-026-02348-y", "url": "https://doi.org/10.1007/s10107-026-02348-y" }, "number": 420 }, { "id": "mor-2026-W7135205807_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2026, "title": "Characterize DDPM low-dimensional adaptivity under Wasserstein or total variation metrics", "topic": "Diffusion Model Sampling", "publicationDate": "2026-03-13", "recordType": "reviewed", "status": "partial_progress", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 6 (Discussion), page 16.", "sourceQuote": "the DDPM sampler adapts to unknown low dimensionality when other metrics (e.g. the total-variation distance or the Wasserstein distance) are used to measure distributional discrepancy.", "problemStatement": "A denoising diffusion probabilistic model (DDPM) samples in ambient dimension d, while the target distribution may have much smaller intrinsic dimension k. The source establishes dimension-adaptive guarantees in Kullback–Leibler divergence. Determine whether iteration complexity can scale with k rather than d when error is measured in total variation or 2-Wasserstein distance; later work settles the total-variation part, while the Wasserstein part remains open.", "statusEvidence": "The total-variation part is settled with intrinsic-dimension-adaptive guarantees; the Wasserstein part remains open.", "verificationNote": "This status is tied to the cited primary-literature result and the precise scope stated above.", "keywords": [ "diffusion models", "DDPM", "low-dimensional structure", "Wasserstein distance", "total variation distance", "convergence rates", "intrinsic dimension" ], "literature": [ { "title": "Low-Dimensional Adaptation of Diffusion Models: Convergence in Total Variation", "url": "https://proceedings.mlr.press/v291/liang25a.html", "relation": "The total-variation part is settled with intrinsic-dimension-adaptive guarantees; the Wasserstein part remains open." } ], "source": { "title": "Denoising Diffusion Probabilistic Models Are Optimally Adaptive to Unknown Low Dimensionality", "authors": [ "Zhihan Huang", "Yuting Wei", "Yuxin Chen" ], "doi": "10.1287/moor.2024.0769", "url": "https://doi.org/10.1287/moor.2024.0769" }, "number": 421 }, { "id": "mp-2026-remove-log-n-dual-treedepth-ip", "journal": "Mathematical Programming", "publicationYear": 2026, "title": "Remove the Log n Factor in Dual Treedepth IP", "topic": "Integer programming", "publicationDate": "2026-03-18", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 6.1, first open problem", "sourceQuote": "is it possible to eliminate the log n factor in the dual algorithm", "problemStatement": "Determine whether the log n factor can be removed from the algorithm for separable convex integer programs parameterized by dual treedepth and the largest matrix coefficient. The source gives time g(tdD(A),‖A‖∞) n log n log‖u−l‖∞, while the comparison-oracle lower bound lacks log n. The task is either a faster algorithm for some or all bounded-parameter classes or a strengthened lower bound matching the extra factor.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact problem wording, title and authors across publisher pages, arXiv, and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "dual treedepth", "separable convex integer programming", "parameterized complexity", "lower bound" ], "literature": [], "source": { "title": "(Near)-Optimal algorithms for sparse separable convex integer programs", "authors": [ "Christoph Hunkenschröder", "Martin Koutecký", "Asaf Levin", "Tung Anh Vu" ], "doi": "10.1007/s10107-026-02341-5", "url": "https://doi.org/10.1007/s10107-026-02341-5" }, "number": 422 }, { "id": "mp-2026-strongly-polynomial-nonlinear-objectives-ip", "journal": "Mathematical Programming", "publicationYear": 2026, "title": "Strongly Polynomial Nonlinear Sparse IP", "topic": "Integer programming", "publicationDate": "2026-03-18", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 6.1, second open problem", "sourceQuote": "for which classes of objectives beyond linear can one design faster, possibly strongly polynomial algorithms?", "problemStatement": "Classify the nonlinear separable convex objectives for which sparse structured integer programs admit faster or strongly polynomial algorithms outside the comparison-oracle lower-bound model. Linear objectives already permit stronger results, while a general nonlinear oracle entails an information-theoretic dependence on the variable ranges. The task is to identify objective representations or regularity classes that bypass that barrier and prove corresponding algorithms.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact problem wording, title and authors across publisher pages, arXiv, and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "strongly polynomial algorithm", "separable convex objective", "sparse integer programming", "oracle model" ], "literature": [], "source": { "title": "(Near)-Optimal algorithms for sparse separable convex integer programs", "authors": [ "Christoph Hunkenschröder", "Martin Koutecký", "Asaf Levin", "Tung Anh Vu" ], "doi": "10.1007/s10107-026-02341-5", "url": "https://doi.org/10.1007/s10107-026-02341-5" }, "number": 423 }, { "id": "siopt-2026-046", "journal": "SIAM Journal on Optimization", "publicationYear": 2026, "title": "Rigid alignment converses and noisy overlap", "topic": "Geometric optimization", "publicationDate": "2026-03-18", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conclusion, items 1-3; extracted preprint lines 1373-1384", "sourceQuote": "necessary and sufficient conditions on the overlapping structure of m > 2 noisy views", "problemStatement": "The paper studies patch alignment through a stress-like matrix L(S), giving sufficient algebraic and rigidity conditions for unique and nondegenerate alignments. It conjectures converses linking uniqueness to global rigidity and noisy nondegeneracy to infinitesimal rigidity. Prove those equivalences and characterize the overlap graphs of more than two noisy views that guarantee nondegenerate or unique optimal alignment.", "statusEvidence": "No forward-citing primary paper or exact-query result explicitly claiming a proof or counterexample was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "rigid alignment", "patch framework", "global rigidity", "overlap graph" ], "literature": [], "source": { "title": "Non-degenerate Rigid Alignment in a Patch Framework", "authors": [ "Dhruv Kohli", "Gal Mishne", "Alexander Cloninger" ], "doi": "10.1137/23m1593280", "url": "https://doi.org/10.1137/23m1593280", "preprintUrl": "https://arxiv.org/abs/2303.11620", "volume": "36", "issue": "1", "pages": "434-465" }, "number": 424 }, { "id": "mor-2026-W7140189588_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2026, "title": "Establish smoothing and gradient consistency for value functions with x-dependent constraints", "topic": "Bilevel Optimization", "publicationDate": "2026-03-24", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 30, Section 5 (Concluding Remarks and Future Directions).", "sourceQuote": "whether the resulting approximation qualifies as a smoothing family in the sense of Definition 2.5 and whether it satisfies gradient consistency.", "problemStatement": "In bilevel optimization, the lower-level value v(x) minimizes g(x,y) over an x-dependent feasible set Y(x). For a fixed feasible set, the paper constructs differentiable approximations that converge to v and whose gradient limits lie in the Clarke subdifferential of v. Extend these smoothing and gradient-consistency guarantees to x-dependent sets by giving verifiable conditions and a concrete approximation construction.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "bilevel optimization", "value function", "smoothing methods", "gradient consistency", "parametric constraints", "Clarke subdifferential" ], "literature": [], "source": { "title": "Theoretical Smoothing Frameworks for Nonsmooth Simple Bilevel Problems", "authors": [ "Jan Harold Alcantara", "Akiko Takeda" ], "doi": "10.1287/moor.2024.0405", "url": "https://doi.org/10.1287/moor.2024.0405" }, "number": 425 }, { "id": "mp-2026-unboundedness-pessimistic-multilevel-complexity", "journal": "Mathematical Programming", "publicationYear": 2026, "title": "Complexity of Pessimistic Multilevel Unboundedness", "topic": "Bilevel and multilevel optimization", "publicationDate": "2026-03-26", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conclusion, first future-research sentence", "sourceQuote": "deriving the computational complexity of deciding unboundedness of mixed-integer pessimistic multilevel problems, or of multi-follower bilevel problems.", "problemStatement": "Determine the computational complexity of deciding unboundedness for mixed-integer pessimistic multilevel optimization and for bilevel models with multiple followers. The paper establishes strong NP-completeness for linear bilevel unboundedness and polynomial-hierarchy hardness as levels and integrality are added, but those two model families are not classified. The task is to give reductions and matching membership results under explicit follower-response conventions.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact problem wording, title and authors across publisher pages, arXiv, and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "unboundedness", "pessimistic bilevel optimization", "multiple followers", "polynomial hierarchy" ], "literature": [], "source": { "title": "Unboundedness in Bilevel Optimization", "authors": [ "Bárbara Rodrigues", "Margarida Carvalho", "Miguel F. Anjos", "Nagisa Sugishita" ], "doi": "10.1007/s10107-026-02344-2", "url": "https://doi.org/10.1007/s10107-026-02344-2" }, "number": 426 }, { "id": "mor-2026-W7143448186_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2026, "title": "Equilibrium-equivalent reduction of zero-sum games to cone-leveled strategy games", "topic": "Zero Sum Games", "publicationDate": "2026-03-30", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 7 (Future directions), page 34.", "sourceQuote": "primal-dual pair of conic linear programs, the following question naturally arises: When is a two-player zero-sum game equilibrium-equivalent to a game with cone-leveled strategy sets?", "problemStatement": "Cone-leveled zero-sum games have strategy sets that are level slices of convex cones and bilinear payoffs, so their equilibria correspond to conic primal-dual programs. The paper establishes this representation for that class. Determine whether a broader zero-sum game can be transformed into an equilibrium-equivalent cone-leveled game, and characterize exactly when such a reduction exists.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "zero-sum games", "Nash equilibrium", "equilibrium equivalence", "cone-leveled sets", "conic programming", "Banach spaces" ], "literature": [], "source": { "title": "On the Equivalence of Zero-Sum Games and Conic Programs", "authors": [ "Nikos Dimou" ], "doi": "10.1287/moor.2025.0926", "url": "https://doi.org/10.1287/moor.2025.0926", "preprintUrl": "https://arxiv.org/pdf/2302.03066" }, "number": 427 }, { "id": "focm-2026-coercive-morse-worst-case-accelerated-gradient", "journal": "Foundations of Computational Mathematics", "publicationYear": 2026, "title": "Coercive Morse Worst Case Accelerated Gradient", "topic": "Nonconvex optimization", "publicationDate": "2026-04-01", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Introduction, worst-case construction discussion", "sourceQuote": "Adapting the construction to coercive Morse functions remains an interesting and nontrivial open question.", "problemStatement": "For the accelerated iteration y_k=x_k+beta_k(x_k-x_(k-1)) and x_(k+1)=x_k-h grad f(y_k), construct a coercive Morse objective with locally Lipschitz Hessian that realizes exponentially small worst-case gradient norms along the iterates. Coercive means bounded sublevel sets, and Morse means every critical point has nonsingular Hessian. Adapting the known noncoercive adversarial construction would give an explicit sharp ambient-dimension dependence for the paper's finite-time small-gradient bound.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "accelerated gradient", "Morse functions", "worst-case complexity", "coercivity" ], "literature": [], "source": { "title": "Accelerated Gradient Methods for Nonconvex Optimization: Escape Trajectories From Strict Saddle Points and Convergence to Local Minima", "authors": [ "Rishabh Dixit", "Mert Gürbüzbalaban", "Waheed U. Bajwa" ], "doi": "10.1007/s10208-026-09745-x", "url": "https://doi.org/10.1007/s10208-026-09745-x" }, "number": 428 }, { "id": "focm-2026-momentum-lyapunov-beta-large", "journal": "Foundations of Computational Mathematics", "publicationYear": 2026, "title": "Momentum Lyapunov Beta Large", "topic": "Nonconvex optimization", "publicationDate": "2026-04-01", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 7.2", "sourceQuote": "constructing such Lyapunov functions for beta in (1/sqrt(2),1] remains an open problem", "problemStatement": "Construct Lyapunov functions proving convergence of accelerated gradient methods with momentum beta between 1/sqrt(2) and one on general gradient-Lipschitz nonconvex objectives. The argument should retain the larger momentum's improved saddle-escape behavior.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "momentum methods", "Lyapunov function", "accelerated gradient", "saddle escape" ], "literature": [], "source": { "title": "Accelerated Gradient Methods for Nonconvex Optimization: Escape Trajectories From Strict Saddle Points and Convergence to Local Minima", "authors": [ "Rishabh Dixit", "Mert Gürbüzbalaban", "Waheed U. Bajwa" ], "doi": "10.1007/s10208-026-09745-x", "url": "https://doi.org/10.1007/s10208-026-09745-x" }, "number": 429 }, { "id": "focm-2026-symmetric-resonance-integrator-long-time-behavior", "journal": "Foundations of Computational Mathematics", "publicationYear": 2026, "title": "Symmetric Resonance Integrator Long Time Behavior", "topic": "Geometric numerical integration", "publicationDate": "2026-04-06", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Introduction", "sourceQuote": "rigorous understanding of their long-time behavior is at the moment a completely open problem", "problemStatement": "For periodic dispersive equations i u_t+L(nabla)u=|nabla|^alpha p(u,conjugate(u)), the paper derives implicit and explicit symmetric resonance integrators from paired decorated-forest expansions that retain the dominant oscillatory phases. Prove long-time error bounds and determine which invariants these specific schemes conserve or nearly conserve for the nonlinear Schrodinger and KdV examples. The theory should explain the observed long-time mass and energy behavior rather than address an unspecified generic structure-preserving method.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "resonance integrators", "long-time behavior", "symmetry", "structure preservation" ], "literature": [], "source": { "title": "Symmetric Resonance Based Integrators and Forest Formulae", "authors": [ "Yvonne Alama Bronsard", "Yvain Bruned", "Georg Maierhofer", "Katharina Schratz" ], "doi": "10.1007/s10208-026-09742-0", "url": "https://doi.org/10.1007/s10208-026-09742-0", "preprintUrl": "https://arxiv.org/abs/2305.16737" }, "number": 430 }, { "id": "mor-2026-W7150839694_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2026, "title": "Existence of interior discount factor where decreasing and increasing value trajectories coincide", "topic": "Dynamic Persuasion", "publicationDate": "2026-04-06", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 3.2.1 (\"On the Analytic Behavior of Two Trajectories\"), page 12 (PDF pagination shown as 12/51).", "sourceQuote": "existence of an example for Case B in which 0 < delta_0 < 1 (where delta_0 was described in Corollary 1) remains an open question.", "problemStatement": "The discounted stochastic game produces one value trajectory that decreases with the discount factor and another that increases, and both agree in the patient limit. If they meet at an interior discount factor, the paper shows both trajectories must remain constant thereafter. Determine whether an irreducible finite-state example with such an interior meeting point exists, or prove it impossible.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "dynamic persuasion", "Markov chain", "discounted value", "monotone trajectories", "concavification", "invariant distribution" ], "literature": [], "source": { "title": "On the Monotonicity and Rate of Convergence of the Markovian Persuasion Value", "authors": [ "Dimitry Shaiderman" ], "doi": "10.1287/moor.2023.0296", "url": "https://doi.org/10.1287/moor.2023.0296" }, "number": 431 }, { "id": "mp-2026-accuracy-independent-gradient-method-parameters", "journal": "Mathematical Programming", "publicationYear": 2026, "title": "Accuracy Independent Gradient Method Parameters", "topic": "First-order optimization", "publicationDate": "2026-04-07", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Concluding remarks, second open question", "sourceQuote": "it will be interesting to study whether an algorithm could be developed in which the parameters do not depend", "problemStatement": "For a smooth convex objective f with L-Lipschitz gradient, design an accumulative-regularization method that reaches ‖∇f(x)‖≤ε without using ε in its regularization weights, stage lengths, or other internal parameters. The paper's fallback repeatedly calls an accuracy-dependent routine with a temporary target equal to a fraction of the current gradient norm. Replace that outer calling scheme by genuinely accuracy-independent parameters while retaining the same optimal worst-case gradient complexity.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact problem wording, title and authors across publisher pages, arXiv, and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "parameter-free optimization", "target accuracy", "gradient minimization", "adaptive method" ], "literature": [], "source": { "title": "Optimal and parameter-free gradient minimization methods for convex and nonconvex optimization", "authors": [ "Guanghui Lan", "Yuyuan Ouyang", "Zhe Zhang" ], "doi": "10.1007/s10107-026-02352-2", "url": "https://doi.org/10.1007/s10107-026-02352-2" }, "number": 432 }, { "id": "mp-2026-single-loop-accumulative-regularization", "journal": "Mathematical Programming", "publicationYear": 2026, "title": "Single Loop Accumulative Regularization", "topic": "First-order optimization", "publicationDate": "2026-04-07", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Concluding remarks, first open question", "sourceQuote": "whether the algorithm framework can be further simplified, e.g., to a single-loop implementation.", "problemStatement": "Simplify the accumulative-regularization framework for optimal gradient-norm minimization to a single-loop implementation. The source obtains optimal convex and strongly convex guarantees by solving a sequence of proximal subproblems, and its parameter-free variants repeatedly call AR, SCAR, or related routines. The task is one iterative loop with the same worst-case gradient complexity and without hidden nested solves.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact problem wording, title and authors across publisher pages, arXiv, and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "accumulative regularization", "single-loop method", "gradient norm", "optimal complexity" ], "literature": [], "source": { "title": "Optimal and parameter-free gradient minimization methods for convex and nonconvex optimization", "authors": [ "Guanghui Lan", "Yuyuan Ouyang", "Zhe Zhang" ], "doi": "10.1007/s10107-026-02352-2", "url": "https://doi.org/10.1007/s10107-026-02352-2" }, "number": 433 }, { "id": "mor-2026-W4404530034_p0", "journal": "Mathematics of Operations Research", "publicationYear": 2026, "title": "Establish a central limit theorem for stochastic mirror descent risk budgeting portfolios", "topic": "Stochastic Approximation", "publicationDate": "2026-04-07", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conclusion (page 25 of 34)", "sourceQuote": "Potential directions for future research include investigating asymptotic error properties by establishing a central limit theorem for the stochastic MD algorithm presented here.", "problemStatement": "Stochastic mirror descent computes a risk-budgeting portfolio from noisy samples and the paper proves almost-sure convergence plus nonasymptotic bounds for weighted averages. It does not describe the limiting fluctuations of the iterates. Establish a central limit theorem, identifying the normalization and asymptotic covariance under verifiable step-size and moment conditions.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "risk budgeting", "stochastic mirror descent", "central limit theorem", "asymptotic normality", "Bregman divergence", "tamed gradients" ], "literature": [], "source": { "title": "Mirror Descent Algorithms for Risk Budgeting Portfolios", "authors": [ "Martín Arnaiz Iglesias", "Adil Rengim Cetingoz", "Noufel Frikha" ], "doi": "10.1287/moor.2024.0847", "url": "https://doi.org/10.1287/moor.2024.0847", "preprintUrl": "https://arxiv.org/pdf/2411.12323" }, "number": 434 }, { "id": "mp-2026-standard-frank-wolfe-linear-rank-two", "journal": "Mathematical Programming", "publicationYear": 2026, "title": "Linear Frank Wolfe Convergence Above Rank One", "topic": "Semidefinite optimization", "publicationDate": "2026-04-13", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Numerical experiments, comparison with standard Frank-Wolfe", "sourceQuote": "while it remained unknown if this is also the case", "problemStatement": "Determine rigorously whether standard line-search Frank-Wolfe converges linearly on smooth convex spectrahedron problems under strict complementarity when the optimal rank r* is at least two. Linear convergence is proved for r*=1, while the source's experiments show sublinear behavior for higher rank even though its new randomized method remains linear. The task is a theorem with sufficient conditions or a strict-complementarity counterexample.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact problem wording, title and authors across publisher pages, arXiv, and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "Frank-Wolfe method", "linear convergence", "optimal rank", "strict complementarity" ], "literature": [], "source": { "title": "A Randomized Linearly Convergent Frank-Wolfe-type Method for Smooth Convex Minimization over the Spectrahedron", "authors": [ "Dan Garber" ], "doi": "10.1007/s10107-026-02343-3", "url": "https://doi.org/10.1007/s10107-026-02343-3" }, "number": 435 }, { "id": "mp-2026-strict-complementarity-all-spectrahedron-optima", "journal": "Mathematical Programming", "publicationYear": 2026, "title": "Strict Complementarity at Every Spectrahedron Optimum", "topic": "Semidefinite optimization", "publicationDate": "2026-04-13", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Introduction, paragraph on nonunique optima", "sourceQuote": "It remains open whether this is mandatory.", "problemStatement": "Decide whether linear convergence of the paper's randomized Frank-Wolfe-type method on a spectrahedron truly requires strict complementarity at every optimal solution when the optimum is not unique. The analysis assumes all optima satisfy the condition, which forces them to share a rank, while earlier work usually imposed uniqueness. The task is either a proof under a weaker local condition or a counterexample showing the global assumption is necessary.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact problem wording, title and authors across publisher pages, arXiv, and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "Frank-Wolfe method", "spectrahedron", "strict complementarity", "nonunique optimum" ], "literature": [], "source": { "title": "A Randomized Linearly Convergent Frank-Wolfe-type Method for Smooth Convex Minimization over the Spectrahedron", "authors": [ "Dan Garber" ], "doi": "10.1007/s10107-026-02343-3", "url": "https://doi.org/10.1007/s10107-026-02343-3" }, "number": 436 }, { "id": "mp-2026-objective-value-clipping-ssqp", "journal": "Mathematical Programming", "publicationYear": 2026, "title": "Clipping Noisy Objective Values in SSQP", "topic": "Stochastic constrained optimization", "publicationDate": "2026-04-13", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 3, Remark 3.5", "sourceQuote": "Whether clipping- or normalization-type ideas can be effectively applied to objective value estimates remains an open question.", "problemStatement": "Develop clipping or normalization mechanisms for noisy objective-value estimates in trust-region stochastic sequential quadratic programming. Heavy-tailed stochastic-gradient methods clip gradients, but the acceptance test here depends on biased and irreducibly noisy differences of objective estimates, so those arguments do not transfer directly. The task is an estimator and analysis that preserves reliable step acceptance and improves tail or moment requirements.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact problem wording, title and authors across publisher pages, arXiv, and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "objective-value oracle", "clipping", "normalization", "stochastic SQP" ], "literature": [], "source": { "title": "High Probability Complexity Bounds of Trust-Region Stochastic Sequential Quadratic Programming with Heavy-Tailed Noise", "authors": [ "Yuchen Fang", "Javad Lavaei", "Sen Na" ], "doi": "10.1007/s10107-026-02357-x", "url": "https://doi.org/10.1007/s10107-026-02357-x" }, "number": 437 }, { "id": "mp-2026-moment-free-heavy-tail-ssqp-bounds", "journal": "Mathematical Programming", "publicationYear": 2026, "title": "Moment Free Heavy Tailed SSQP Bounds", "topic": "Stochastic constrained optimization", "publicationDate": "2026-04-13", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 3, heavy-tailed oracle discussion", "sourceQuote": "whether one can strengthen our high-probability complexity bounds by completely removing the moment conditions", "problemStatement": "Prove high-probability first- and second-order complexity bounds for trust-region stochastic sequential quadratic programming without moment assumptions on the noisy objective-value oracle. The paper handles finite 1+δ moments and obtains only a liminf-type almost-sure consequence when irreducible noise vanishes. The target is comparable finite-time bounds, and ideally lim-type convergence, under substantially weaker zeroth-order noise assumptions.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact problem wording, title and authors across publisher pages, arXiv, and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "stochastic SQP", "heavy-tailed noise", "high-probability complexity", "moment assumptions" ], "literature": [], "source": { "title": "High Probability Complexity Bounds of Trust-Region Stochastic Sequential Quadratic Programming with Heavy-Tailed Noise", "authors": [ "Yuchen Fang", "Javad Lavaei", "Sen Na" ], "doi": "10.1007/s10107-026-02357-x", "url": "https://doi.org/10.1007/s10107-026-02357-x" }, "number": 438 }, { "id": "mor-2026-manual-10.1287-moor.2024.0582-1", "journal": "Mathematics of Operations Research", "publicationYear": 2026, "title": "Restrict the P2GDR rank loop while preserving Bouligand stationarity", "topic": "Low-Rank Optimization", "publicationDate": "2026-04-20", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 19, discussion after the P2GDR complexity table", "sourceQuote": "whether it is possible to restrict (conditionally or not) the range of values in the for-loop while preserving the B-stationarity", "problemStatement": "P2GDR is a low-rank optimization method that tries several candidate ranks and whose accumulation points are Bouligand stationary. Its rank loop may range over every value from 0 to the ambient rank bound, creating potentially large overhead. Determine whether the candidate-rank range can be reduced while retaining the Bouligand-stationarity guarantee, or prove that no such restriction is possible.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "low-rank optimization", "Bouligand stationarity", "projected gradient", "rank adaptation" ], "literature": [], "source": { "title": "Low-Rank Optimization Methods Based on Projected Projected-Gradient Descent That Accumulate at Bouligand Stationary Points", "authors": [ "Guillaume Olikier", "Kyle A. Gallivan", "P.-A. Absil" ], "doi": "10.1287/moor.2024.0582", "url": "https://doi.org/10.1287/moor.2024.0582", "preprintUrl": "https://arxiv.org/pdf/2201.03962" }, "number": 439 }, { "id": "mor-2026-manual-10.1287-moor.2025.0939-2", "journal": "Mathematics of Operations Research", "publicationYear": 2026, "title": "Limit behavior of operator Sinkhorn scaling on unscalable instances", "topic": "Matrix Scaling", "publicationDate": "2026-04-24", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 31, discussion of the operator Sinkhorn limit", "sourceQuote": "it is reasonable to conjecture that it oscillates between orbits", "problemStatement": "For an unscalable tuple of matrices, the operator Sinkhorn algorithm alternates left and right normalizations and is expected to reveal the generalized Dulmage–Mendelsohn block structure. The paper proves convergence for a related gradient-descent scaling sequence and identifies two limiting unitary orbits. Prove that the operator Sinkhorn sequence oscillates between those two orbits, or characterize its actual limit set.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "operator Sinkhorn", "operator scaling", "Dulmage–Mendelsohn decomposition", "limit set" ], "literature": [], "source": { "title": "Gradient Descent for Unbounded Convex Functions on Hadamard Manifolds and Its Applications to Scaling Problems", "authors": [ "Hiroshi Hirai", "Keiya Sakabe" ], "doi": "10.1287/moor.2025.0939", "url": "https://doi.org/10.1287/moor.2025.0939", "preprintUrl": "https://arxiv.org/pdf/2404.09746" }, "number": 440 }, { "id": "mor-2026-manual-10.1287-moor.2025.0939-1", "journal": "Mathematics of Operations Research", "publicationYear": 2026, "title": "Nonexpansiveness of convex gradient-flow semigroups on Hadamard manifolds", "topic": "Riemannian Optimization", "publicationDate": "2026-04-24", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 12, discussion of equation (3.13)", "sourceQuote": "to establish the contraction property d(phi_i(x),phi_i(y))≤ d(x,y) for the semigroup phi_i of (3.11)", "problemStatement": "For an L-smooth convex function on a Hadamard manifold, one gradient-descent step maps x to expₓ(−∇f(x)/L), and φᵢ denotes its i-fold iterate. In Euclidean space every φᵢ is nonexpansive in distance. Determine whether d(φᵢ(x),φᵢ(y))≤d(x,y) holds for all points and every positive integer i on an arbitrary Hadamard manifold.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "Hadamard manifold", "gradient flow", "nonexpansive semigroup", "geodesic convexity" ], "literature": [], "source": { "title": "Gradient Descent for Unbounded Convex Functions on Hadamard Manifolds and Its Applications to Scaling Problems", "authors": [ "Hiroshi Hirai", "Keiya Sakabe" ], "doi": "10.1287/moor.2025.0939", "url": "https://doi.org/10.1287/moor.2025.0939", "preprintUrl": "https://arxiv.org/pdf/2404.09746" }, "number": 441 }, { "id": "mp-2026-integral-online-hypergraph-better-than-greedy", "journal": "Mathematical Programming", "publicationYear": 2026, "title": "Beat Greedy for Integral Online 3-Hypergraph Matching", "topic": "Online algorithms", "publicationDate": "2026-04-27", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Introduction and concluding remarks", "sourceQuote": "whether there exists an integral algorithm better than greedy on 3-uniform hypergraphs.", "problemStatement": "Find an online integral matching algorithm under vertex arrivals on general 3-uniform hypergraphs with competitive ratio strictly greater than the greedy ratio 1/3. The source settles the fractional problem at (e−1)/(e+1), which is also an upper bound for randomized integral algorithms, and improves on greedy only under bounded online degree. The open task is any general-case ratio in the gap (1/3,(e−1)/(e+1)] or a matching impossibility result.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact problem wording, title and authors across publisher pages, arXiv, and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "online matching", "3-uniform hypergraph", "competitive ratio", "randomized algorithm" ], "literature": [], "source": { "title": "Online matching on 3-uniform hypergraphs", "authors": [ "Sander Borst", "Danish Kashaev", "Zhuan Khye Koh" ], "doi": "10.1007/s10107-026-02360-2", "url": "https://doi.org/10.1007/s10107-026-02360-2" }, "number": 442 }, { "id": "focm-2026-integral-affine-base-contractible-complement", "journal": "Foundations of Computational Mathematics", "publicationYear": 2026, "title": "Vine Decomposition of Persistence-Diagram Bundles", "topic": "Computational and symplectic geometry", "publicationDate": "2026-04-28", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 7.3, future research", "sourceQuote": "I conjecture that if B minus B* is contractible, then the proposed reconstruction is unique.", "problemStatement": "For a fibered filtration f:K x B->R, its persistence-diagram bundle assigns to each parameter b the persistence diagram of f_b, and B* is the set where a diagram has a point of multiplicity greater than one. A decomposition of form (3) means expressing the nonsingular bundle globally as continuous sections, or vines, that track individual diagram points. Prove the conjecture that if B minus B* is contractible then such a global vine decomposition exists, or determine the additional obstruction; the question is not about integral-affine reconstruction.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "integral affine geometry", "singularities", "contractibility", "reconstruction" ], "literature": [], "source": { "title": "Persistence Diagram Bundles: A Multidimensional Generalization of Vineyards", "authors": [ "Abigail Hickok" ], "doi": "10.1007/s10208-026-09747-9", "url": "https://doi.org/10.1007/s10208-026-09747-9", "preprintUrl": "https://arxiv.org/abs/2210.05124" }, "number": 443 }, { "id": "mor-2026-manual-10.1287-moor.2024.0729-1", "journal": "Mathematics of Operations Research", "publicationYear": 2026, "title": "Closed-form characterization of revenue-maximizing Groves mechanisms", "topic": "Mechanism Design", "publicationDate": "2026-05-12", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 32, Conclusions and Future Research", "sourceQuote": "is there a closed form characterization of the revenue-maximizing Groves mechanism?", "problemStatement": "With a known prior over agents' multidimensional types, a Groves mechanism chooses the welfare-maximizing allocation and adjusts each payment by a function of the other agents' reports only. Expected revenue therefore depends on those report-independent functions, subject to incentive-compatibility and individual-rationality constraints. Find a closed-form characterization of the expected-revenue-maximizing Groves mechanism and determine when it is computationally tractable.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "Groves mechanism", "revenue maximization", "multidimensional mechanism design", "side information" ], "literature": [], "source": { "title": "Bicriteria Multidimensional Mechanism Design with Side Information", "authors": [ "Maria-Florina Balcan", "Siddharth Prasad", "Tuomas Sandholm" ], "doi": "10.1287/moor.2024.0729", "url": "https://doi.org/10.1287/moor.2024.0729", "preprintUrl": "https://arxiv.org/pdf/2302.14234" }, "number": 444 }, { "id": "siopt-2026-069", "journal": "SIAM Journal on Optimization", "publicationYear": 2026, "title": "Hoffman kappa identity", "topic": "Condition measures", "publicationDate": "2026-05-22", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Introduction; extracted preprint lines 124-131", "sourceQuote": "We conjecture that, with an appropriate choice of norms, an identity like the one for the chi measures also holds.", "problemStatement": "The paper identifies certain Hoffman error-bound constants with classical chi condition measures for polyhedral systems. The kappa circuit-imbalance measure is a newer condition number governing modern linear-programming algorithms. Find norms for which the corresponding Hoffman constant equals kappa, or show that no exact identity is possible.", "statusEvidence": "No forward-citing primary paper or exact-query result explicitly claiming a proof or counterexample was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "Hoffman constant", "circuit imbalance", "condition number", "linear programming" ], "literature": [], "source": { "title": "Duality of Hoffman Constants", "authors": [ "Javier F. Peña", "Juan C. Vera", "Luis F. Zuluaga" ], "doi": "10.1137/25m1760544", "url": "https://doi.org/10.1137/25m1760544", "preprintUrl": "https://arxiv.org/abs/2312.09858", "volume": "36", "issue": "2", "pages": "866-886" }, "number": 445 }, { "id": "siopt-2026-070", "journal": "SIAM Journal on Optimization", "publicationYear": 2026, "title": "Nonconic hoffman duality", "topic": "Condition measures", "publicationDate": "2026-05-22", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conclusion; extracted preprint lines 1147-1151", "sourceQuote": "We conjecture that similar identities hold for other non-conic pairs of polyhedral feasibility problems.", "problemStatement": "The paper proves dual identities between Hoffman constants for conic primal-dual feasibility systems and for one box-constrained nonconic pair. General polyhedral formulations need not share the same cone structure. Characterize nonconic primal-dual pairs whose Hoffman constants obey an analogous identity and determine the correct dual norms.", "statusEvidence": "No forward-citing primary paper or exact-query result explicitly claiming a proof or counterexample was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "Hoffman constant", "polyhedral feasibility", "duality", "error bounds" ], "literature": [], "source": { "title": "Duality of Hoffman Constants", "authors": [ "Javier F. Peña", "Juan C. Vera", "Luis F. Zuluaga" ], "doi": "10.1137/25m1760544", "url": "https://doi.org/10.1137/25m1760544", "preprintUrl": "https://arxiv.org/abs/2312.09858", "volume": "36", "issue": "2", "pages": "866-886" }, "number": 446 }, { "id": "siopt-2026-071", "journal": "SIAM Journal on Optimization", "publicationYear": 2026, "title": "Outer facial distance hoffman duality", "topic": "Polyhedral geometry", "publicationDate": "2026-05-22", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conclusion; extracted preprint lines 1179-1186", "sourceQuote": "We conjecture that the outer facial distance is a Hoffman constant as well", "problemStatement": "Inner and outer facial distances quantify polytope geometry and control Frank-Wolfe convergence; the inner distance is known to equal a Hoffman constant. Prove that the outer facial distance also admits a Hoffman-constant representation. Then determine whether the inner and outer distances are dual under an appropriate primal-polar construction.", "statusEvidence": "No forward-citing primary paper or exact-query result explicitly claiming a proof or counterexample was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "facial distance", "Hoffman constant", "Frank-Wolfe", "duality" ], "literature": [], "source": { "title": "Duality of Hoffman Constants", "authors": [ "Javier F. Peña", "Juan C. Vera", "Luis F. Zuluaga" ], "doi": "10.1137/25m1760544", "url": "https://doi.org/10.1137/25m1760544", "preprintUrl": "https://arxiv.org/abs/2312.09858", "volume": "36", "issue": "2", "pages": "866-886" }, "number": 447 }, { "id": "mp-2026-finite-sdp-flat-torus-embedding", "journal": "Mathematical Programming", "publicationYear": 2026, "title": "Finite SDP for Flat Torus Embeddings", "topic": "Metric embeddings and geometry", "publicationDate": "2026-05-26", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 9, first open question", "sourceQuote": "this infinite-dimensional semidefinite program can in fact be turned into a finite-dimensional semidefinite program.", "problemStatement": "Convert the infinite-dimensional semidefinite program for the least Euclidean distortion of a flat torus ℝⁿ/L into a finite-dimensional semidefinite program. The paper proves that a finite-dimensional optimal embedding exists, yet its optimization formulation still has infinitely many Fourier variables and constraints. The goal is a finite exact representation, or a convergent finite hierarchy with certified accuracy, enabling algorithmic computation for arbitrary lattices.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact problem wording, title and authors across publisher pages, arXiv, and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "flat torus", "semidefinite programming", "Euclidean distortion", "finite formulation" ], "literature": [], "source": { "title": "A semidefinite program for least distortion embeddings of flat tori into Hilbert spaces", "authors": [ "Arne Heimendahl", "Moritz Lücke", "Frank Vallentin", "Marc Christian Zimmermann" ], "doi": "10.1007/s10107-026-02369-7", "url": "https://doi.org/10.1007/s10107-026-02369-7" }, "number": 448 }, { "id": "mp-2026-finite-support-bound-torus-sdp", "journal": "Mathematical Programming", "publicationYear": 2026, "title": "Finite Support Bounds for the Torus SDP", "topic": "Metric embeddings and geometry", "publicationDate": "2026-05-26", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 9, finite-support question", "sourceQuote": "We also do not know how to restrict the variable", "problemStatement": "Bound the dual-lattice vector lengths needed in the support of an optimal Fourier variable z for the flat-torus distortion program. Although the paper proves that some finite-dimensional least-distortion embedding exists, it gives no a priori finite search region inside each coset of L*/2L*. A computable support-radius bound would reduce z to finitely many coordinates and would help produce a finite SDP algorithm.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact problem wording, title and authors across publisher pages, arXiv, and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "Fourier support", "dual lattice", "flat torus", "semidefinite program" ], "literature": [], "source": { "title": "A semidefinite program for least distortion embeddings of flat tori into Hilbert spaces", "authors": [ "Arne Heimendahl", "Moritz Lücke", "Frank Vallentin", "Marc Christian Zimmermann" ], "doi": "10.1007/s10107-026-02369-7", "url": "https://doi.org/10.1007/s10107-026-02369-7" }, "number": 449 }, { "id": "mp-2026-finite-termination-flat-torus-support-reduction", "journal": "Mathematical Programming", "publicationYear": 2026, "title": "Finite Termination of Torus Support Reduction", "topic": "Metric embeddings and geometry", "publicationDate": "2026-05-26", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 4, support-reduction lemma", "sourceQuote": "We do not know how to turn this procedure into an algorithm that terminates after finitely many steps", "problemStatement": "For a lattice L, a feasible pair consists of C≥0 and an even nonnegative summable weight z on the dual lattice L*, with Dz(x)=2Σu z(u)(1−cos(2πuᵀx)) satisfying ‖x‖²≤Dz(x)≤C‖x‖² on the Voronoi cell. Turn the support-reduction proof into a finite algorithm that preserves feasibility and leaves at most one primitive dual-lattice vector in each coset u+2L*. The present repeated local replacements converge only in the limit, so a pivot rule or discrete potential proving finite termination is missing.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact problem wording, title and authors across publisher pages, arXiv, and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "flat torus", "support reduction", "finite termination", "dual lattice" ], "literature": [], "source": { "title": "A semidefinite program for least distortion embeddings of flat tori into Hilbert spaces", "authors": [ "Arne Heimendahl", "Moritz Lücke", "Frank Vallentin", "Marc Christian Zimmermann" ], "doi": "10.1007/s10107-026-02369-7", "url": "https://doi.org/10.1007/s10107-026-02369-7" }, "number": 450 }, { "id": "mp-2026-finitely-many-contracted-pairs-flat-torus", "journal": "Mathematical Programming", "publicationYear": 2026, "title": "Finiteness of Most Contracted Torus Pairs", "topic": "Metric embeddings and geometry", "publicationDate": "2026-05-26", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 9, most-contracted-pair discussion", "sourceQuote": "We do not even know whether there are only finitely many most contracted pairs.", "problemStatement": "Decide whether a least-distortion embedding of an arbitrary flat torus has only finitely many most-contracted displacement pairs, after translation symmetry reduces them to pairs of the form (0,y). Two-dimensional computations produce finitely many Voronoi-face centers, but no general characterization is known. The task is a dimension-independent finiteness theorem with a bound, or an example whose optimal embedding has infinitely many maximally contracted displacements.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact problem wording, title and authors across publisher pages, arXiv, and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "most contracted pair", "flat torus", "finiteness", "Voronoi geometry" ], "literature": [], "source": { "title": "A semidefinite program for least distortion embeddings of flat tori into Hilbert spaces", "authors": [ "Arne Heimendahl", "Moritz Lücke", "Frank Vallentin", "Marc Christian Zimmermann" ], "doi": "10.1007/s10107-026-02369-7", "url": "https://doi.org/10.1007/s10107-026-02369-7" }, "number": 451 }, { "id": "mp-2026-maximal-distortion-lattices", "journal": "Mathematical Programming", "publicationYear": 2026, "title": "Lattices of Maximal Torus Distortion", "topic": "Metric embeddings and geometry", "publicationDate": "2026-05-26", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 9, final open problem", "sourceQuote": "Another interesting problem is to determine n-dimensional lattices which maximize the distortion among all n-dimensional lattices.", "problemStatement": "For each dimension n, determine the lattices L whose flat tori ℝⁿ/L maximize the least possible Euclidean embedding distortion among all n-dimensional lattices, under an appropriate scale normalization. The source improves universal lower bounds and solves several symmetric examples but does not identify the extremizers. The task includes proving existence or compactness, computing the extremal value, and classifying maximizing lattices up to equivalence.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact problem wording, title and authors across publisher pages, arXiv, and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "lattice", "flat torus", "maximal distortion", "extremal geometry" ], "literature": [], "source": { "title": "A semidefinite program for least distortion embeddings of flat tori into Hilbert spaces", "authors": [ "Arne Heimendahl", "Moritz Lücke", "Frank Vallentin", "Marc Christian Zimmermann" ], "doi": "10.1007/s10107-026-02369-7", "url": "https://doi.org/10.1007/s10107-026-02369-7" }, "number": 452 }, { "id": "mp-2026-contracted-pairs-voronoi-boundary", "journal": "Mathematical Programming", "publicationYear": 2026, "title": "Most Contracted Pairs on the Voronoi Boundary", "topic": "Metric embeddings and geometry", "publicationDate": "2026-05-26", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 9, most-contracted-pair discussion", "sourceQuote": "we do not know whether such a y can only lie on the Voronoi cell’s boundary.", "problemStatement": "Determine whether every most-contracted displacement y for a least-distortion embedding of a flat torus lies on the boundary of the lattice's Voronoi cell. The authors expect most-contracted pairs to have the form (0,y), with y at the center of a Voronoi face, but cannot exclude interior centers. The task is a geometric characterization valid for arbitrary dimension and lattice, or a counterexample with an interior maximizer.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact problem wording, title and authors across publisher pages, arXiv, and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "Voronoi cell", "most contracted pair", "flat torus", "least distortion" ], "literature": [], "source": { "title": "A semidefinite program for least distortion embeddings of flat tori into Hilbert spaces", "authors": [ "Arne Heimendahl", "Moritz Lücke", "Frank Vallentin", "Marc Christian Zimmermann" ], "doi": "10.1007/s10107-026-02369-7", "url": "https://doi.org/10.1007/s10107-026-02369-7" }, "number": 453 }, { "id": "mp-2026-uniform-root-weights-an-star", "journal": "Mathematical Programming", "publicationYear": 2026, "title": "Uniform Root Weights for A Star Lattices", "topic": "Metric embeddings and geometry", "publicationDate": "2026-05-26", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 7, conjecture after the D*n analysis", "sourceQuote": "However, we conjecture that there is also an optimal solution", "problemStatement": "Prove that the least-distortion semidefinite program for the flat torus associated with the dual lattice A*n has an optimal solution assigning equal weights to all roots of Aₙ. The analogous symmetry argument works for D*n, but the paper cannot transfer it directly to A*n. The conjecture is known for n=2, and the task is a proof for every dimension or an explicit higher-dimensional counterexample.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact problem wording, title and authors across publisher pages, arXiv, and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "A-star lattice", "flat torus", "uniform weights", "least distortion" ], "literature": [], "source": { "title": "A semidefinite program for least distortion embeddings of flat tori into Hilbert spaces", "authors": [ "Arne Heimendahl", "Moritz Lücke", "Frank Vallentin", "Marc Christian Zimmermann" ], "doi": "10.1007/s10107-026-02369-7", "url": "https://doi.org/10.1007/s10107-026-02369-7" }, "number": 454 }, { "id": "mor-2026-manual-10.1287-moor.2024.0681-1", "journal": "Mathematics of Operations Research", "publicationYear": 2026, "title": "Global well-posedness of the nonlocal equilibrium PDE without smallness conditions", "topic": "Mean-Field Control", "publicationDate": "2026-05-26", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 3, Introduction, discussion of equation (5.10)", "sourceQuote": "the global well-posedness remains an open problem.", "problemStatement": "A time-inconsistent, nonquadratic extended mean-field control problem with common noise has an equilibrium Hamilton–Jacobi–Bellman equation on Wasserstein space. An equilibrium ansatz reduces it to a nonlocal nonlinear parabolic equation for a function of the planning time, current time, and state. Establish global existence and uniqueness of a classical solution without the coefficient-smallness condition used by the current fixed-point proof, or identify an obstruction.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "mean-field control", "nonlocal PDE", "global well-posedness", "time inconsistency" ], "literature": [], "source": { "title": "On Time-Inconsistent Extended Mean-Field Control Problems with Common Noise", "authors": [ "Zongxia Liang", "Xiang Yu", "Keyu Zhang" ], "doi": "10.1287/moor.2024.0681", "url": "https://doi.org/10.1287/moor.2024.0681", "preprintUrl": "https://arxiv.org/pdf/2409.07219" }, "number": 455 }, { "id": "mp-2026-moment-problem-alternating-projection-convergence", "journal": "Mathematical Programming", "publicationYear": 2026, "title": "Alternating Projections for Higher Codimension Moments", "topic": "Projection methods", "publicationDate": "2026-06-05", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 4, Open problems and remarks", "sourceQuote": "it remains unknown if the alternating projection sequences converges in norm for a general moment problem of codimension k>1.", "problemStatement": "Determine whether von Neumann alternating projections converge in norm for every moment problem formed by a finite-codimensional linear subspace A and a Hilbert lattice cone B when codim(A)>1. Norm convergence is known in codimension one, and recent work supplies sufficient conditions in some higher-codimension configurations. The open task is a general convergence theorem or an explicit higher-codimension counterexample.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact problem wording, title and authors across publisher pages, arXiv, and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "alternating projections", "moment problem", "Hilbert lattice", "norm convergence" ], "literature": [], "source": { "title": "Characterization of regularity via variational stability of alternating projection sequences", "authors": [ "F. Battistoni", "A. Daniilidis", "C. A. De Bernardi", "E. Miglierina" ], "doi": "10.1007/s10107-026-02374-w", "url": "https://doi.org/10.1007/s10107-026-02374-w" }, "number": 456 }, { "id": "mor-2026-manual-10.1287-moor.2025.1239-1", "journal": "Mathematics of Operations Research", "publicationYear": 2026, "title": "Convergence rate of non-SOS Positivstellensatz relaxations", "topic": "Polynomial Optimization", "publicationDate": "2026-06-08", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 24, comparison of hierarchies (H2m) and (Pm)", "sourceQuote": "the exact convergence rate of f⋆₂ₘ to f⋆ remains unknown", "problemStatement": "The hierarchy (H2m) gives non-SOS semidefinite relaxations for polynomial optimization over sets defined by polynomial matrix inequalities, and its optimal values increase to the true optimum f⋆. The matrix sizes in this hierarchy avoid dependence on the number of decision variables but grow exponentially with the relaxation order. Determine quantitative upper and lower bounds on the convergence rate f⋆ − f⋆₂ₘ as a function of m and the problem data.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "Positivstellensatz", "non-SOS hierarchy", "polynomial matrix inequalities", "convergence rate" ], "literature": [], "source": { "title": "Non-SOS Positivstellensätze for Semialgebraic Sets Defined by Polynomial Matrix Inequalities", "authors": [ "Feng Guo" ], "doi": "10.1287/moor.2025.1239", "url": "https://doi.org/10.1287/moor.2025.1239", "preprintUrl": "https://arxiv.org/pdf/2509.01159" }, "number": 457 }, { "id": "mor-2026-manual-10.1287-moor.2024.0465-1", "journal": "Mathematics of Operations Research", "publicationYear": 2026, "title": "Tie-breaking independence in scale-free cascading-failure limits", "topic": "Network Reliability", "publicationDate": "2026-06-08", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 26, Conjecture 17", "sourceQuote": "The result of Theorem 2 holds independent of the choice of the exceedance tie-breaking rule.", "problemStatement": "Network nodes have independent Pareto-tailed demands; direct-current optimal power flow schedules generation, and a cascade repeatedly removes an edge with maximal flow-to-capacity exceedance. For a fixed tie-breaking rule, the total lost demand has a tail asymptotic to C times x to the power −α when disconnection occurs with positive probability, and of order x to the power −2α otherwise; the constant and regime may appear rule-dependent. Prove that the cascade-size law is actually independent of how tied exceedance maximizers are selected, or construct a counterexample.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "cascading failures", "power networks", "tie-breaking", "scale-free limit" ], "literature": [], "source": { "title": "Scale-Free Cascading Failures: Generalized Approach for All Simple, Connected Graphs", "authors": [ "Agnieszka Janicka", "Fiona Sloothaak", "Maria Vlasiou" ], "doi": "10.1287/moor.2024.0465", "url": "https://doi.org/10.1287/moor.2024.0465", "preprintUrl": "https://arxiv.org/pdf/2403.11727" }, "number": 458 }, { "id": "mor-2026-manual-10.1287-moor.2024.0830-1", "journal": "Mathematics of Operations Research", "publicationYear": 2026, "title": "Polynomial-time solvability of integer programs with bounded subdeterminants", "topic": "Integer Programming", "publicationDate": "2026-06-09", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Page 1, Abstract", "sourceQuote": "whether integer programs (IPs) with an integer coefficient matrix M whose subdeterminants are all bounded by a constant Δ can be solved in polynomial time.", "problemStatement": "A matrix is totally Δ-modular when every square subdeterminant has absolute value at most the fixed constant Δ. Integer programming is polynomial-time solvable for Δ = 1 and Δ = 2, and the source paper proves tractability for a cographic nearly totally-unimodular subclass at arbitrary fixed Δ. Determine whether every integer program with a totally Δ-modular constraint matrix is polynomial-time solvable for each fixed Δ.", "statusEvidence": "Targeted title, exact-statement, cited-by, and forward-literature searches located no later primary paper claiming a complete resolution through 2026-07-13. This bounded negative result is not proof that no resolution exists.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "bounded subdeterminants", "totally Delta-modular", "integer programming", "polynomial time" ], "literature": [], "source": { "title": "Integer Programs with Nearly Totally Unimodular Matrices: The Cographic Case", "authors": [ "Manuel Aprile", "Samuel Fiorini", "Gwenaël Joret", "Stefan Kober", "Michał Seweryn", "Stefan Weltge", "Yelena Yuditsky" ], "doi": "10.1287/moor.2024.0830", "url": "https://doi.org/10.1287/moor.2024.0830", "preprintUrl": "https://arxiv.org/pdf/2407.09477" }, "number": 459 }, { "id": "focm-2026-transformer-attention-limit-location", "journal": "Foundations of Computational Mathematics", "publicationYear": 2026, "title": "Transformer Attention Limit Location", "topic": "Mathematics of machine learning", "publicationDate": "2026-06-10", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Introduction", "sourceQuote": "characterizing the location of the limit point mass for general measures is an open problem", "problemStatement": "Characterize the location of the Dirac mass selected by continuous transformer self-attention dynamics from a general initial probability measure. Existing results establish concentration but identify the limit only for special symmetric or discrete inputs.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "transformers", "self-attention", "measure dynamics", "concentration" ], "literature": [], "source": { "title": "Measure-to-Measure Interpolation Using Transformers", "authors": [ "Borjan Geshkovski", "Philippe Rigollet", "Domènec Ruiz-Balet" ], "doi": "10.1007/s10208-026-09754-w", "url": "https://doi.org/10.1007/s10208-026-09754-w", "preprintUrl": "https://arxiv.org/abs/2411.04551" }, "number": 460 }, { "id": "focm-2026-wasserstein-flow-analysis-beyond-lmc", "journal": "Foundations of Computational Mathematics", "publicationYear": 2026, "title": "Wasserstein Flow Analysis Beyond LMC", "topic": "Sampling and optimization", "publicationDate": "2026-06-11", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 2.3, methodological discussion", "sourceQuote": "it remains open if this analysis technique can be used for more general algorithms or settings", "problemStatement": "Extend the Wasserstein-gradient-flow convex-optimization analysis beyond Langevin Monte Carlo and mirror LMC to more general sampling algorithms or geometries. The extension should retain nonasymptotic discretization and bias guarantees.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "Wasserstein gradient flow", "Langevin Monte Carlo", "sampling algorithms", "convexity" ], "literature": [], "source": { "title": "Shifted Composition III: Local Error Framework for KL Divergence", "authors": [ "Jason M. Altschuler", "Sinho Chewi" ], "doi": "10.1007/s10208-026-09753-x", "url": "https://doi.org/10.1007/s10208-026-09753-x", "preprintUrl": "https://arxiv.org/abs/2412.17997" }, "number": 461 }, { "id": "mp-2026-silver-stepsize-asymptotic-optimality", "journal": "Mathematical Programming", "publicationYear": 2026, "title": "Asymptotic Optimality of Silver Stepsizes", "topic": "First-order optimization", "publicationDate": "2026-06-13", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Introduction, stepsize-accelerated gradient descent", "sourceQuote": "conjectured it to be asymptotically optimal among all possible stepsize schedules.", "problemStatement": "Prove that the silver convergence exponent log₂(1+√2) is asymptotically optimal among all predetermined stepsize schedules for gradient descent on smooth convex functions. The source extends silver-step acceleration to projected and proximal gradient methods but uses the same conjectural benchmark inherited from unconstrained optimization. The task is a matching lower bound over arbitrary schedules or a schedule with a strictly better exponent.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact problem wording, title and authors across publisher pages, arXiv, and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "silver stepsizes", "gradient descent", "asymptotic optimality", "lower bound" ], "literature": [], "source": { "title": "Optimized methods for composite optimization: a reduction perspective", "authors": [ "Jinho Bok", "Jason M. Altschuler" ], "doi": "10.1007/s10107-026-02377-7", "url": "https://doi.org/10.1007/s10107-026-02377-7" }, "number": 462 }, { "id": "mp-2026-unified-rank-one-proofs-optimized-methods", "journal": "Mathematical Programming", "publicationYear": 2026, "title": "Unified Rank One Proofs for Optimized Methods", "topic": "First-order optimization", "publicationDate": "2026-06-13", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Discussion, Further understanding of optimized methods", "sourceQuote": "It is an interesting open question to understand the generality of this phenomenon and whether it hints at an underlying unified theory", "problemStatement": "Characterize when a performance-estimation proof for a first-order method has a sum-of-squares quadratic form of rank zero or one. The source exploits this simple structure for several optimized algorithms, while it is unknown whether ordinary constant-step gradient descent or accelerated gradient methods admit comparable certificates. The task is a general equivalence, classification, or obstruction theorem linking proof rank to algorithmic optimality.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact problem wording, title and authors across publisher pages, arXiv, and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "performance estimation", "sum of squares", "rank-one certificate", "optimized method" ], "literature": [], "source": { "title": "Optimized methods for composite optimization: a reduction perspective", "authors": [ "Jinho Bok", "Jason M. Altschuler" ], "doi": "10.1007/s10107-026-02377-7", "url": "https://doi.org/10.1007/s10107-026-02377-7" }, "number": 463 }, { "id": "siopt-2026-072", "journal": "SIAM Journal on Optimization", "publicationYear": 2026, "title": "Minimal degree matrix LME", "topic": "Polynomial optimization", "publicationDate": "2026-06-15", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 3; extracted preprint lines 582-587", "sourceQuote": "For general cases, it is mostly an open question for determining the minimal degree of L(x).", "problemStatement": "For matrix polynomial optimization, a Lagrange-multiplier expression L(x) encodes matrix multipliers polynomially and can make moment-SOS relaxations tight. Constant-degree expressions exist in a dimension-dominant linear case, but the degree needed for general matrix size, variable dimension, and polynomial degree is unknown. Determine the minimal degree or derive sharp upper and lower bounds in these parameters.", "statusEvidence": "No forward-citing primary paper or exact-query result explicitly claiming a proof or counterexample was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "matrix polynomial", "Lagrange multiplier expression", "minimal degree", "moment-SOS" ], "literature": [], "source": { "title": "Lagrange Multiplier Expressions for Matrix Polynomial Optimization and Tight Relaxations", "authors": [ "Lei Huang", "Jiawang Nie", "Jiajia Wang", "Lingling Xie" ], "doi": "10.1137/25m1788907", "url": "https://doi.org/10.1137/25m1788907", "preprintUrl": "https://www.semanticscholar.org/paper/a1865cdc385adb913fd66a52ec6058b27c1f13c2", "volume": "36", "issue": "2", "pages": "1017-1042" }, "number": 464 }, { "id": "siopt-2026-064", "journal": "SIAM Journal on Optimization", "publicationYear": 2026, "title": "Accelerated bilevel tradeoff without structure", "topic": "Bilevel optimization", "publicationDate": "2026-06-19", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conclusion; extracted preprint lines 1728-1734", "sourceQuote": "It remains an open question whether the accelerated trade-off obtained by IRE-APG can be achieved by any method", "problemStatement": "The bilevel model is minω(x):x∈argmin φ, where ω=f₁+g₁ and φ=f₂+g₂, each f_i is convex with Lipschitz gradient, each g_i is proper closed convex, and the bilevel solution set is nonempty and bounded. IRE-APG applies acceleration to φ+σ_kω with σ_k=k^(−β); when the combined proximal step is directly computable and 0<β<1, its inner gap is O(k^(−β)) and outer gap O(k^(−(2−β))), with O(log k/k) and O(1/k) at β=1. When prox(g₂+σ_kg₁) is intractable, the equivalent lifted surrogate φ̃(x,p)=φ(x)+(ρ/2)‖x−p‖², ω̃(x,p)=ω(p) separates the proximal steps, but translating back yields only O(log k/k) inner and O(√(log k/k)) outer rates when ω is real-valued (or O(√(log k/k)) inner and O(1/k) outer when φ is real-valued), reducing the exponent tradeoff from 2 to 3/2. Determine whether the full accelerated tradeoff is attainable on the surrogate without extra structure such as a Hölder error bound for φ or norm-like ω.", "statusEvidence": "No forward-citing primary paper or exact-query result explicitly claiming a proof or counterexample was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "iterative regularization", "bilevel optimization", "acceleration", "rate tradeoff" ], "literature": [], "source": { "title": "On the Convergence Rates of Iterative Regularization Algorithms for Composite Bilevel Optimization", "authors": [ "Shimrit Shtern", "Adeolu Taiwo" ], "doi": "10.1137/24m1703355", "url": "https://doi.org/10.1137/24m1703355", "preprintUrl": "https://arxiv.org/abs/2506.16382", "volume": "36", "issue": "2", "pages": "1043-1074" }, "number": 465 }, { "id": "mp-2026-bounded-regret-robust-matroid", "journal": "Mathematical Programming", "publicationYear": 2026, "title": "Bounded Regret Robust Matroid", "topic": "Robust combinatorial optimization", "publicationDate": "2026-06-22", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conclusion, final paragraph", "sourceQuote": "the complexity of the bounded-regret version is open.", "problemStatement": "Determine the complexity of the bounded-regret version of Robust Matroid, in which the first-stage independent set must satisfy a prescribed regret bound under each deletion scenario. Ordinary Robust Matroid is efficiently solvable for k≤ℓ, but that does not directly yield a polynomial bounded-regret subroutine. The task is an exact polynomial algorithm or a hardness proof, with the deletion and recourse parameters stated explicitly.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact problem wording, title and authors across publisher pages, arXiv, and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "bounded regret", "robust matroid", "recoverable robustness", "complexity" ], "literature": [], "source": { "title": "Recoverable robust optimization with commitment", "authors": [ "Felix Hommelsheim", "Nicole Megow", "Komal Muluk", "Britta Peis" ], "doi": "10.1007/s10107-026-02385-7", "url": "https://doi.org/10.1007/s10107-026-02385-7" }, "number": 466 }, { "id": "mp-2026-general-robust-interval-scheduling-complexity", "journal": "Mathematical Programming", "publicationYear": 2026, "title": "General Robust Interval Scheduling Complexity", "topic": "Robust combinatorial optimization", "publicationDate": "2026-06-22", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conclusion and future research", "sourceQuote": "the complexity status of (k,ℓ)-Robust Interval Scheduling and also the (k,ℓ)-Robust Stable Set remain open.", "problemStatement": "In recoverable robustness with commitment, choose a feasible interval set S; an adversary deletes at most k intervals, after which at most ℓ undeleted intervals may be added while every surviving member of S remains selected. Determine the computational complexity of maximizing the worst-case weight when feasibility means pairwise-disjoint intervals and k,ℓ are arbitrary. The paper gives efficient algorithms only for (1,p) with fixed p, so the general exact complexity is open.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact problem wording, title and authors across publisher pages, arXiv, and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "robust interval scheduling", "recoverable robustness", "commitment", "complexity" ], "literature": [], "source": { "title": "Recoverable robust optimization with commitment", "authors": [ "Felix Hommelsheim", "Nicole Megow", "Komal Muluk", "Britta Peis" ], "doi": "10.1007/s10107-026-02385-7", "url": "https://doi.org/10.1007/s10107-026-02385-7" }, "number": 467 }, { "id": "mp-2026-general-robust-stable-set-complexity", "journal": "Mathematical Programming", "publicationYear": 2026, "title": "General Robust Stable Set Complexity", "topic": "Robust combinatorial optimization", "publicationDate": "2026-06-22", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conclusion and future research", "sourceQuote": "the complexity status of (k,ℓ)-Robust Interval Scheduling and also the (k,ℓ)-Robust Stable Set remain open.", "problemStatement": "In recoverable robustness with commitment, choose an initial stable set S; an adversary deletes at most k vertices, after which at most ℓ undeleted vertices may be added while every survivor in S remains selected. Determine the computational complexity of maximizing the worst-case final weight on an arbitrary graph for general k and ℓ. The task is the exact polynomial-time versus hardness classification of this committed robust stable-set problem.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact problem wording, title and authors across publisher pages, arXiv, and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "robust stable set", "recoverable robustness", "commitment", "complexity" ], "literature": [], "source": { "title": "Recoverable robust optimization with commitment", "authors": [ "Felix Hommelsheim", "Nicole Megow", "Komal Muluk", "Britta Peis" ], "doi": "10.1007/s10107-026-02385-7", "url": "https://doi.org/10.1007/s10107-026-02385-7" }, "number": 468 }, { "id": "mp-2026-robust-matroid-k-greater-l-polynomial", "journal": "Mathematical Programming", "publicationYear": 2026, "title": "Robust Matroid Complexity When k Exceeds l", "topic": "Robust combinatorial optimization", "publicationDate": "2026-06-22", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 3, Robust matroid", "sourceQuote": "It remains open whether this problem admits a polynomial-time algorithm", "problemStatement": "Determine whether recoverable (k,ℓ)-Robust Matroid with commitment is polynomial-time solvable when the adversary may delete k elements and only ℓℓ. The task is either a different polynomial algorithm or an NP-hardness proof for the unresolved regime.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact problem wording, title and authors across publisher pages, arXiv, and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "robust matroid", "recoverable robustness", "polynomial time", "recourse" ], "literature": [], "source": { "title": "Recoverable robust optimization with commitment", "authors": [ "Felix Hommelsheim", "Nicole Megow", "Komal Muluk", "Britta Peis" ], "doi": "10.1007/s10107-026-02385-7", "url": "https://doi.org/10.1007/s10107-026-02385-7" }, "number": 469 }, { "id": "mp-2026-snl-sparse-graph-benign-landscape", "journal": "Mathematical Programming", "publicationYear": 2026, "title": "Benign SNL Landscapes on Sparse Graphs", "topic": "Nonconvex optimization and localization", "publicationDate": "2026-06-22", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 8, Perspectives, Incomplete graph", "sourceQuote": "Extending this result to substantially sparser graphs remains an open problem.", "problemStatement": "Give theoretical conditions under which a small dimension relaxation produces a benign sensor-network-localization landscape on incomplete measurement graphs. Existing results cover complete graphs and slightly incomplete random instances, with k depending on the number of missing edges and the ground-truth condition number. The task is a substantially sparser family such as expanders, geometric graphs, unit-disk graphs, or trilateration graphs with sharp relaxation bounds.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact problem wording, title and authors across publisher pages, arXiv, and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "sensor network localization", "sparse graph", "benign landscape", "graph structure" ], "literature": [], "source": { "title": "Sensor network localization has a benign landscape after low-dimensional relaxation", "authors": [ "Christopher Criscitiello", "Andrew D. McRae", "Quentin Rebjock", "Nicolas Boumal" ], "doi": "10.1007/s10107-026-02380-y", "url": "https://doi.org/10.1007/s10107-026-02380-y" }, "number": 470 }, { "id": "mp-2026-snl-one-extra-dimension-benign", "journal": "Mathematical Programming", "publicationYear": 2026, "title": "One Extra Dimension for Benign SNL", "topic": "Nonconvex optimization and localization", "publicationDate": "2026-06-22", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 8, Perspectives, Relaxing by +1", "sourceQuote": "Based on numerical simulations, we conjecture that relaxing to", "problemStatement": "Prove or refute that complete-graph sensor network localization in ground-truth dimension ℓ has a benign nonconvex landscape for every configuration after relaxing the factor dimension to k=ℓ+1. The paper proves benignness with substantially larger k and reports strong numerical evidence for one extra dimension. A counterexample would need a genuine spurious second-order local minimum at k=ℓ+1, not merely a nongeneric degeneracy.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact problem wording, title and authors across publisher pages, arXiv, and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "sensor network localization", "benign landscape", "dimension relaxation", "spurious local minimum" ], "literature": [], "source": { "title": "Sensor network localization has a benign landscape after low-dimensional relaxation", "authors": [ "Christopher Criscitiello", "Andrew D. McRae", "Quentin Rebjock", "Nicolas Boumal" ], "doi": "10.1007/s10107-026-02380-y", "url": "https://doi.org/10.1007/s10107-026-02380-y" }, "number": 471 }, { "id": "mp-2026-bounded-subdeterminant-ip-polynomial", "journal": "Mathematical Programming", "publicationYear": 2026, "title": "Polynomial Time for Bounded Subdeterminant IP", "topic": "Integer programming", "publicationDate": "2026-06-22", "recordType": "reviewed", "status": "partial_progress", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Abstract, opening sentence", "sourceQuote": "on constraint matrices with bounded subdeterminants are conjectured to be solvable in polynomial time.", "problemStatement": "Prove that integer programming is polynomial-time solvable whenever every subdeterminant of the constraint matrix is bounded by a fixed constant Δ. Polynomial algorithms are known for Δ=2 and for several structural subclasses, including the source paper's matrices with at most two nonzeros per row after deleting constantly many rows and columns. The task is an algorithm for general totally Δ-modular matrices, already open for Δ=3, or a complexity-theoretic refutation.", "statusEvidence": "Gives a polynomial algorithm for bounded-domain instances with bounded totally Delta-modular treewidth, adding a tractable subclass without settling general totally Delta-modular integer programming.", "verificationNote": "This status is tied to the cited primary-literature result and the precise scope stated above.", "keywords": [ "bounded subdeterminants", "totally Delta-modular", "integer programming", "polynomial time" ], "literature": [ { "title": "Totally Delta-Modular Tree Decompositions of Graphic Matrices for Integer Programming", "url": "https://arxiv.org/abs/2602.01499", "relation": "Gives a polynomial algorithm for bounded-domain instances with bounded totally Delta-modular treewidth, adding a tractable subclass without settling general totally Delta-modular integer programming." } ], "source": { "title": "Totally Δ-modular IPs with two non-zeros in most rows", "authors": [ "Stefan Kober" ], "doi": "10.1007/s10107-026-02388-4", "url": "https://doi.org/10.1007/s10107-026-02388-4" }, "number": 472 }, { "id": "siopt-2026-068", "journal": "SIAM Journal on Optimization", "publicationYear": 2026, "title": "Concatenated stepsize schedule optimality", "topic": "First-order methods", "publicationDate": "2026-06-24", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conclusion; extracted preprint lines 1578-1581", "sourceQuote": "Whether these SSs are truly optimal for memoryless fixed-step gradient descent remains an open question.", "problemStatement": "For L-smooth convex f, an n-step memoryless schedule h=(h_0,…,h_(n−1)) runs x_(k+1)=x_k−(h_k/L)∇f(x_k); writing r=Σ_ih_i and Q_(i,*)=f_i−f*−⟨∇f(x_i),x_i−x*⟩+‖∇f(x_i)‖²/(2L), primitiveness means L‖x_0−x*‖²/2+Σ_ih_iQ_(i,*)−r(f_n−f*)≥L‖x_n−x*‖²/2+r(r+1)‖∇f(x_n)‖²/(2L) for every f and x_0. ConPP(h^a,h^b) inserts α=ϕ(1ᵀh^a,1ᵀh^b) between two primitive schedules, where ϕ(x,y)=[−x−y+√((x+y+2)²+4(x+1)(y+1))]/2; let h°_n maximize 1ᵀh among schedules of length n recursively generated by ConPP from the empty schedule. Prove 1ᵀh°_n=1ᵀConPP(h°_⌊(n−1)/2⌋,h°_⌈(n−1)/2⌉) and the additional odd-factor identities in Conjecture 4.17. Finally, prove that the resulting primitive, dominant, and gradient-norm schedules are globally optimal among all memoryless fixed-step schedules, or exhibit a better schedule.", "statusEvidence": "No forward-citing primary paper or exact-query result explicitly claiming a proof or counterexample was located in the recorded search.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "gradient descent", "stepsize schedule", "concatenation", "global optimality" ], "literature": [], "source": { "title": "Accelerated Gradient Descent by Concatenation of Stepsize Schedules", "authors": [ "Zehao Zhang", "Rujun Jiang" ], "doi": "10.1137/25m173898x", "url": "https://doi.org/10.1137/25m173898x", "preprintUrl": "https://arxiv.org/abs/2410.12395", "volume": "36", "issue": "2", "pages": "1182-1210" }, "number": 473 }, { "id": "focm-2026-tracefem-evolving-surfaces", "journal": "Foundations of Computational Mathematics", "publicationYear": 2026, "title": "TraceFEM Evolving Surfaces", "topic": "Finite element methods on surfaces", "publicationDate": "2026-06-25", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Conclusion", "sourceQuote": "Extending the analysis to evolving surfaces remains an open problem.", "problemStatement": "Extend the time-derivative-stabilized parabolic TraceFEM and its error analysis from stationary embedded surfaces to evolving surfaces. The theory must handle changing active meshes and geometric transport terms. TraceFEM restricts bulk finite-element functions to a surface that cuts the mesh; when the surface moves, both the cut cells and the discrete trace space change between time levels.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "TraceFEM", "evolving surfaces", "time derivative stabilization", "error analysis" ], "literature": [], "source": { "title": "Inf-Sup Stability of Parabolic TraceFEM", "authors": [ "Lucas Bouck", "Ricardo H. Nochetto", "Mansur Shakipov", "Vladimir Yushutin" ], "doi": "10.1007/s10208-026-09757-7", "url": "https://doi.org/10.1007/s10208-026-09757-7" }, "number": 474 }, { "id": "focm-2026-tracefem-uniform-infsup-weaker-meshes", "journal": "Foundations of Computational Mathematics", "publicationYear": 2026, "title": "TraceFEM Uniform Inf-Sup Weaker Meshes", "topic": "Finite element methods on surfaces", "publicationDate": "2026-06-25", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Introduction", "sourceQuote": "uniform bounds on the inf-sup constant with respect to h remain delicate and unresolved", "problemStatement": "Prove a mesh-uniform inf-sup bound for the parabolic TraceFEM under weaker, possibly strongly graded bulk-mesh assumptions. Equivalently, establish the required uniform stability of the discrete projection and extension operators.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "TraceFEM", "inf-sup stability", "graded meshes", "parabolic PDE" ], "literature": [], "source": { "title": "Inf-Sup Stability of Parabolic TraceFEM", "authors": [ "Lucas Bouck", "Ricardo H. Nochetto", "Mansur Shakipov", "Vladimir Yushutin" ], "doi": "10.1007/s10208-026-09757-7", "url": "https://doi.org/10.1007/s10208-026-09757-7" }, "number": 475 }, { "id": "mp-2026-goemans-unsplittable-flow-conjecture", "journal": "Mathematical Programming", "publicationYear": 2026, "title": "Goemans' Unsplittable Flow Conjecture", "topic": "Network flows and rounding", "publicationDate": "2026-06-29", "recordType": "reviewed", "status": "partial_progress", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Introduction and Note 1", "sourceQuote": "Goemans’ original conjecture asserts that, for any arc costs, an unsplittable flow can be found that satisfies (2)", "problemStatement": "Prove Goemans' conjecture for arbitrary single-source unsplittable-flow instances: round a feasible fractional flow to an unsplittable one whose arc load is less than the fractional load plus the largest demand and whose total cost does not increase. The source proves the equivalent convex-combination form for series-parallel digraphs. General acyclic digraphs remain open despite additional special cases and reductions in later primary work.", "statusEvidence": "Proves Goemans' convex-combination formulation for series-parallel digraphs, the first nontrivial digraph class covered, but not for arbitrary single-source digraphs.", "verificationNote": "This status is tied to the cited primary-literature result and the precise scope stated above.", "keywords": [ "unsplittable flow", "Goemans conjecture", "additive congestion", "cost-preserving rounding" ], "literature": [ { "title": "Integer and unsplittable multiflows in series-parallel digraphs", "url": "https://doi.org/10.1007/s10107-026-02392-8", "relation": "Proves Goemans' convex-combination formulation for series-parallel digraphs, the first nontrivial digraph class covered, but not for arbitrary single-source digraphs." } ], "source": { "title": "Integer and unsplittable multiflows in series-parallel digraphs", "authors": [ "Mohammed Majthoub Almoghrabi", "Martin Skutella", "Philipp Warode" ], "doi": "10.1007/s10107-026-02392-8", "url": "https://doi.org/10.1007/s10107-026-02392-8" }, "number": 476 }, { "id": "mp-2026-morell-skutella-two-sided-flow", "journal": "Mathematical Programming", "publicationYear": 2026, "title": "Morell Skutella Two Sided Flow Conjecture", "topic": "Network flows and rounding", "publicationDate": "2026-06-29", "recordType": "reviewed", "status": "partial_progress", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Introduction, Conjecture 1", "sourceQuote": "For the single-source unsplittable flow problem on acyclic digraphs, any fractional flow", "problemStatement": "Prove the Morell-Skutella conjecture on arbitrary acyclic single-source digraphs: every feasible fractional flow is a convex combination of unsplittable flows whose load on each arc differs from the fractional load by at most the maximum demand. The source proves this stronger two-sided guarantee for series-parallel digraphs, even with multiple sources and sinks. Later work settles further special cases, but the general directed-acyclic case remains open.", "statusEvidence": "Proves the two-sided convex-combination conjecture for series-parallel digraphs, including general multiflow instances, without settling arbitrary acyclic digraphs. Proves the two-sided discrepancy guarantee for a further special class, while explicitly presenting the full Morell-Skutella statement as a conjecture.", "verificationNote": "This status is tied to the cited primary-literature result and the precise scope stated above.", "keywords": [ "unsplittable flow", "Morell-Skutella conjecture", "discrepancy", "acyclic digraph" ], "literature": [ { "title": "Integer and unsplittable multiflows in series-parallel digraphs", "url": "https://doi.org/10.1007/s10107-026-02392-8", "relation": "Proves the two-sided convex-combination conjecture for series-parallel digraphs, including general multiflow instances, without settling arbitrary acyclic digraphs." }, { "title": "Weighted Chairman Assignment and Flow-Time Scheduling", "url": "https://doi.org/10.4230/LIPIcs.ITCS.2026.98", "relation": "Proves the two-sided discrepancy guarantee for a further special class, while explicitly presenting the full Morell-Skutella statement as a conjecture." } ], "source": { "title": "Integer and unsplittable multiflows in series-parallel digraphs", "authors": [ "Mohammed Majthoub Almoghrabi", "Martin Skutella", "Philipp Warode" ], "doi": "10.1007/s10107-026-02392-8", "url": "https://doi.org/10.1007/s10107-026-02392-8" }, "number": 477 }, { "id": "focm-2026-hadamard-squared-distance-universal-self-concordance", "journal": "Foundations of Computational Mathematics", "publicationYear": 2026, "title": "Hadamard Squared Distance Universal Self Concordance", "topic": "Riemannian optimization", "publicationDate": "2026-07-07", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Introduction", "sourceQuote": "Is squared distance universally C times kappa self-concordant on every Hadamard manifold?", "problemStatement": "Determine whether squared distance on every Hadamard manifold with sectional curvature in [-kappa,0] is self-concordant with a universal constant proportional to kappa. Find the best universal constant or a geometric counterexample.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "Hadamard manifolds", "sectional curvature", "self-concordance", "squared distance" ], "literature": [], "source": { "title": "Interior-Point Methods on Manifolds: Theory and Applications", "authors": [ "Hiroshi Hirai", "Harold Nieuwboer", "Michael Walter" ], "doi": "10.1007/s10208-026-09756-8", "url": "https://doi.org/10.1007/s10208-026-09756-8" }, "number": 478 }, { "id": "focm-2026-interior-point-well-centered-start", "journal": "Foundations of Computational Mathematics", "publicationYear": 2026, "title": "Interior Point Well Centered Start", "topic": "Riemannian optimization", "publicationDate": "2026-07-07", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Section 7, outlook", "sourceQuote": "Can one give a general procedure for finding such a starting point from an arbitrary feasible point?", "problemStatement": "Give a general procedure that converts an arbitrary feasible point into a well-centered starting point for the Riemannian interior-point method. Analyze the procedure's complexity so the paper's main-stage guarantees extend to an end-to-end algorithm.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "interior-point methods", "initialization", "central path", "feasible point" ], "literature": [], "source": { "title": "Interior-Point Methods on Manifolds: Theory and Applications", "authors": [ "Hiroshi Hirai", "Harold Nieuwboer", "Michael Walter" ], "doi": "10.1007/s10208-026-09756-8", "url": "https://doi.org/10.1007/s10208-026-09756-8" }, "number": 479 }, { "id": "focm-2026-pd-complex-squared-distance-eight-self-concordant", "journal": "Foundations of Computational Mathematics", "publicationYear": 2026, "title": "Positive-Definite Complex Squared Distance Eight Self Concordant", "topic": "Riemannian optimization", "publicationDate": "2026-07-07", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Introduction and Remark 5.14", "sourceQuote": "We conjecture that the squared distance is actually 8-self-concordant.", "problemStatement": "Prove that squared Riemannian distance on the manifold of complex positive-definite matrices is 8-self-concordant. Establish whether the constant eight is optimal. Self-concordance bounds the third covariant derivative by a fixed power of the Hessian norm, providing the dimension-independent control needed for Newton and interior-point complexity.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "self-concordance", "positive-definite matrices", "squared distance", "interior-point methods" ], "literature": [], "source": { "title": "Interior-Point Methods on Manifolds: Theory and Applications", "authors": [ "Hiroshi Hirai", "Harold Nieuwboer", "Michael Walter" ], "doi": "10.1007/s10208-026-09756-8", "url": "https://doi.org/10.1007/s10208-026-09756-8" }, "number": 480 }, { "id": "focm-2026-reverse-agm-optimal-constant", "journal": "Foundations of Computational Mathematics", "publicationYear": 2026, "title": "Reverse AGM Optimal Constant", "topic": "Matrix inequalities", "publicationDate": "2026-07-07", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Remark 5.14", "sourceQuote": "We conjecture, based on numerical evidence, that the optimal constant is C = 1/sqrt(2).", "problemStatement": "Let A(x,y)=(x+y)/2 and G(x,y)=sqrt(xy). Prove for all real a and b that A(a^2 coth(a)^2,b^2 coth(b)^2)/G(a^2 coth(a)^2,b^2 coth(b)^2) is at most 1+((a-b)tanh(a-b))/2, equivalently obtain constant C=1/sqrt(2) in the paper's coefficient inequality of Lemma 5.13. The proved value is C=sqrt(2), while the conjectured sharp value would yield the corresponding optimal self-concordance constant for squared matrix distance.", "statusEvidence": "On 2026-07-13, searched Crossref, OpenAlex, Semantic Scholar citation metadata, arXiv, and exact-title and statement-keyword web queries using the source DOI. No later primary paper claiming a proof, counterexample, or complete resolution was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "arithmetic-geometric mean", "reverse inequality", "optimal constant", "self-concordance" ], "literature": [], "source": { "title": "Interior-Point Methods on Manifolds: Theory and Applications", "authors": [ "Hiroshi Hirai", "Harold Nieuwboer", "Michael Walter" ], "doi": "10.1007/s10208-026-09756-8", "url": "https://doi.org/10.1007/s10208-026-09756-8" }, "number": 481 }, { "id": "mp-2026-oertel-mixed-integer-grunbaum", "journal": "Mathematical Programming", "publicationYear": 2026, "title": "Oertel's Mixed Integer Grünbaum Conjecture", "topic": "Mixed-integer geometry", "publicationDate": "2026-07-07", "recordType": "reviewed", "status": "partial_progress", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Abstract, status sentence after Conjecture 1", "sourceQuote": "While the conjecture remains open", "problemStatement": "Prove Oertel's mixed-integer analogue of Grünbaum's inequality for every convex body C in ℝⁿ⁺ᵈ. It asks for a mixed-integer point x in S=C∩(ℤⁿ×ℝᵈ) such that every halfspace containing x captures at least a 1/(2ⁿe) fraction of the d-dimensional Hausdorff measure of S. The source lowers a sufficient lattice-width threshold from exponential to polynomial, and later work lowers it to linear under a strong projection-radius condition, but full generality is unresolved.", "statusEvidence": "Verifies the conjecture for a substantially larger family whose integer projection contains a sufficiently large infinity-norm ball, but does not prove the unrestricted conjecture.", "verificationNote": "This status is tied to the cited primary-literature result and the precise scope stated above.", "keywords": [ "Oertel conjecture", "Grünbaum inequality", "mixed-integer volume", "halfspace depth" ], "literature": [ { "title": "Linear Threshold for Oertel's Conjecture on the Mixed-Integer Volume", "url": "https://arxiv.org/abs/2603.00286", "relation": "Verifies the conjecture for a substantially larger family whose integer projection contains a sufficiently large infinity-norm ball, but does not prove the unrestricted conjecture." } ], "source": { "title": "Reducing the large set threshold for Oertel’s conjecture on the mixed-integer volume", "authors": [ "Andrés Cristi", "David Salas" ], "doi": "10.1007/s10107-026-02391-9", "url": "https://doi.org/10.1007/s10107-026-02391-9" }, "number": 482 }, { "id": "mp-2026-multivariate-finite-difference-tensors", "journal": "Mathematical Programming", "publicationYear": 2026, "title": "Multivariate Finite Differences for Higher Order Methods", "topic": "High-order optimization", "publicationDate": "2026-07-08", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Discussion, lower-order recursive implementation", "sourceQuote": "it remains unclear to us how to systematically construct the corresponding finite-difference formulas together with explicit error bounds", "problemStatement": "Construct multivariate finite-difference formulas, with explicit uniform error bounds, that recursively approximate derivative tensors of orders q+1 through p using only qth-order derivative oracles. The paper's lower-order implementations achieve complexity bounds along certain order pairs, but the recursive multivariate tensor construction needed for all pairs is missing. The bounds must be explicit enough to choose the finite-difference scale and control cumulative approximation error.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact problem wording, title and authors across publisher pages, arXiv, and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "finite differences", "higher-order method", "derivative tensor", "oracle complexity" ], "literature": [], "source": { "title": "On the complexity of lower-order implementations of higher-order methods", "authors": [ "Nikita Doikov", "Geovani Nunes Grapiglia" ], "doi": "10.1007/s10107-026-02386-6", "url": "https://doi.org/10.1007/s10107-026-02386-6" }, "number": 483 }, { "id": "mp-2026-tight-lower-order-higher-order-bounds", "journal": "Mathematical Programming", "publicationYear": 2026, "title": "Tightness of Lower Order Implementations", "topic": "High-order optimization", "publicationDate": "2026-07-08", "recordType": "reviewed", "status": "no_resolution_found", "statusCheckedThrough": "2026-07-13", "sourceClaim": "explicit_open_problem", "sourceLocation": "Discussion, complexity-bound paragraph", "sourceQuote": "outside the main diagonal (q=p) it is unclear whether they can be improved, as the question of their tightness remains open.", "problemStatement": "Determine whether the oracle-complexity bounds for implementing pth-order optimization methods with qth-order information are tight when q≠p. The paper supplies constructive upper bounds for several lower-order simulations and identifies the diagonal case as understood, but lacks matching lower bounds off the diagonal. The task is either sharper algorithms or adversarial function families proving the existing dependence on dimension, accuracy, p, and q cannot be improved.", "statusEvidence": "On 2026-07-13, searched the source DOI, exact problem wording, title and authors across publisher pages, arXiv, and scholarly web indexes. No later primary paper claiming a complete resolution or materially sharper result was located.", "verificationNote": "No resolution found is an evidence-bounded search result, not proof that no resolution exists.", "keywords": [ "oracle complexity", "higher-order method", "lower bound", "finite differences" ], "literature": [], "source": { "title": "On the complexity of lower-order implementations of higher-order methods", "authors": [ "Nikita Doikov", "Geovani Nunes Grapiglia" ], "doi": "10.1007/s10107-026-02386-6", "url": "https://doi.org/10.1007/s10107-026-02386-6" }, "number": 484 } ] }