{ "journal": "Mathematical Programming", "issn": "0025-5610", "corpus_basis": "Crossref publication date, checked against publisher metadata", "date_from": "2023-01-01", "date_to": "2026-07-13", "generated_at": "2026-07-13T09:30:22Z", "screening_note": "Automated high-recall phrase screen followed by manual passage review. A no-match paper is retained in the audit rather than inferred from metadata.", "summary": { "crossref_records": 469, "publisher_pages_screened": 469, "arxiv_fallbacks_screened": 0, "full_text_unavailable": 0, "research_articles": 454, "non_research_items": 15, "papers_with_candidate_passages": 184 }, "articles": [ { "doi": "10.1007/s10107-022-01919-z", "title": "A unified single-loop alternating gradient projection algorithm for nonconvex–concave and convex–nonconcave minimax problems", "authors": [ "Zi Xu", "Huiling Zhang", "Yang Xu", "Guanghui Lan" ], "published_online": "2023-01-02", "volume": "201", "issue": "1-2", "pages": "635-706", "publisher_url": "https://link.springer.com/article/10.1007/s10107-022-01919-z/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-022-01921-5", "title": "Strong valid inequalities for a class of concave submodular minimization problems under cardinality constraints", "authors": [ "Qimeng Yu", "Simge Küçükyavuz" ], "published_online": "2023-01-09", "volume": "201", "issue": "1-2", "pages": "803-861", "publisher_url": "https://link.springer.com/article/10.1007/s10107-022-01921-5/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-022-01916-2", "title": "Accelerating inexact successive quadratic approximation for regularized optimization through manifold identification", "authors": [ "Ching-pei Lee" ], "published_online": "2023-01-12", "volume": "201", "issue": "1-2", "pages": "599-633", "publisher_url": "https://link.springer.com/article/10.1007/s10107-022-01916-2/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "An efficient inexact SQA method with superlinear convergence in running time", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "ISQA \\(^{+}\\) has two stages, separated by the event of identifying the active manifold \\(\\mathcal {M}\\) of a cluster point \\(x^*\\). Our analysis showed that iterates converging to \\(x^*\\) will eventually identify \\(\\mathcal {M}\\), but since neither \\(x^*\\) nor \\(\\mathcal {M}\\) is known a priori, the conjecture of identification can only be made when \\(\\mathcal {M}\\) remains unchanged for \\(S > 0\\) iterations." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-022-01920-6", "title": "Softmax policy gradient methods can take exponential time to converge", "authors": [ "Gen Li", "Yuting Wei", "Yuejie Chi", "Yuxin Chen" ], "published_online": "2023-01-23", "volume": "201", "issue": "1-2", "pages": "707-802", "publisher_url": "https://link.springer.com/article/10.1007/s10107-022-01920-6/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Discussion", "element": "p", "matched_phrases": [ "future_work" ], "passage": "Our work relies heavily on proper exploitation of the structural properties of the MDP in algorithm-dependent analysis, which might shed light on lower bound construction for other algorithms as well. For instance, if the objective function (i.e., the value function) is augmented by a regularization term, how does the choice of regularization affect the global convergence behavior? While Agarwal et al. [1] demonstrated polynomial-time convergence of PG methods in the presence of log-barrier regularization, non-asymptotic analysis of PG methods with other popular regularization—particularly entropy regularization—remains unavailable in existing literature. How to understand the (in)-effectiveness of entropy-regularized PG methods is of fundamental importance in the theory of policy optimization. Additionally, the current paper concentrates on the use of constant learning rates; it falls short of accommodating more adaptive learning rates, which might be a potential solution to accelerate vanilla PG methods. Furthermore, our strategy for lower bound construction might be extended to unveil algorithmic bottlenecks of policy optimization in multi-agent Markov games as well. All this is worthy of future investigation." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01925-9", "title": "Data perturbations in stochastic generalized equations: statistical robustness in static and sample average approximated models", "authors": [ "Shaoyan Guo", "Huifu Xu" ], "published_online": "2023-01-27", "volume": "202", "issue": "1-2", "pages": "135-168", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01925-9/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01924-w", "title": "$$\\mathbf {2\\times 2}$$-Convexifications for convex quadratic optimization with indicator variables", "authors": [ "Shaoning Han", "Andrés Gómez", "Alper Atamtürk" ], "published_online": "2023-01-28", "volume": "202", "issue": "1-2", "pages": "95-134", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01924-w/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-022-01922-4", "title": "Constrained composite optimization and augmented Lagrangian methods", "authors": [ "Alberto De Marchi", "Xiaoxi Jia", "Christian Kanzow", "Patrick Mehlitz" ], "published_online": "2023-02-08", "volume": "201", "issue": "1-2", "pages": "863-896", "publisher_url": "https://link.springer.com/article/10.1007/s10107-022-01922-4/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-022-01923-3", "title": "Smooth over-parameterized solvers for non-smooth structured optimization", "authors": [ "Clarice Poon", "Gabriel Peyré" ], "published_online": "2023-02-08", "volume": "201", "issue": "1-2", "pages": "897-952", "publisher_url": "https://link.springer.com/article/10.1007/s10107-022-01923-3/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01928-6", "title": "On the resolution of cross-liabilities", "authors": [ "Gabrielle Demange" ], "published_online": "2023-02-08", "volume": "203", "issue": "1-2", "pages": "827-856", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01928-6/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01930-y", "title": "A novel reformulation for the single-sink fixed-charge transportation problem", "authors": [ "Robin Legault", "Jean-François Côté", "Bernard Gendron" ], "published_online": "2023-02-11", "volume": "202", "issue": "1-2", "pages": "169-198", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01930-y/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01932-w", "title": "Efficient joint object matching via linear programming", "authors": [ "Antonio De Rosa", "Aida Khajavirad" ], "published_online": "2023-02-16", "volume": "202", "issue": "1-2", "pages": "1-46", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01932-w/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01936-6", "title": "Lyapunov stability of the subgradient method with constant step size", "authors": [ "Cédric Josz", "Lexiao Lai" ], "published_online": "2023-02-16", "volume": "202", "issue": "1-2", "pages": "387-396", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01936-6/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01933-9", "title": "A new perspective on low-rank optimization", "authors": [ "Dimitris Bertsimas", "Ryan Cory-Wright", "Jean Pauphilet" ], "published_online": "2023-02-18", "volume": "202", "issue": "1-2", "pages": "47-92", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01933-9/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "The Matrix Perspective Function and its Applications", "element": "p", "matched_phrases": [ "open_question" ], "passage": "Theorem 1 only uses the fact that f is an operator function with \\(\\omega \\) satisfying Assumption 1, not the fact that f is matrix convex. In other words, (19) is always an equivalent reformulation of (18). An interesting question is to identify the set of necessary conditions for the objective of (19) to be convex in \\((\\varvec{X},\\varvec{Y})\\) – f being matrix convex is clearly sufficient. The objective in (19) is convex only as long as \\({\\text {tr}}\\left( g_f \\right) \\) is. Interestingly, this is not equivalent to the convexity of \\(\\textrm{tr}(f)\\). See Appendix B.3 for a counter-example. It is, however, an open question whether a weaker notion than matrix convexity could ensure the joint convexity of \\(\\textrm{tr}(g_f)\\). It would also be interesting to investigate the benefits and the tractability of non-convex penalties (either by having f not matrix convex or \\(\\omega \\) non-convex), given the successes of non-convex penalty functions in sparse regression problems [30, 79]." }, { "section": "The Matrix Perspective Function and its Applications", "element": "p", "matched_phrases": [ "future_work" ], "passage": "Note that, in the above formulation, we extended our results for symmetric matrices to rectangular matrices \\(\\varvec{X} \\in \\mathbb {R}^{n \\times m}\\) without justification. We rigorously derive this extension for \\(f(\\varvec{X}) = \\varvec{X}^\\top \\varvec{X}\\) in Appendix D and defer the study of the general case to future research." }, { "section": "Conclusion", "element": "p", "matched_phrases": [ "future_work" ], "passage": "Future work could take three directions: (1) automatically detecting structures where the MPRT could be applied, as is already done for perspective reformulations in the MIO case by |CPLEX| and |Gurobi|, (2) developing scalable semidefinite-free techniques for solving the semidefinite relaxations proposed in this paper, and (3) combining the ideas in this paper and in our prior work [11] with custom branching strategies to solve low-rank problems to optimality at scale." }, { "section": "Appendices", "element": "p", "matched_phrases": [ "future_work" ], "passage": "where \\(\\varvec{X}=\\varvec{U}\\textrm{Diag}\\left( \\sigma _1^x, \\dots , \\sigma _m^x\\right) \\varvec{V}^\\top \\) is a singular value decomposition of \\(\\varvec{X}\\) and \\(\\omega \\) is a convex function satisfying Assumption 1. Again, \\(f_\\omega \\) could arbitrarily be defined as preserving \\(\\varvec{U}\\) or \\(\\varvec{V}\\). For these functions, the perspective \\(g_{f_\\omega }(\\varvec{X},\\varvec{Y},\\varvec{X})\\) is well defined for \\(\\varvec{Y},\\varvec{Z} \\succ \\varvec{0}\\). Unlike in the quadratic case, however, its value will depend on both \\(\\varvec{Y}\\) and \\(\\varvec{Z}\\). Developing the theoretical tools necessary to extend the MPRT to rectangular matrices, is therefore a question for future research." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01929-5", "title": "A simple and fast linear-time algorithm for divisor methods of apportionment", "authors": [ "Raphael Reitzig", "Sebastian Wild" ], "published_online": "2023-02-21", "volume": "203", "issue": "1-2", "pages": "187-205", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01929-5/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [ { "section": "Conclusion", "element": "p", "matched_phrases": [ "future_work" ], "passage": "A closely related area where this quest has not conclusively been achieved is bi-proportional apportionment (double proportionality) [1, 2, 9]. We leave the question whether new insights from the one-dimensional version can be put to good use in the two-dimensional variant for future work." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01937-5", "title": "Global convergence of the gradient method for functions definable in o-minimal structures", "authors": [ "Cédric Josz" ], "published_online": "2023-02-23", "volume": "202", "issue": "1-2", "pages": "355-383", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01937-5/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01935-7", "title": "Inequality constrained stochastic nonlinear optimization via active-set sequential quadratic programming", "authors": [ "Sen Na", "Mihai Anitescu", "Mladen Kolar" ], "published_online": "2023-03-02", "volume": "202", "issue": "1-2", "pages": "279-353", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01935-7/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Conclusion", "element": "p", "matched_phrases": [ "remains_open" ], "passage": "The extension of this work includes studying more advanced StoSQP schemes. As mentioned in Sect. 3.5, the proposed StoSQP scheme has to solve the SQP system exactly. We note that, recently, [18] designed a StoSQP scheme where an inexact Newton direction is employed, and [3] designed a StoSQP scheme to relax LICQ condition. It is still open how to design related schemes to achieve relaxations with inequality constraints. In addition, some advanced SQP schemes solve inequality constrained problems by mixing IQP with EQP: one solves a convex IQP to obtain an active set, and then solves an EQP to obtain the search direction. See the “SQP+\" scheme in [37] for example. Investigating this kind of mixed scheme with a stochastic objective is promising. Besides SQP, there are other classical methods for solving nonlinear problems that can be exploited to deal with stochastic objectives, such as the augmented Lagrangian methods and interior point methods. Different methods have different benefits and all of them deserve studying in the setup where the model can only be accessed with certain noise." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01931-x", "title": "A gradient sampling algorithm for stratified maps with applications to topological data analysis", "authors": [ "Jacob Leygonie", "Mathieu Carrière", "Théo Lacombe", "Steve Oudot" ], "published_online": "2023-03-04", "volume": "202", "issue": "1-2", "pages": "199-239", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01931-x/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01926-8", "title": "Phragmén’s voting methods and justified representation", "authors": [ "Markus Brill", "Rupert Freeman", "Svante Janson", "Martin Lackner" ], "published_online": "2023-03-06", "volume": "203", "issue": "1-2", "pages": "47-76", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01926-8/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [ { "section": "Related work", "element": "p", "matched_phrases": [ "open_question" ], "passage": "Computing the outcome of PAV is NP-hard [4, 60] and thus not feasible in polynomial time unless \\(\\text {P}=\\text {NP}\\). Prior to our work, it had remained an open question whether there exist polynomial-time computable rules satisfying EJR or PJR. Phragmén ’s sequential rule, as we show in this paper, is polynomial-time computable and satisfies PJR." }, { "section": "Computational aspects", "element": "p", "matched_phrases": [ "remains_to_determine" ], "passage": "It remains to be shown that leximax-Phragmén and var-Phragmén are contained in NP. This is not immediate as a witness for a Yes-Instance (i.e., a load distribution) may not have a polynomially-sized bit representation. In other words, the fractions in the load distribution may have very large numerators and denominators. To resolve this issue, we encode leximax-Phragmén as a mixed-integer linear program (see the discussion following this proof). Solving a mixed-integer linear program (i.e., its corresponding decision problem) is known to be NP-complete [59].Footnote 8 For showing NP-membership of var-Phragmén, we proceed in a similar fashion: we encode it as a mixed-integer quadratic program (see Theorem 5.3). NP-membership then follows from a result by Pia et al. [48]. \\(\\square \\)" }, { "section": "Conclusion", "element": "p", "matched_phrases": [ "open_problem" ], "passage": "Since seq-Phragmén violates EJR, it remains an open problem whether EJR is compatible with committee monotonicity.Footnote 17 Further, the intricate nature of Example 6.9 seems to suggest that instances on which seq-Phragmén violates EJR are rare. It would be interesting to see whether seq-Phragmén satisfies EJR for realistic distributions of preferences and/or for reasonable domain restrictions.Footnote 18 Finally, it would be of great interest to find axiomatic characterizations of Phragmén ’s rules, i.e., to find sets of axiomatic properties that uniquely define leximax-Phragmén, var-Phragmén, seq-Phragmén, and Eneström-Phragmén." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01938-4", "title": "Analysis of the optimization landscape of Linear Quadratic Gaussian (LQG) control", "authors": [ "Yujie Tang", "Yang Zheng", "Na Li" ], "published_online": "2023-03-20", "volume": "202", "issue": "1-2", "pages": "399-444", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01938-4/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01927-7", "title": "No-regret algorithms in on-line learning, games and convex optimization", "authors": [ "Sylvain Sorin" ], "published_online": "2023-03-23", "volume": "203", "issue": "1-2", "pages": "645-686", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01927-7/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01939-3", "title": "Graphs with $$G^p$$-connected medians", "authors": [ "Laurine Bénéteau", "Jérémie Chalopin", "Victor Chepoi", "Yann Vaxès" ], "published_online": "2023-03-23", "volume": "203", "issue": "1-2", "pages": "369-420", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01939-3/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01940-w", "title": "Sample average approximation with heavier tails II: localization in stochastic convex optimization and persistence results for the Lasso", "authors": [ "Roberto I. Oliveira", "Philip Thompson" ], "published_online": "2023-03-23", "volume": "199", "issue": "1-2", "pages": "49-86", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01940-w/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01942-8", "title": "First- and second-order optimality conditions for second-order cone and semidefinite programming under a constant rank condition", "authors": [ "Roberto Andreani", "Gabriel Haeser", "Leonardo M. Mito", "Héctor Ramírez", "Thiago P. Silveira" ], "published_online": "2023-03-23", "volume": "202", "issue": "1-2", "pages": "473-513", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01942-8/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01947-3", "title": "Publisher Correction to: A new perspective on low-rank optimization", "authors": [ "Dimitris Bertsimas", "Ryan Cory-Wright", "Jean Pauphilet" ], "published_online": "2023-03-23", "volume": "202", "issue": "1-2", "pages": "93-94", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01947-3/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01948-2", "title": "Publisher Correction to: Lyapunov stability of the subgradient method with constant step size", "authors": [ "Cédric Josz", "Lexiao Lai" ], "published_online": "2023-03-23", "volume": "202", "issue": "1-2", "pages": "397-398", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01948-2/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01941-9", "title": "A trust region method for noisy unconstrained optimization", "authors": [ "Shigeng Sun", "Jorge Nocedal" ], "published_online": "2023-03-24", "volume": "202", "issue": "1-2", "pages": "445-472", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01941-9/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01943-7", "title": "Gradient regularization of Newton method with Bregman distances", "authors": [ "Nikita Doikov", "Yurii Nesterov" ], "published_online": "2023-03-24", "volume": "204", "issue": "1-2", "pages": "1-25", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01943-7/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01934-8", "title": "Discrete potential mean field games: duality and numerical resolution", "authors": [ "J. Frédéric Bonnans", "Pierre Lavigne", "Laurent Pfeiffer" ], "published_online": "2023-03-28", "volume": "202", "issue": "1-2", "pages": "241-278", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01934-8/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01946-4", "title": "Shapes and recession cones in mixed-integer convex representability", "authors": [ "Ilias Zadik", "Miles Lubin", "Juan Pablo Vielma" ], "published_online": "2023-03-30", "volume": "204", "issue": "1-2", "pages": "739-752", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01946-4/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01952-6", "title": "Solving graph equipartition SDPs on an algebraic variety", "authors": [ "Tianyun Tang", "Kim-Chuan Toh" ], "published_online": "2023-04-01", "volume": "204", "issue": "1-2", "pages": "299-347", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01952-6/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01954-4", "title": "Geometry of vectorial martingale optimal transportations and duality", "authors": [ "Tongseok Lim" ], "published_online": "2023-04-08", "volume": "204", "issue": "1-2", "pages": "349-383", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01954-4/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01958-0", "title": "Hyperbolicity cones are amenable", "authors": [ "Bruno F. Lourenço", "Vera Roshchina", "James Saunderson" ], "published_online": "2023-04-08", "volume": "204", "issue": "1-2", "pages": "753-764", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01958-0/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Introduction", "element": "p", "matched_phrases": [ "unresolved", "conjecture" ], "passage": "The generalized Lax conjecture asserts that every hyperbolicity cone is a spectrahedral cone. This is known to hold (in a stronger form) in dimension three [5, 6], but is unresolved in general. Examples of hyperbolic polynomials with hyperbolicity cones that are not currently known to be spectrahedral include mixed discriminants of partially specified tuples of symmetric matrices (see, e.g., [14, Section 6]), certain hyperbolic polynomials associated with (hyper)graphs [1], and certain cubic hyperbolic polynomials associated with the size of the largest clique in a graph [13]." }, { "section": "Introduction", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "If the generalized Lax conjecture were true, then any geometric property of spectrahedral cones should also hold for hyperbolicity cones. In this paper, we consider properties related to the faces (see Sect. 2.1 for a definition) of hyperbolicity cones that are known to hold for spectrahedral cones." }, { "section": "Introduction", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "It was recently established that spectrahedral cones are amenable [8, Corollary 3.5]. In light of the Lax conjecture, it is natural to ask whether hyperbolicity cones are also amenable. Our main result is the following." }, { "section": "Introduction", "element": "p", "matched_phrases": [ "unknown_whether" ], "passage": "Prior to this, it was known that hyperbolicity cones are facially exposed [11, Theorem 23], but it was not known whether every hyperbolicity cone is nice, let alone amenable. A simple consequence of Theorem 1.1 and [7, Proposition 13] is that every hyperbolicity cone is nice." }, { "section": "Discussion", "element": "p", "matched_phrases": [ "remains_open" ], "passage": "It could be fruitful to examine in more depth the geometry and facial structure of hyperbolicity cones and check how it compares with spectrahedral cones. One recent result in this style is that the tangent cone to a hyperbolicity cone at a point is, again, a hyperbolicity cone [14, Theorem 5.9], which parallels the analogous result for spectrahedral cones. For a direction where many questions remain open, it might be interesting to see how spectrahedral cones and hyperbolicity cones fare with regards to projectional exposedness." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01960-6", "title": "FISTA is an automatic geometrically optimized algorithm for strongly convex functions", "authors": [ "J.-F. Aujol", "Ch. Dossal", "A. Rondepierre" ], "published_online": "2023-04-08", "volume": "204", "issue": "1-2", "pages": "449-491", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01960-6/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01945-5", "title": "Fair division of graphs and of tangled cakes", "authors": [ "Ayumi Igarashi", "William S. Zwicker" ], "published_online": "2023-04-15", "volume": "203", "issue": "1-2", "pages": "931-975", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01945-5/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [ { "section": "Introduction", "element": "p", "matched_phrases": [ "open_problem" ], "passage": "\\([\\clubsuit ]\\) “For the case of three or more agents, it is a challenging open problem to find an infinite class of non-traceable graphs that guarantee EF1\" [from [4], introduction]." }, { "section": "Introduction", "element": "p", "matched_phrases": [ "open_problem" ], "passage": "Our investigations can be seen as an immediate follow-up to [4]—one aimed directly at addressing this open problem. Here, we provide two somewhat different answers (which are restated more precisely in Sect. 9):" }, { "section": "Introduction", "element": "p", "matched_phrases": [ "we_do_not_know" ], "passage": "Bei et al. [2] showed that the complete bipartite graph \\(K_{a,b}\\) guarantees EF1 for n agents whenever \\(a, b \\ge n\\). As \\(K_{a,b}\\) is non-traceable whenever \\(|a - b| \\ge 2\\), classes of bipartite graphs provide an alternative to \\([\\clubsuit \\), Answer #1]. It is open whether or not these results extend to their topological classes – we do not know whether adding degree 2 subdivision vertices along edges of \\(K_{a,b}\\) always preserves its EF1 guarantee – but the question in [4] was not restricted to topological classes. Note that when \\(a - b \\ge 2\\), \\(K_{a,b}\\) fails to guarantee EF1 for \\(b+1\\) agents; this follows because some agent gets none of the vertices on the b side, limiting that agent’s share to a single vertex on the a side. Some other agent must get at least two vertices on the a side along with one on the b side to connect those two, for a total of three or more vertices." }, { "section": "Conclusions: envy-freeness for tangles and graphs", "element": "p", "matched_phrases": [ "open_problem" ], "passage": "\\([\\clubsuit ]\\) “For the case of three or more agents, it is a challenging open problem to find an infinite class of non-traceable graphs that guarantee EF1.\"" }, { "section": "Conclusions: envy-freeness for tangles and graphs", "element": "p", "matched_phrases": [ "left_open" ], "passage": "Important questions left open by our investigations here include:" }, { "section": "Notes", "element": "li", "matched_phrases": [ "we_do_not_know" ], "passage": "The definition does not say, “guaranteed for n or fewer agents.\" In fact, we do not know if the following “population monotonicity\" holds: if a tangle guarantees envy-freeness for n agents, then it guarantees envy-freeness for \\(n-1\\) agents." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01959-z", "title": "Diverse collections in matroids and graphs", "authors": [ "Fedor V. Fomin", "Petr A. Golovach", "Fahad Panolan", "Geevarghese Philip", "Saket Saurabh" ], "published_online": "2023-04-15", "volume": "204", "issue": "1-2", "pages": "415-447", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01959-z/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Conclusion", "element": "p", "matched_phrases": [ "open_question" ], "passage": "We conclude with a list of open questions:" } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01950-8", "title": "Small separations in pinch-graphic matroids", "authors": [ "Bertrand Guenin", "Cheolwon Heo" ], "published_online": "2023-04-19", "volume": "204", "issue": "1-2", "pages": "81-111", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01950-8/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01953-5", "title": "Optimal algorithms for differentially private stochastic monotone variational inequalities and saddle-point problems", "authors": [ "Digvijay Boob", "Cristóbal Guzmán" ], "published_online": "2023-04-19", "volume": "204", "issue": "1-2", "pages": "255-297", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01953-5/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Introduction", "element": "li", "matched_phrases": [ "open_problem" ], "passage": "1. Stochastic Variational Inequalities and Saddle-Point Problems: Variational inequalities and saddle-point problems are classical topics in applied mathematics, operations research and engineering (e.g., [3, 18, 33, 38, 40,41,42,43, 45]). Their stochastic counterparts have only gained traction recently, mainly motivated by their applications in machine learning (e.g., [24, 25, 29, 30, 34] and references therein). For the stochastic version of (SP(f)), [39] proposed a robust stochastic approximation method. The first optimal algorithm for SVI with monotone Lipschitz operators was obtained by Juditsky, Nemirovski and Tauvel [30], and very recently Kotsalis, Lan and Li [34] developed optimal variants for the strongly monotone case (in terms of distance to the optimum criterion, rather than VI gap). It is important to note that naive adaptation of these methods to the DP setting requires adding noise to the operator evaluations at every iteration, which substantially degrades the accuracy of the obtained solution. A careful privacy accounting and minibatch schedule can lead to optimal guarantees for single-pass methods [19], however this requires accuracy guarantees for the last iterate, which is currently an open problem for SVI and SSP (aside from specific cases, typically involving strong monotonicity conditions, e.g., [24, 34]). We circumvent this problem by providing population risk guarantees for multipass methods." }, { "section": "Introduction", "element": "p", "matched_phrases": [ "open_problem" ], "passage": "It is important to note that naive adaptation of these methods to the DP setting requires adding noise to the operator evaluations at every iteration, which substantially degrades the accuracy of the obtained solution. A careful privacy accounting and minibatch schedule can lead to optimal guarantees for single-pass methods [19], however this requires accuracy guarantees for the last iterate, which is currently an open problem for SVI and SSP (aside from specific cases, typically involving strong monotonicity conditions, e.g., [24, 34]). We circumvent this problem by providing population risk guarantees for multipass methods." }, { "section": "Introduction", "element": "li", "matched_phrases": [ "future_work" ], "passage": "3. Differential Privacy: Differential privacy is the gold standard for private data analysis, and it has been studied for nearly 20 years [15, 16]. Beyond its classical definition, multiple variants have been introduced, including local [14, 31], concentrated [10], Rényi [37], and Gaussian [13]. Relevant to the optimization community are the applications of differential privacy to combinatorial optimization [22]. Differentially private empirical risk minimization and stochastic convex optimization have been extensively studied for over a decade (see, e.g. [5, 7, 11, 12, 19, 26, 27, 32, 47]). Relevant to our work are the first optimal risk algorithms for DP-ERM [7] and DP-SCO [5]. Non-Euclidean extensions have also been obtained recently [2, 6]. To the best of our knowledge, our work is the first to address DP algorithms for SVI and SSP. Our approach for generalization of multipass algorithms is inspired by the noisy SGD analysis in [4]. However, our stability analysis differs crucially from [4]: in the case of NSEG, we need to carefully address the double operator evaluation of the extragradient step, which is done by using the fact that the extragradient operator is approximately nonexpansive. In the case of NISPP, we leverage the contraction properties of strongly monotone VI solutions. By contrast, SGD in the nonsmooth case is far from nonexpansive [4]. Alternative approaches to obtain optimal risk in DP-SCO, including privacy amplification by iteration [19, 20], and phased regularization or phased SGD [19], appear to run into fundamental limitations when applied to DP-SVI and DP-SSP. It is an interesting future research direction to obtain faster running times with optimal population risk in DP-SVI and DP-SSP, which may benefit from these alternative approaches." }, { "section": "Introduction", "element": "p", "matched_phrases": [ "future_work" ], "passage": "Differentially private empirical risk minimization and stochastic convex optimization have been extensively studied for over a decade (see, e.g. [5, 7, 11, 12, 19, 26, 27, 32, 47]). Relevant to our work are the first optimal risk algorithms for DP-ERM [7] and DP-SCO [5]. Non-Euclidean extensions have also been obtained recently [2, 6]. To the best of our knowledge, our work is the first to address DP algorithms for SVI and SSP. Our approach for generalization of multipass algorithms is inspired by the noisy SGD analysis in [4]. However, our stability analysis differs crucially from [4]: in the case of NSEG, we need to carefully address the double operator evaluation of the extragradient step, which is done by using the fact that the extragradient operator is approximately nonexpansive. In the case of NISPP, we leverage the contraction properties of strongly monotone VI solutions. By contrast, SGD in the nonsmooth case is far from nonexpansive [4]. Alternative approaches to obtain optimal risk in DP-SCO, including privacy amplification by iteration [19, 20], and phased regularization or phased SGD [19], appear to run into fundamental limitations when applied to DP-SVI and DP-SSP. It is an interesting future research direction to obtain faster running times with optimal population risk in DP-SVI and DP-SSP, which may benefit from these alternative approaches." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01951-7", "title": "Recognizing pinch-graphic matroids", "authors": [ "Bertrand Guenin", "Cheolwon Heo" ], "published_online": "2023-04-20", "volume": "204", "issue": "1-2", "pages": "113-134", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01951-7/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01957-1", "title": "A J-symmetric quasi-newton method for minimax problems", "authors": [ "Azam Asl", "Haihao Lu", "Jinwen Yang" ], "published_online": "2023-04-20", "volume": "204", "issue": "1-2", "pages": "207-254", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01957-1/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01944-6", "title": "Recognizing even-cycle and even-cut matroids", "authors": [ "Bertrand Guenin", "Cheolwon Heo" ], "published_online": "2023-04-26", "volume": "202", "issue": "1-2", "pages": "515-542", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01944-6/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01955-3", "title": "Modeling combinatorial disjunctive constraints via junction trees", "authors": [ "Bochuan Lyu", "Illya V. Hicks", "Joey Huchette" ], "published_online": "2023-04-26", "volume": "204", "issue": "1-2", "pages": "385-413", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01955-3/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01961-5", "title": "Approximation algorithms for flexible graph connectivity", "authors": [ "Sylvia Boyd", "Joseph Cheriyan", "Arash Haddadan", "Sharat Ibrahimpur" ], "published_online": "2023-04-26", "volume": "204", "issue": "1-2", "pages": "493-516", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01961-5/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01956-2", "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" ], "published_online": "2023-04-29", "volume": "204", "issue": "1-2", "pages": "135-206", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01956-2/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Abstract", "element": "p", "matched_phrases": [ "open_question" ], "passage": "Following the breakthrough work of Tardos (Oper Res 34:250–256, 1986) in the bit-complexity model, Vavasis and Ye (Math Program 74(1):79–120, 1996) gave the first exact algorithm for linear programming in the real model of computation with running time depending only on the constraint matrix. For solving a linear program (LP) \\(\\max \\, c^\\top x,\\, Ax = b,\\, x \\ge 0,\\, A \\in \\mathbb {R}^{m \\times n}\\), Vavasis and Ye developed a primal-dual interior point method using a ‘layered least squares’ (LLS) step, and showed that \\(O(n^{3.5} \\log (\\bar{\\chi }_A+n))\\) iterations suffice to solve (LP) exactly, where \\(\\bar{\\chi }_A\\) is a condition measure controlling the size of solutions to linear systems related to A. Monteiro and Tsuchiya (SIAM J Optim 13(4):1054–1079, 2003), noting that the central path is invariant under rescalings of the columns of A and c, asked whether there exists an LP algorithm depending instead on the measure \\(\\bar{\\chi }^*_A\\), defined as the minimum \\(\\bar{\\chi }_{AD}\\) value achievable by a column rescaling AD of A, and gave strong evidence that this should be the case. We resolve this open question affirmatively. Our first main contribution is an \\(O(m^2 n^2 + n^3)\\) time algorithm which works on the linear matroid of A to compute a nearly optimal diagonal rescaling D satisfying \\(\\bar{\\chi }_{AD} \\le n(\\bar{\\chi }_A^*)^3\\). This algorithm also allows us to approximate the value of \\(\\bar{\\chi }_A\\) up to a factor \\(n (\\bar{\\chi }_A^*)^2\\). This result is in surprising contrast to that of Tunçel (Math Program 86(1):219–223, 1999), who showed NP-hardness for approximating \\(\\bar{\\chi }_A\\) to within \\(2^{\\textrm{poly}(\\textrm{rank}(A))}\\). The key insight for our algorithm is to work with ratios \\(g_i/g_j\\) of circuits of A—i.e., minimal linear dependencies \\(Ag=0\\)—which allow us to approximate the value of \\(\\bar{\\chi }_A^*\\) by a maximum geometric mean cycle computation in what we call the ‘circuit ratio digraph’ of A. While this resolves Monteiro and Tsuchiya’s question by appropriate preprocessing, it falls short of providing either a truly scaling invariant algorithm or an improvement upon the base LLS analysis. In this vein, as our second main contribution we develop a scaling invariant LLS algorithm, which uses and dynamically maintains improving estimates of the circuit ratio digraph, together with a refined potential function based analysis for LLS algorithms in general. With this analysis, we derive an improved \\(O(n^{2.5} \\log (n)\\log (\\bar{\\chi }^*_A+n))\\) iteration bound for optimally solving (LP) using our algorithm. The same argument also yields a factor \\(n/\\log n\\) improvement on the iteration complexity bound of the original Vavasis–Ye algorithm." }, { "section": "Introduction", "element": "p", "matched_phrases": [ "open_question" ], "passage": "Khachiyan [23] used the ellipsoid method to give the first polynomial time LP algorithm in the bit-complexity model, that is, polynomial in the bit description length of (A, b, c). An outstanding open question is the existence of a strongly polynomial algorithm for LP, listed by Smale as one of the most prominent mathematical challenges for the 21st century [46]. Such an algorithm amounts to solving LP using \\(\\textrm{poly}(n,m)\\) basic arithmetic operations in the real model of computation.Footnote 1 Known strongly polynomially solvable LP problems classes include: feasibility for two variable per inequality systems [33], the minimum-cost circulation problem [50], the maximum generalized flow problem [41, 61], and discounted Markov decision problems [65, 67]." }, { "section": "Introduction", "element": "p", "matched_phrases": [ "open_problem" ], "passage": "A related open problem to the above is whether it is possible to compute a near-optimal rescaling D for program (1)? This would give an alternate pathway to the desired LP algorithm by simply preprocessing the matrix A. The related question of approximating \\(\\bar{\\chi }_A\\) was already studied by Tunçel [54], who showed NP-hardness for approximating \\(\\bar{\\chi }_A\\) to within a \\(2^{\\textrm{poly}(\\textrm{rank}(A))}\\) factor. Taken at face value, this may seem to suggest that approximating the rescaling D should be hard." }, { "section": "Introduction", "element": "p", "matched_phrases": [ "open_question" ], "passage": "A further open question is whether Vavasis and Ye’s cross-over analysis can be improved. Ye showed in [66] that the iteration complexity can be reduced to \\(O(n^{2.5} \\log (\\bar{\\chi }_A+n))\\) for feasibility problems and further to \\(O(n^{1.5} \\log (\\bar{\\chi }_A+n))\\) for homogeneous systems, though the \\(O(n^{3.5} \\log (\\bar{\\chi }_A+n))\\) bound for optimization has not been improved since [63]." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01963-3", "title": "On the complexity of separating cutting planes for the knapsack polytope", "authors": [ "Alberto Del Pia", "Jeff Linderoth", "Haoran Zhu" ], "published_online": "2023-05-04", "volume": "206", "issue": "1-2", "pages": "33-59", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01963-3/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [ { "section": "Abstract", "element": "p", "matched_phrases": [ "open_problem" ], "passage": "We close three open problems on the separation complexity of valid inequalities for the knapsack polytope. Specifically, we establish that the separation problems for extended cover inequalities, (1, k)-configuration inequalities, and weight inequalities are all \\(\\mathcal{N}\\mathcal{P}\\)-complete. We also show that, when the number of constraints of the LP relaxation is constant and its optimal solution is an extreme point, then the separation problems of both extended cover inequalities and weight inequalities can be solved in polynomial time. Moreover, we provide a natural generalization of (1, k)-configuration inequality which is easier to separate and contains the original (1, k)-configuration inequality as a strict sub-family." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01966-0", "title": "Comparing solution paths of sparse quadratic minimization with a Stieltjes matrix", "authors": [ "Ziyu He", "Shaoning Han", "Andrés Gómez", "Ying Cui", "Jong-Shi Pang" ], "published_online": "2023-05-04", "volume": "204", "issue": "1-2", "pages": "517-566", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01966-0/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Some details of the problem setting", "element": "li", "matched_phrases": [ "open_question" ], "passage": "The optimal objective value function \\(f_1(\\gamma )\\) of the \\(\\ell _1\\)-problem is concave, nondecreasing, once continuously differentiable, and piecewise linear-quadratic composed of finitely many quadratic functions over an exponential number of non-overlapping intervals. The evaluation of \\(f_1(\\gamma )\\) for each fixed \\(\\gamma > 0\\) can be accomplished by a strongly polynomially bounded algorithm. The reference [35] contains a class of \\(\\ell _1\\)-regularized problems where the number of smooth pieces of the solution path is exponential; yet Q in these problems cannot be a Z-matrix. While it remains an open question to date whether the number of such pieces of the solution path is an exponential or polynomial function of n when Q is a Stieltjes matrix, there are cases [29, Sect. 6] where this number is linear in n." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01969-x", "title": "The maximum measure of non-trivial 3-wise intersecting families", "authors": [ "Norihide Tokushige" ], "published_online": "2023-05-09", "volume": "204", "issue": "1-2", "pages": "643-676", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01969-x/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01964-2", "title": "On the maximal number of columns of a $$\\Delta $$-modular integer matrix: bounds and computations", "authors": [ "Gennadiy Averkov", "Matthias Schymura" ], "published_online": "2023-05-13", "volume": "206", "issue": "1-2", "pages": "61-89", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01964-2/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [ { "section": "Inline Recommendations", "element": "h3", "matched_phrases": [ "conjecture" ], "passage": "A Survey of the Marcus–de Oliveira Conjecture" }, { "section": "Introduction", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "where \\(e_i\\) denotes the ith coordinate unit vector. This is a natural generalization of the unimodular matrix (\\(\\varDelta = 1\\)) that attains Heller’s result \\({{\\,\\textrm{h}\\,}}(1,m) = m^2+m+1\\). With this perspective and their precise result (1), for \\(\\varDelta \\le 2\\) or \\(m \\le 2\\), Lee et al. [11] conjecture that the lower bound in (2) is actually the correct value of \\({{\\,\\textrm{h}\\,}}(\\varDelta ,m)\\), for any choice of \\(\\varDelta ,m \\in \\mathbb {Z}_{>0}\\)." }, { "section": "Introduction", "element": "h3", "matched_phrases": [ "conjecture" ], "passage": "Conjecture 1" }, { "section": "Introduction", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "On the qualitative side, Conjecture 1 implies that \\({{\\,\\textrm{h}\\,}}(\\varDelta ,m) \\le a(\\varDelta ) m^2 + b(\\varDelta ) m+c\\) holds with \\(a(\\varDelta ) \\in \\mathcal {O}(1)\\) and \\(b(\\varDelta ) \\in \\mathcal {O}(\\varDelta )\\). If this is true, we would also have \\({{\\,\\textrm{h}\\,}}(\\varDelta ,m) \\le \\mathcal {O}(\\varDelta ) m^2\\), but even this estimate has not yet been confirmed since the currently available bounds are asymptotically too large for \\(\\varDelta \\rightarrow \\infty \\). The authors of [11, p. 24] ask if there exists a bound of the form \\(\\mathcal {O}(m^d) \\varDelta \\), for some constant \\(d \\in \\mathbb {Z}_{>0}\\). As our main result, we answer this question in the affirmative by establishing a bound of \\(\\mathcal {O}(m^4) \\varDelta \\)." }, { "section": "Introduction", "element": "p", "matched_phrases": [ "open_question" ], "passage": "It remains an open question whether our bound can be improved, for all \\(\\varDelta \\), to a bound of \\(\\mathcal {O}(m^d) \\varDelta \\) for some exponent \\(d < 4\\)." }, { "section": "Introduction", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "Based on computational experiments for small values of m and \\(\\varDelta \\), we found counterexamples to Conjecture 1 for \\(\\varDelta \\in \\{4,8,16\\}\\) and sufficiently large m." }, { "section": "Introduction", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "These lower bounds exceed the conjecture for \\({{\\,\\textrm{h}\\,}}(4,m),{{\\,\\textrm{h}\\,}}(8,m)\\) and \\({{\\,\\textrm{h}\\,}}(16,m)\\) by Lee et al. by the additive terms \\(2(m-2)\\), \\(4(m-3)\\), and \\(2(m-9)\\), respectively. This means that the qualitative side of Conjecture 1 as described above still stands." }, { "section": "Introduction", "element": "p", "matched_phrases": [ "open_problem", "conjecture" ], "passage": "Organization of the paper The paper is organized as follows. In Sect. 2, we describe our geometric idea that explains the linearity in \\(\\varDelta \\) of the bounds in Theorem 1, and we introduce two variants of the Heller constant \\({{\\,\\textrm{h}\\,}}(1,m)\\) which we polynomially bound in Sects. 3 and 4. In Sect. 5, we describe our approach to compute the generalized Heller constant \\({{\\,\\textrm{h}\\,}}(\\varDelta ,m)\\) for small parameters \\(\\varDelta ,m\\). We also discuss the results of our computer experiments, and identify counterexamples to Conjecture 1 whose structure leads us to construct the lower bounds in Theorem 2. Finally, in Sect. 6, we pose some natural open problems that result from our investigations." }, { "section": "Counting by residue classes", "element": "li", "matched_phrases": [ "future_work", "conjecture" ], "passage": "(iii) The separation of \\({{\\,\\textrm{h}\\,}}(1,m)\\) and \\({{\\,\\mathrm{h_s}\\,}}^\\varDelta (m)\\) in Lemma 1 is possible because we excluded \\(t = \\textbf{0}\\) in the definition of both \\({{\\,\\mathrm{h_s}\\,}}(m)\\) and \\({\\text {h}_\\text {s}}^{\\delta }{(m)}\\). This is motivated by a comparison to Conjecture 1; we know the exact value of \\({{\\,\\textrm{h}\\,}}(1,m)\\), while we may hope to bound \\({{\\,\\mathrm{h_s}\\,}}^\\varDelta (m)\\) subquadratically in future work." }, { "section": "Counting by residue classes", "element": "p", "matched_phrases": [ "future_work", "conjecture" ], "passage": "The separation of \\({{\\,\\textrm{h}\\,}}(1,m)\\) and \\({{\\,\\mathrm{h_s}\\,}}^\\varDelta (m)\\) in Lemma 1 is possible because we excluded \\(t = \\textbf{0}\\) in the definition of both \\({{\\,\\mathrm{h_s}\\,}}(m)\\) and \\({\\text {h}_\\text {s}}^{\\delta }{(m)}\\). This is motivated by a comparison to Conjecture 1; we know the exact value of \\({{\\,\\textrm{h}\\,}}(1,m)\\), while we may hope to bound \\({{\\,\\mathrm{h_s}\\,}}^\\varDelta (m)\\) subquadratically in future work." }, { "section": "Computational experiments and the proof of Theorem 2", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "In this section, we describe a computational approach to determine so far unknown values of \\({{\\,\\textrm{h}\\,}}(\\varDelta ,m)\\) for small parameters \\(m,\\varDelta \\in \\mathbb {Z}_{>0}\\). The results of our computations led us to identify a family of counterexamples to Conjecture 1 that lie behind the lower bounds in Theorem 2. Our approach is based on the sandwich factory classification scheme described and utilized by Averkov, Borger and Soprunov [3, Sect. 7.3]." }, { "section": "Computational experiments and the proof of Theorem 2", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "Table 1 determines the previously unknown exact values of \\({{\\,\\textrm{h}\\,}}(\\varDelta ,m)\\), for \\(m=3\\) and \\(3 \\le \\varDelta \\le 13\\), as well as for \\((\\varDelta ,m) = (3,4)\\). It also reveals a counterexample to Conjecture 1 for the case \\((\\varDelta ,m) = (4,3)\\), whose structure we used to construct counterexamples for every \\((\\varDelta ,m) \\in \\{ (4,4) , (4,5) , (8,4) , (8,5) \\}\\) as well. The construction is discussed further below in the next section." }, { "section": "Computational experiments and the proof of Theorem 2", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "Based on this lemma we can now construct series of examples that exceed the conjectured value of \\({{\\,\\textrm{h}\\,}}(4,m),{{\\,\\textrm{h}\\,}}(8,m)\\), and \\({{\\,\\textrm{h}\\,}}(16,m)\\) in Conjecture 1 by an additive term that is linear in m, and for m large enough. Our construction is based on the unique (up to unimodular equivalence) set attaining the value \\({{\\,\\textrm{h}\\,}}(4,3) = 33\\), resulting from our enumeration approach described earlier (see Tables 1 and 3). This set can be written as" }, { "section": "Computational experiments and the proof of Theorem 2", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "The conjectured value in Conjecture 1 is \\({{\\,\\textrm{h}\\,}}(2^\\ell ,m) = m^2 + (2^{\\ell +1} - 1) m + 1\\). Using \\(|H_\\ell | = \\ell ^2 + \\ell + 1\\) and (9), this means that \\(\\mathcal {D}(C_m^\\ell )\\) is a counterexample to Conjecture 1, for fixed \\(\\ell \\) and large enough m, if and only if" }, { "section": "Open problems", "element": "p", "matched_phrases": [ "open_problem" ], "passage": "The determination of the exact value of \\({{\\,\\textrm{h}\\,}}(\\varDelta ,m)\\) remains the major open problem. Note that the bounds from other sources and the bound we prove here are incomparable when both m and \\(\\varDelta \\) vary. In order to understand the limits of our method for upper bounding \\({{\\,\\textrm{h}\\,}}(\\varDelta ,m)\\), it is necessary to determine the exact asymptotic behavior of \\({{\\,\\mathrm{h_s}\\,}}(m)\\) or \\({\\text {h}_\\text {s}}^{\\varDelta }{(m)}\\). Proposition 2 suggests that the following may have an affirmative answer:" }, { "section": "Open problems", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "Does Conjecture 1 hold for every \\(m \\in \\mathbb {Z}_{>0}\\), and every prime \\(\\varDelta \\in \\mathbb {Z}_{>0}\\)? In particular, does it hold for \\({{\\,\\textrm{h}\\,}}(3,m)\\)?" }, { "section": "Open problems", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "Moreover, from the data in Table 1 one may also suspect that for any given \\(m \\in \\mathbb {Z}_{>0}\\) there are only finitely many \\(\\varDelta \\in \\mathbb {Z}_{>0}\\) that possibly violate Conjecture 1. For instance, it could very well be that the value \\({{\\,\\textrm{h}\\,}}(4,3) = 33\\) is the only exception from Conjecture 1 in dimension \\(m=3\\)." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01968-y", "title": "On computing small variable disjunction branch-and-bound trees", "authors": [ "Max Gläser", "Marc E. Pfetsch" ], "published_online": "2023-05-13", "volume": "206", "issue": "1-2", "pages": "145-173", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01968-y/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [ { "section": "Hardness of approximating the size of a smallest branch-and-bound tree using variable disjunctions", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "The exponential time hypothesis (ETH) formulated by Impagliazzo and Paturi [20] states \\(s_3 > 0\\) and the strong exponential time hypothesis (SETH) states \\(s_\\infty {:}{=}\\lim _{k \\rightarrow \\infty } s_k = 1\\). Note that \\(\\text {ETH}\\) not only rules out polynomial algorithms for NP-hard problems, but even subexponential algorithms and therefore is a strengthening of the \\(\\text {P} \\ne \\text {NP} \\)-conjecture." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01971-3", "title": "A colorful Steinitz Lemma with application to block-structured integer programs", "authors": [ "Timm Oertel", "Joseph Paat", "Robert Weismantel" ], "published_online": "2023-05-17", "volume": "204", "issue": "1-2", "pages": "677-702", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01971-3/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01949-1", "title": "A combinatorial algorithm for computing the entire sequence of the maximum degree of minors of a generic partitioned polynomial matrix with $$2 \\times 2$$ submatrices", "authors": [ "Yuni Iwamasa" ], "published_online": "2023-05-19", "volume": "204", "issue": "1-2", "pages": "27-79", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01949-1/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01978-w", "title": "On the centralization of the circumcentered-reflection method", "authors": [ "Roger Behling", "Yunier Bello-Cruz", "Alfredo N. Iusem", "Luiz-Rafael Santos" ], "published_online": "2023-05-19", "volume": "205", "issue": "1-2", "pages": "337-371", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01978-w/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01967-z", "title": "Total dual dyadicness and dyadic generating sets", "authors": [ "Ahmad Abdi", "Gérard Cornuéjols", "Bertrand Guenin", "Levent Tunçel" ], "published_online": "2023-05-22", "volume": "206", "issue": "1-2", "pages": "125-143", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01967-z/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01979-9", "title": "On the stationarity for nonlinear optimization problems with polyhedral constraints", "authors": [ "Daniela di Serafino", "William W. Hager", "Gerardo Toraldo", "Marco Viola" ], "published_online": "2023-05-22", "volume": "205", "issue": "1-2", "pages": "107-134", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01979-9/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Conclusions", "element": "p", "matched_phrases": [ "future_work" ], "passage": "As an example of this, we have introduced an active-set algorithm for the solution of convex quadratic programs, and proved its finite convergence in the case of possibly degenerate strictly convex problems. Numerical experiments on synthetic QP problems with a small number of dense constraints show the efficiency of the proposed strategy over the gradient projection method and over a tailored interior point method in the case of number of constraints between 2 and 20. Future work will deal with the implementation of an efficient C-based version of the PSAQP method and its extension to the non-quadratic case." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01962-4", "title": "Faster goal-oriented shortest path search for bulk and incremental detailed routing", "authors": [ "Markus Ahrens", "Dorothee Henke", "Stefan Rabenstein", "Jens Vygen" ], "published_online": "2023-05-24", "volume": "206", "issue": "1-2", "pages": "3-32", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01962-4/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [ { "section": "Practical aspects", "element": "p", "matched_phrases": [ "future_work" ], "passage": "The column Standard Dijkstra calls in Table 3 excludes these situations and hence shows an even larger gain than the column All Dijkstra calls. The question how to model rip-up costs efficiently when computing potentials remains for future research." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01974-0", "title": "Radial duality part II: applications and algorithms", "authors": [ "Benjamin Grimmer" ], "published_online": "2023-05-26", "volume": "205", "issue": "1-2", "pages": "69-105", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01974-0/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01965-1", "title": "On SOCP-based disjunctive cuts for solving a class of integer bilevel nonlinear programs", "authors": [ "Elisabeth Gaar", "Jon Lee", "Ivana Ljubić", "Markus Sinnl", "Kübra Tanınmış" ], "published_online": "2023-05-27", "volume": "206", "issue": "1-2", "pages": "91-124", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01965-1/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [ { "section": "Conclusions and outlook", "element": "p", "matched_phrases": [ "open_question", "remains_open", "future_work" ], "passage": "There are still many open questions for future research. The proposed B &C could be enhanced by bilevel-specific preprocessing, or bilevel-specific valid inequalities (as this has been done for MIBLPs in e.g., [18, 19]). Problem-specific strengthening inequalities could be used within disjunctions to obtain stronger DCs, and finally outer-approximation could be used as an alternative to SOCP-based separation. It also remains open to study problem generalizations involving (discrete) follower problems with (multiple) conic constraints." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01970-4", "title": "Weak notions of nondegeneracy in nonlinear semidefinite programming", "authors": [ "Roberto Andreani", "Gabriel Haeser", "Leonardo M. Mito", "Héctor Ramírez" ], "published_online": "2023-05-27", "volume": "205", "issue": "1-2", "pages": "1-32", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01970-4/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01980-2", "title": "Globally convergent coderivative-based generalized Newton methods in nonsmooth optimization", "authors": [ "Pham Duy Khanh", "Boris S. Mordukhovich", "Vo Thanh Phat", "Dat Ba Tran" ], "published_online": "2023-05-27", "volume": "205", "issue": "1-2", "pages": "373-429", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01980-2/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01973-1", "title": "Branch-and-bound performance estimation programming: a unified methodology for constructing optimal optimization methods", "authors": [ "Shuvomoy Das Gupta", "Bart P. G. Van Parys", "Ernest K. Ryu" ], "published_online": "2023-06-07", "volume": "204", "issue": "1-2", "pages": "567-639", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01973-1/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01981-1", "title": "Worst-case complexity of an SQP method for nonlinear equality constrained stochastic optimization", "authors": [ "Frank E. Curtis", "Michael J. O’Neill", "Daniel P. Robinson" ], "published_online": "2023-06-07", "volume": "205", "issue": "1-2", "pages": "431-483", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01981-1/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01982-0", "title": "On the convex hull of convex quadratic optimization problems with indicators", "authors": [ "Linchuan Wei", "Alper Atamtürk", "Andrés Gómez", "Simge Küçükyavuz" ], "published_online": "2023-06-07", "volume": "204", "issue": "1-2", "pages": "703-737", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01982-0/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01985-x", "title": "Convex hull results on quadratic programs with non-intersecting constraints", "authors": [ "Alexander Joyce", "Boshi Yang" ], "published_online": "2023-06-07", "volume": "205", "issue": "1-2", "pages": "539-558", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01985-x/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01988-8", "title": "Swarm gradient dynamics for global optimization: the mean-field limit case", "authors": [ "Jérôme Bolte", "Laurent Miclo", "Stéphane Villeneuve" ], "published_online": "2023-06-07", "volume": "205", "issue": "1-2", "pages": "661-701", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01988-8/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Abstract", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "Using jointly geometric and stochastic reformulations of nonconvex problems and exploiting a Monge–Kantorovich (or Wasserstein) gradient system formulation with vanishing forces, we formally extend the simulated annealing method to a wide range of global optimization methods. Due to the built-in combination of a gradient-like strategy and particle interactions, we call them swarm gradient dynamics. As in the original paper by Holley–Kusuoka–Stroock, a functional inequality is the key to the existence of a schedule that ensures convergence to a global minimizer. One of our central theoretical contributions is proving such an inequality for one-dimensional compact manifolds. We conjecture that the inequality holds true in a much broader setting. Additionally, we describe a general method for global optimization that highlights the essential role of functional inequalities la Łojasiewicz." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01990-0", "title": "Special Issue: International Symposium on Mathematical Programming 2022", "authors": [ "Damek Shea Davis", "Oktay Günlük", "Volker Kaibel", "Giacomo Nannicini", "Xa-Xiang Yuan" ], "published_online": "2023-06-07", "volume": "200", "issue": "2", "pages": "629-631", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01990-0/fulltext.html", "article_kind": "non_research_item", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01972-2", "title": "Correction: Global convergence of the gradient method for functions definable in o-minimal structures", "authors": [ "Cédric Josz" ], "published_online": "2023-06-12", "volume": "202", "issue": "1-2", "pages": "385-385", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01972-2/fulltext.html", "article_kind": "non_research_item", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01984-y", "title": "A solution algorithm for chance-constrained problems with integer second-stage recourse decisions", "authors": [ "Andrea Lodi", "Enrico Malaguti", "Michele Monaci", "Giacomo Nannicini", "Paolo Paronuzzi" ], "published_online": "2023-06-15", "volume": "205", "issue": "1-2", "pages": "269-301", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01984-y/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01986-w", "title": "A preconditioned iterative interior point approach to the conic bundle subproblem", "authors": [ "Christoph Helmberg" ], "published_online": "2023-06-15", "volume": "205", "issue": "1-2", "pages": "559-615", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01986-w/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Introduction", "element": "p", "matched_phrases": [ "remains_open" ], "passage": "The projection onto a random subspace may be motivated geometrically by interpreting the Gram matrix \\(VV^\\top \\) as the inner products of the row vectors of V. The result of Johnson–Lindenstrauss, cf. [1, 9], allows to approximate this with low distortion by a projection onto a low dimensional subspace. In matrix approximations this idea seems to have first appeared in [35]. In connection with preconditioning a recent probabilistic approach is described in [28] in the context of controlling the error of a LU preconditioner. [17] gives an excellent introduction to probabilistic algorithms for constructing approximate matrix decompositions and provides useful bounds. Building directly on their techniques we provide deterministic and probabilistic bounds on the condition number of the random subspace preconditioned system in Theorems 6 and 7. In comparison to the moment analysis of the Ritz values of the preconditioned matrix presented in Theorem 4, the bounds seem to fall below expectation and are maybe still improvable. Random projections do not require any problem specific structural insights, but it remains open how to choose the subspace dimension in order to obtain an efficient preconditioner." }, { "section": "Numerical experiments", "element": "p", "matched_phrases": [ "future_work" ], "passage": "Based on this analysis, a hybrid approach seems advisable that switches dynamically between the solvers depending on precision, model size and number of interior point iterations. In implementing these ideas a number of further design aspects would have to be reconsidered as outlined before. The true advantage of iterative solvers, however, is that dynamic model adaptations become feasible during the solution of the subproblem, because there is no need to recompute the Schur complement each time. This allows for entirely new strategies such as combining the ideas of [4, 34] and [25] in order to cut down on the number null steps at an early stage. This remains to be addressed in future work." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01976-y", "title": "No-regret dynamics in the Fenchel game: a unified framework for algorithmic convex optimization", "authors": [ "Jun-Kun Wang", "Jacob Abernethy", "Kfir Y. Levy" ], "published_online": "2023-06-22", "volume": "205", "issue": "1-2", "pages": "203-268", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01976-y/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01987-9", "title": "On the minimum $$s-t$$ cut problem with budget constraints", "authors": [ "Hassene Aissi", "A. Ridha Mahjoub" ], "published_online": "2023-06-23", "volume": "203", "issue": "1-2", "pages": "421-442", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01987-9/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01977-x", "title": "A theoretical and computational analysis of full strong-branching", "authors": [ "Santanu S. Dey", "Yatharth Dubey", "Marco Molinaro", "Prachi Shah" ], "published_online": "2023-06-30", "volume": "205", "issue": "1-2", "pages": "303-336", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01977-x/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01983-z", "title": "Pareto Adaptive Robust Optimality via a Fourier–Motzkin Elimination lens", "authors": [ "Dimitris Bertsimas", "Stefan C. M. ten Eikelder", "Dick den Hertog", "Nikolaos Trichakis" ], "published_online": "2023-06-30", "volume": "205", "issue": "1-2", "pages": "485-538", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01983-z/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01991-z", "title": "An abstract model for branch and cut", "authors": [ "Aleksandr M. Kazachkov", "Pierre Le Bodic", "Sriram Sankaranarayanan" ], "published_online": "2023-06-30", "volume": "206", "issue": "1-2", "pages": "175-202", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01991-z/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01992-y", "title": "Simple odd $$\\beta $$-cycle inequalities for binary polynomial optimization", "authors": [ "Alberto Del Pia", "Matthias Walter" ], "published_online": "2023-07-07", "volume": "206", "issue": "1-2", "pages": "203-238", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01992-y/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [ { "section": "Redundancy", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "Note that Proposition 11 is only stated for repetition of edges whose signs are both \\(+\\). However, we have evidence (based on computations for small instances) that also other types of closed walks yield redundant inequalities. Examples are those with repetitions of edges of arbitrary sign, those with two subsequent equal nodes, those in which two nodes are repeated and the four involved edges all have the same sign. The strongest redundancy statement that we can think of is captured in the following conjecture." }, { "section": "Redundancy", "element": "h3", "matched_phrases": [ "conjecture" ], "passage": "Conjecture 12" }, { "section": "Redundancy", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "We recall that a \\(\\beta \\)-cycle of length k, for some \\(k \\ge 3\\), is a sequence \\(v_1\\)-\\(e_1\\)-\\(v_2\\)-\\(e_2\\)-\\(\\cdots \\)-\\(v_{k}\\)-\\(e_{k}\\)-\\(v_1\\) such that \\(v_1\\), \\(v_2\\), \\(\\ldots \\), \\(v_k\\) are pairwise distinct nodes, \\(e_1\\), \\(e_2\\), \\(\\ldots \\), \\(e_k\\) are pairwise distinct edges, and \\(v_i\\) belongs to \\(e_{i-1}, e_i\\) and no other \\(e_j\\) for all \\(i = 1,\\ldots ,k\\), where \\(e_0:= e_k\\). Conjecture 12 justifies that although inequalities (5) are defined for closed walks, we call them simple odd \\(\\beta \\)-cycle inequalities. The main reason for considering closed walks instead of \\(\\beta \\)-cycles is the separation algorithm described in Sect. 4." }, { "section": "Discussion and conclusions", "element": "p", "matched_phrases": [ "open_question" ], "passage": "We would like to conclude our paper with a reflection of our computational results as well as open questions that could be investigated." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01996-8", "title": "Tropical medians by transportation", "authors": [ "Andrei Comăneci", "Michael Joswig" ], "published_online": "2023-07-07", "volume": "205", "issue": "1-2", "pages": "813-839", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01996-8/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01997-7", "title": "Optimal gradient tracking for decentralized optimization", "authors": [ "Zhuoqing Song", "Lei Shi", "Shi Pu", "Ming Yan" ], "published_online": "2023-07-07", "volume": "207", "issue": "1-2", "pages": "1-53", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01997-7/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01993-x", "title": "Exploiting ideal-sparsity in the generalized moment problem with application to matrix factorization ranks", "authors": [ "Milan Korda", "Monique Laurent", "Victor Magron", "Andries Steenkamp" ], "published_online": "2023-07-08", "volume": "205", "issue": "1-2", "pages": "703-744", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01993-x/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Application to the completely positive rank", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "It is known that \\(\\textrm{c}(G)\\le n^2/4\\) [28]. In analogy, it has been a long standing conjecture by Drew et al. [26] that the cp-rank of an \\(n\\times n\\) completely positive matrix is at most \\(n^2/4\\). This conjecture, however, was disproved in [11, 12] for any \\(n\\ge 7\\). In particular, it is shown in [12] that the maximum cp-rank of an \\(n\\times n\\) cp-matrix is of the order \\(n^2/2 + O(n^{3/2})\\)." }, { "section": "Application to the completely positive rank", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "The matrix ex1 (from [6]) is supported on the 5-cycle \\(C_5\\) and the matrix ex2 (from [68]) is supported on the bipartite graph \\(K_{3,2}\\). In both cases, we have \\(\\xi ^{\\textrm{cp,isp}}_1(A) = \\text {rank}_{\\textrm{cp}}(A) = |E_A|\\) (combining Lemma 13 and the results of [26] mentioned earlier at the end of Sect. 4.2). The matrices ex3 and ex4 were constructed, respectively, in [11, 12] as examples of matrices having a large cp-rank exceeding the value \\(n^2/4\\) (thus refuting the conjecture by Drew et al. [26]). The matrix ex3 is supported on \\(\\overline{C_{11}}\\), the complement of an 11-cycle, and matrix ex4 is supported on the complete tripartite graph \\(K_{4,4,4}\\). One can verify that the edge clique-cover number is equal to 8 for \\(\\overline{C_{11}}\\) and to 16 for \\(K_{4,4,4}\\)." }, { "section": "Concluding remarks", "element": "p", "matched_phrases": [ "left_open" ], "passage": "In this paper we have introduced a new sparsity approach for GMP, which arises when in the formulation of GMP one has explicit ideal-type constraints that require the support of the measure to be contained in the variety of an ideal generated by monomials \\(x_ix_j\\) corresponding to (the non-edges of) a graph G. We compared it to the more classic correlative sparsity structure that requires a chordal structure on the graph G, while our new ideal-sparse hierarchy does not need it. We explored its application to the problem of bounding the nonnnegative rank and the cp-rank of a matrix and illustrate the new approach on some classes of examples. There are several natural extensions and further research directions that are left open by this work. We now sketch some of them." }, { "section": "Concluding remarks", "element": "p", "matched_phrases": [ "we_leave", "future_work" ], "passage": "Application to other matrix factorization ranks. We have explored the application to nonnegative and completely positive matrix factorization ranks. We have not considered their non-commutative analogs for the positive semidefinite (psd) rank and the completely positive semidefinite (cpsd) rank, where, respectively, given \\(M\\in \\mathbb {R}^{m\\times n}_+\\) one wants psd matrices \\(X_i,Y_j\\in \\mathcal S^r\\) such that \\(M=(\\langle X_i,Y_j\\rangle )_{i\\in [m], j\\in [n]}\\), and given \\(A\\in \\mathcal S^n\\) one wants psd matrices \\(X_i\\in \\mathcal S^r_+\\) such that \\(A=(\\langle X_i,X_j\\rangle )_{i,j\\in [n]}\\), with r smallest possible. One recovers the nonnegative rank and the cp-rank when restricting the factors \\(X_i,Y_j\\) to be diagonal matrices. We refer the reader to [33], where a common polynomial optimization framework is offered to treat all these four matrix factorization ranks. In the noncommutative setting of the psd- and cpsd-ranks, zero entries of M (or A) also imply ideal-type constraints of the form \\(X_iY_j=0\\) (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." }, { "section": "Concluding remarks", "element": "p", "matched_phrases": [ "we_leave" ], "passage": "Finally, instead of considering an ideal generated by quadratic monomials, one may consider an ideal generated by a set of monomials \\(x^S=\\prod _{i\\in S} x_i\\) (\\(S\\in \\mathcal S\\)), where \\(\\mathcal S\\) is a given collection of subsets of \\(V=[n].\\) The treatment extends naturally to this more general setting, where in the definition (2) of the set K, we replace the constraints \\(x_ix_j=0\\) (\\(\\{i,j\\}\\in {\\overline{E}}\\)) by \\(\\prod _{i\\in S}x_i=0\\) (\\(S\\in \\mathcal S\\)). Indeed, let \\(V_1,\\ldots ,V_p\\) denote the maximal subsets of V that do not contain any set \\(S\\in \\mathcal S\\). Then, for the dense formulation (1) of GMP, one can again show an equivalent sparse reformulation as in (11), which involves p measures supported on the subspaces \\(\\mathbb {R}^{|V_1|},\\ldots ,\\mathbb {R}^{|V_p|}\\) instead of a single measure on \\(\\mathbb {R}^{|V|}\\). We leave it for further research to explore applications of this more general ideal-sparsity setting and possible further extensions to other types of varieties." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01998-6", "title": "Publisher Correction: Branch-and-bound performance estimation programming: a unified methodology for constructing optimal optimization methods", "authors": [ "Shuvomoy Das Gupta", "Bart P. G. Van Parys", "Ernest K. Ryu" ], "published_online": "2023-07-12", "volume": "204", "issue": "1-2", "pages": "641-641", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01998-6/fulltext.html", "article_kind": "non_research_item", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01975-z", "title": "Asymptotic linear convergence of fully-corrective generalized conditional gradient methods", "authors": [ "Kristian Bredies", "Marcello Carioni", "Silvio Fanzon", "Daniel Walter" ], "published_online": "2023-07-13", "volume": "205", "issue": "1-2", "pages": "135-202", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01975-z/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Introduction", "element": "p", "matched_phrases": [ "future_work" ], "passage": "In this paper, we mainly focus on the theoretical aspects of the FC-GCG method and, in particular, its convergence. Naturally, from the practical standpoint, the applicability of our method relies on the knowledge of the set \\({\\text {Ext}}(B)\\), as well as on the efficient solution of the constrained linear problem in (1.1). Especially the computational cost of the latter greatly varies between different instances of Problem (\\(\\mathcal {P}_{\\mathcal {M}}\\)). To illustrate the working of our method and the role of assumptions (B1)–(B5) we present in Sect. 4 several examples of applications where our algorithm can be implemented. For the first three examples we provide a natural and easy to verify set of assumptions that imply (B1)–(B5), and for all of them we discuss the computational burden of computing a solution for the constrained linear problem in (1.1). We first consider the problem of identifying the initial source of a heat equation from given temperature measurements, see Sect. 4.1. For this example we also demonstrate numerically the expected linear convergence of Algorithm 1, discussing the stopping criterion and the advantages of our algorithm compared to classical GCG methods. Then, in Sect. 4.2, we consider the trace regularization of linear operators from a Hilbert space into itself, which favours the reconstruction of rank-one operators. In Sect. 4.3, we discuss minimum effort problems, where the regularization enforced by the supremum norm favours binary solutions. Finally, in Sect. 4.4, we briefly deal with the optimal transport regularization of dynamic inverse problems [17], showing that the algorithm introduced in [16] is, in some instances, a particular case of Algorithm 1. For this example the verification of the hypotheses (B1)–(B5), necessary to ensure fast convergence, is non-trivial and is left to future work." }, { "section": "Examples", "element": "p", "matched_phrases": [ "future_work" ], "passage": "where \\(p_k = - K_* \\nabla F(K \\rho _k) \\in C(X)\\) is the dual variable at the k-th iteration. The new point inserted is then \\(\\widehat{v}_k=(\\rho _{{\\widehat{\\gamma }_{{}_k}}},m_{{\\widehat{\\gamma }_{{}_k}}})\\). Thanks to the validity of Assumption (A1)–(A3), Theorem 3.3 guarantees the sublinear convergence of Algorithm 1. On the other hand, the verification of hypotheses (B1)–(B5), necessary for ensuring fast convergence of Algorithm 1, is non-trivial and is left to future work. We remark that an implementable version of Algorithm 1 for solving (4.27) under specific choices of the fidelity term F and the operator K has been recently proposed in [16] (see also [38]). We refer to these papers for more details about the practical implementation and the modifications needed to deal with time dependent measurement operators. Similarly to [16], solving (3.1) amounts to computing an absolutely continuous curve \\(\\gamma \\) by solving (4.32) at every iteration of the algorithm. This is a challenging non-concave variational problem in the space of curves. In [16], the authors proposed to solve (4.32) by a multistart gradient descent approach in the space of curves. The initialization curves are chosen to be a linear interpolation of randomly generated points \\(\\{x^h\\}\\) in \\(\\varOmega \\). Moreover several heuristic rules are employed to optimally select \\(\\{x^h\\}\\) and to reduce the computational time of the multistart gradient descent routine. Together with further acceleration strategies, this method is shown to be computationally feasible and very accurate for the task of tracking several dynamic sources in presence of high noise and severe spatial undersampling. In [38], the authors proposed to speed up the algorithm in [16] by considering inexact subproblems (3.1) and solving them using known algorithms for computing shortest paths on directed acyclic graphs." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01994-w", "title": "The role of rationality in integer-programming relaxations", "authors": [ "Manuel Aprile", "Gennadiy Averkov", "Marco Di Summa", "Christopher Hojny" ], "published_online": "2023-07-13", "volume": "205", "issue": "1-2", "pages": "745-771", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01994-w/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Abstract", "element": "p", "matched_phrases": [ "open_question" ], "passage": "For a finite set \\(X \\subset \\mathbb {Z}^d\\) that can be represented as \\(X = Q \\cap \\mathbb {Z}^d\\) for some polyhedron Q, we call Q a relaxation of X and define the relaxation complexity \\({\\text {rc}}(X)\\) of X as the least number of facets among all possible relaxations Q of X. The rational relaxation complexity \\({\\text {rc}}_\\mathbb {Q}(X)\\) restricts the definition of \\({\\text {rc}}(X)\\) to rational polyhedra Q. In this article, we focus on \\(X = \\Delta _d\\), the vertex set of the standard simplex, which consists of the null vector and the standard unit vectors in \\(\\mathbb {R}^d\\). We show that \\({\\text {rc}}(\\Delta _d) \\le d\\) for every \\(d \\ge 5\\). That is, since \\({\\text {rc}}_\\mathbb {Q}(\\Delta _d)=d+1\\), irrationality can reduce the minimal size of relaxations. This answers an open question posed by Kaibel and Weltge (Math Program 154(1):407–425, 2015). Moreover, we prove the asymptotic statement \\({\\text {rc}}(\\Delta _d) \\in O(\\nicefrac {d}{\\sqrt{\\log (d)}})\\), which shows that the ratio \\(\\nicefrac {{\\text {rc}}(\\Delta _d)}{{\\text {rc}}_\\mathbb {Q}(\\Delta _d)}\\) goes to 0, as \\(d \\rightarrow \\infty \\)." }, { "section": "Introduction", "element": "p", "matched_phrases": [ "open_problem" ], "passage": "The remainder of this article is structured as follows. First, we mention related literature (Sect. 1.1) and introduce the notation and terminology that we use throughout the article (Sect. 1.2). In Sect. 2, we prove Theorem 1 and Corollary 1, and, based on the proof of Theorem 1, we derive our strategy for proving Theorem 2. The proof of the latter theorem is then presented in Sect. 4. Finally, we list some open problems in Sect. 5." }, { "section": "Outlook", "element": "p", "matched_phrases": [ "open_problem" ], "passage": "In light of our new results, we formulate the following open problems:" } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-02000-z", "title": "Complexity of a projected Newton-CG method for optimization with bounds", "authors": [ "Yue Xie", "Stephen J. Wright" ], "published_online": "2023-07-13", "volume": "207", "issue": "1-2", "pages": "107-144", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-02000-z/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01989-7", "title": "Graph isomorphism: physical resources, optimization models, and algebraic characterizations", "authors": [ "Laura Mančinska", "David E. Roberson", "Antonios Varvitsiotis" ], "published_online": "2023-07-14", "volume": "205", "issue": "1-2", "pages": "617-660", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01989-7/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01995-9", "title": "A new extension of Chubanov’s method to symmetric cones", "authors": [ "Shin-ichi Kanoh", "Akiko Yoshise" ], "published_online": "2023-07-18", "volume": "205", "issue": "1-2", "pages": "773-812", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01995-9/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-02002-x", "title": "Short-step methods are not strongly polynomial-time", "authors": [ "Manru Zong", "Yin Tat Lee", "Man-Chung Yue" ], "published_online": "2023-07-22", "volume": "207", "issue": "1-2", "pages": "733-746", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-02002-x/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-01999-5", "title": "First- and second-order high probability complexity bounds for trust-region methods with noisy oracles", "authors": [ "Liyuan Cao", "Albert S. Berahas", "Katya Scheinberg" ], "published_online": "2023-07-29", "volume": "207", "issue": "1-2", "pages": "55-106", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-01999-5/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-02001-y", "title": "A unified stochastic approximation framework for learning in games", "authors": [ "Panayotis Mertikopoulos", "Ya-Ping Hsieh", "Volkan Cevher" ], "published_online": "2023-08-04", "volume": "203", "issue": "1-2", "pages": "559-609", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-02001-y/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-02004-9", "title": "LP-based approximations for disjoint bilinear and two-stage adjustable robust optimization", "authors": [ "Omar El Housni", "Ayoub Foussoul", "Vineet Goyal" ], "published_online": "2023-08-10", "volume": "206", "issue": "1-2", "pages": "239-281", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-02004-9/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-02005-8", "title": "A simple method for convex optimization in the oracle model", "authors": [ "Daniel Dadush", "Christopher Hojny", "Sophie Huiberts", "Stefan Weltge" ], "published_online": "2023-08-10", "volume": "206", "issue": "1-2", "pages": "283-304", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-02005-8/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [ { "section": "Computational experiments", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "An explanation for this behavior is explained by the integral values, which are provided by Table 1. Among all tested algorithms, our method beforms best in finding an optimal solution as the primal integrals are comparably very small. On the dual side, the picture is less clear, but in particular for matching instances our algorithm is able to quickly find the true dual value. For LPboost, however, our method converges much slower on the dual side than the other methods. We conjecture that this behavior is due to the special structure of the objective for LPboost instances. While the other problem classes have a completely dense objective, only one variable contributes to the objective for LPboost." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-02006-7", "title": "Radial duality part I: foundations", "authors": [ "Benjamin Grimmer" ], "published_online": "2023-08-10", "volume": "205", "issue": "1-2", "pages": "33-68", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-02006-7/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-02011-w", "title": "The simultaneous semi-random model for TSP", "authors": [ "Eric Balkanski", "Yuri Faenza", "Mathieu Kubik" ], "published_online": "2023-08-11", "volume": "206", "issue": "1-2", "pages": "305-332", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-02011-w/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-02008-5", "title": "A new complexity metric for nonconvex rank-one generalized matrix completion", "authors": [ "Haixiang Zhang", "Baturalp Yalcin", "Javad Lavaei", "Somayeh Sojoudi" ], "published_online": "2023-08-16", "volume": "207", "issue": "1-2", "pages": "227-268", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-02008-5/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-02009-4", "title": "A polynomial-size extended formulation for the multilinear polytope of beta-acyclic hypergraphs", "authors": [ "Alberto Del Pia", "Aida Khajavirad" ], "published_online": "2023-08-28", "volume": "207", "issue": "1-2", "pages": "269-301", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-02009-4/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-02010-x", "title": "New characterizations of strategy-proofness under single-peakedness", "authors": [ "Andrew B. Jennings", "Rida Laraki", "Clemens Puppe", "Estelle M. Varloot" ], "published_online": "2023-08-28", "volume": "203", "issue": "1-2", "pages": "207-238", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-02010-x/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-02003-w", "title": "Stochastic algorithms with geometric step decay converge linearly on sharp functions", "authors": [ "Damek Davis", "Dmitriy Drusvyatskiy", "Vasileios Charisopoulos" ], "published_online": "2023-09-05", "volume": "207", "issue": "1-2", "pages": "145-190", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-02003-w/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-02013-8", "title": "Deterministic enumeration of all minimum cut-sets and k-cut-sets in hypergraphs for fixed k", "authors": [ "Calvin Beideman", "Karthekeyan Chandrasekaran", "Weihang Wang" ], "published_online": "2023-09-12", "volume": "207", "issue": "1-2", "pages": "329-367", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-02013-8/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-02016-5", "title": "Neural networks with linear threshold activations: structure and algorithms", "authors": [ "Sammy Khalife", "Hongyu Cheng", "Amitabh Basu" ], "published_online": "2023-09-12", "volume": "206", "issue": "1-2", "pages": "333-356", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-02016-5/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-02017-4", "title": "Homotopic policy mirror descent: policy convergence, algorithmic regularization, and improved sample complexity", "authors": [ "Yan Li", "Guanghui Lan", "Tuo Zhao" ], "published_online": "2023-09-19", "volume": "207", "issue": "1-2", "pages": "457-513", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-02017-4/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-02019-2", "title": "Sparse multi-term disjunctive cuts for the epigraph of a function of binary variables", "authors": [ "Rui Chen", "James Luedtke" ], "published_online": "2023-09-21", "volume": "206", "issue": "1-2", "pages": "357-388", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-02019-2/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-02014-7", "title": "Residuals-based distributionally robust optimization with covariate information", "authors": [ "Rohit Kannan", "Güzin Bayraksan", "James R. Luedtke" ], "published_online": "2023-09-26", "volume": "207", "issue": "1-2", "pages": "369-425", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-02014-7/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-02018-3", "title": "Better-than-$$\\frac{4}{3}$$-approximations for leaf-to-leaf tree and connectivity augmentation", "authors": [ "Federica Cecchetto", "Vera Traub", "Rico Zenklusen" ], "published_online": "2023-09-26", "volume": "207", "issue": "1-2", "pages": "515-549", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-02018-3/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "A matching-based approach", "element": "p", "matched_phrases": [ "remains_to_determine" ], "passage": "is integral and hence also (dir-LP) is integral. Hence, all that remains to be shown is that the coordinates of extreme points are not just integral but each one of them is either zero or one. Observe first, that (dir-LP) contains non-negativity constraints, thus each coordinate of any extreme point is non-negative. Finally, we show that any point \\(x\\in \\mathbb {R}^{\\vec {L}}_{\\ge 0}\\) feasible for (dir-LP) with \\(x(\\ell )>1\\) for some \\(\\ell \\in \\vec {L}\\) cannot be an extreme point. Indeed, for such a point x, both increasing or decreasing \\(x(\\ell )\\) by a small quantity will lead to other feasible points in (dir-LP), which contradicts the assumption of x being an extreme point." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-02015-6", "title": "Multiplicative updates for symmetric-cone factorizations", "authors": [ "Yong Sheng Soh", "Antonios Varvitsiotis" ], "published_online": "2023-09-30", "volume": "207", "issue": "1-2", "pages": "427-456", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-02015-6/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-02012-9", "title": "The density of planar sets avoiding unit distances", "authors": [ "Gergely Ambrus", "Adrián Csiszárik", "Máté Matolcsi", "Dániel Varga", "Pál Zsámboki" ], "published_online": "2023-10-06", "volume": "207", "issue": "1-2", "pages": "303-327", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-02012-9/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Abstract", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "By improving upon previous estimates on a problem posed by L. Moser, we prove a conjecture of Erdős that the density of any measurable planar set avoiding unit distances is less than 1/4. Our argument implies the upper bound of 0.2470." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-02020-9", "title": "Global stability of first-order methods for coercive tame functions", "authors": [ "Cédric Josz", "Lexiao Lai" ], "published_online": "2023-10-06", "volume": "207", "issue": "1-2", "pages": "551-576", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-02020-9/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-02021-8", "title": "Global optimization for cardinality-constrained minimum sum-of-squares clustering via semidefinite programming", "authors": [ "Veronica Piccialli", "Antonio M. Sudoso" ], "published_online": "2023-10-12", "volume": "211", "issue": "1-2", "pages": "245-279", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-02021-8/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [ { "section": "Introduction", "element": "p", "matched_phrases": [ "future_work" ], "passage": "The rest of this paper is organized as follows. Section 2 reviews the literature related to the ccMSSC problem. In Sect. 3, the existing SDP relaxation of ccMSSC and the new SDP bound are described. In Sect. 4, problem-specific post-processing techniques yielding valid lower bounds are described. In Sect. 5, the cutting-plane algorithm used for computing the bound is proposed. A problem-specific branching rule is shown in Sect. 6, and a primal heuristic is introduced in Sect. 7. Computational results are reported in Sect. 8 and some directions for future work conclude the paper in Sect. 9." }, { "section": "Conclusions", "element": "p", "matched_phrases": [ "future_work" ], "passage": "In this paper, we proposed an SDP-based branch-and-cut algorithm for solving MSSC with strict cardinality constraints. For computing the lower bound, we used the SDP relaxation recently proposed in [4] for small-scale instances, whereas for large-scale ones we derived a new SDP relaxation. We implemented a cutting-plane algorithm to strengthen the bounds provided by both relaxation by adding polyhedral cuts. For the upper bound computation, we designed a constrained variant of k-means algorithm and we initialized it with the solution of the SDP relaxation solved at each node. Numerical results impressively exhibit the efficiency of our solver: when using the new SDP bound, we can solve real-world instances up to 1000 data points, whereas when using the bound in [4] we can only solve small-size problems and guarantee an optimality gap smaller than 2% on larger instances. To the best of our knowledge, no other exact solution methods can handle real-world instances of that size. An interesting research direction to improve our methodology is to study facial reduction for both the VL-SDP and ML-SDP relaxations, following [43]. This should make the solution of the SDPs faster and more robust, leading to a more efficient branch-and-cut algorithm. Furthermore, an immediate future research direction is to design global algorithms for clustering problems with fairness constraints. As introduced in [58], the goal of fair clustering is to find a partition where all the clusters are balanced with respect to some protected attributes such as gender or religion." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-02022-7", "title": "Relaxations and duality for multiobjective integer programming", "authors": [ "Alex Dunbar", "Saumya Sinha", "Andrew J. Schaefer" ], "published_online": "2023-10-12", "volume": "207", "issue": "1-2", "pages": "577-616", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-02022-7/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-02025-4", "title": "Inapproximability of shortest paths on perfect matching polytopes", "authors": [ "Jean Cardinal", "Raphael Steiner" ], "published_online": "2023-10-21", "volume": "210", "issue": "1-2", "pages": "147-163", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-02025-4/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [ { "section": "Abstract", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "We consider the computational problem of finding short paths in the skeleton of the perfect matching polytope of a bipartite graph. We prove that unless \\({\\textsf {P}}={\\textsf {NP}}\\), there is no polynomial-time algorithm that computes a path of constant length between two vertices at distance two of the perfect matching polytope of a bipartite graph. Conditioned on \\({\\textsf {P}}\\ne {\\textsf {NP}}\\), this disproves a conjecture by Ito et al. (SIAM J Discrete Math 36(2):1102–1123, 2022). Assuming the Exponential Time Hypothesis we prove the stronger result that there exists no polynomial-time algorithm computing a path of length at most \\(\\left( \\frac{1}{4}-o(1)\\right) \\log N / \\log \\log N\\) between two vertices at distance two of the perfect matching polytope of an N-vertex bipartite graph. These results remain true if the bipartite graph is restricted to be of maximum degree three. The above has the following interesting implication for the performance of pivot rules for the simplex algorithm on simply-structured combinatorial polytopes: If \\({\\textsf {P}}\\ne {\\textsf {NP}}\\), then for every simplex pivot rule executable in polynomial time and every constant \\(k \\in {\\mathbb {N}}\\) there exists a linear program on a perfect matching polytope and a starting vertex of the polytope such that the optimal solution can be reached in two monotone non-degenerate steps from the starting vertex, yet the pivot rule will require at least k non-degenerate steps to reach the optimal solution. This result remains true in the more general setting of pivot rules for so-called circuit-augmentation algorithms." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-02026-3", "title": "The limits of local search for weighted k-set packing", "authors": [ "Meike Neuwohner" ], "published_online": "2023-10-28", "volume": "206", "issue": "1-2", "pages": "389-427", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-02026-3/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-02027-2", "title": "Optimal item pricing in online combinatorial auctions", "authors": [ "José Correa", "Andrés Cristi", "Andrés Fielbaum", "Tristan Pollner", "S. Matthew Weinberg" ], "published_online": "2023-10-28", "volume": "206", "issue": "1-2", "pages": "429-460", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-02027-2/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-02007-6", "title": "Scalable adaptive cubic regularization methods", "authors": [ "Jean-Pierre Dussault", "Tangi Migot", "Dominique Orban" ], "published_online": "2023-10-31", "volume": "207", "issue": "1-2", "pages": "191-225", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-02007-6/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-02024-5", "title": "State polynomials: positivity, optimization and nonlinear Bell inequalities", "authors": [ "Igor Klep", "Victor Magron", "Jurij Volčič", "Jie Wang" ], "published_online": "2023-11-03", "volume": "207", "issue": "1-2", "pages": "645-691", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-02024-5/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-02028-1", "title": "New lower bounds on crossing numbers of $$K_{m,n}$$ from semidefinite programming", "authors": [ "Daniel Brosch", "Sven C. Polak" ], "published_online": "2023-11-20", "volume": "207", "issue": "1-2", "pages": "693-715", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-02028-1/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Introduction", "element": "p", "matched_phrases": [ "open_problem", "conjecture" ], "passage": "Computing the crossing number \\(\\mathop {\\textrm{cr}}\\limits (K_{m,n})\\) of the complete bipartite graph \\(K_{m,n}\\) is a long-standing open problem, which goes back to Turán in the 1940s. In 1956, Zarankiewicz [28] conjectured that \\(\\mathop {\\textrm{cr}}\\limits (K_{m,n})= Z(m,n)\\), where Z(m, n) is the Zarankiewicz number" }, { "section": "Introduction", "element": "p", "matched_phrases": [ "open_problem", "conjecture" ], "passage": "Zarankiewicz claimed to have a proof for his conjecture, but this turned out to be false. The conjecture thus remains a notorious open problem. As Erdős and Guy [9] wrote in 1973: ‘Almost all questions that one can ask about crossing numbers remain unsolved’, which is still true today. It is known that \\(\\mathop {\\textrm{cr}}\\limits (K_{m,n}) \\le Z(m,n)\\), by exhibiting an explicit drawing of \\(K_{m,n}\\) in the plane with Z(m, n) crossings—see Fig. 1 for an example." }, { "section": "Introduction", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "This theorem and Corollary 1 below yield the best known lower bounds on all fixed \\(\\mathop {\\textrm{cr}}\\limits (K_{m,n})\\) with \\(m,n \\ge 10\\). The best previously known lower bounds for \\(m,n\\ge 10\\) are \\(\\mathop {\\textrm{cr}}\\limits (K_{m,n}) \\ge \\tfrac{(m-1)m}{72} (3.86760n^2-8n)\\), cf. [6]. For an overview of known results regarding Zarankiewicz’s conjecture, see the survey by Székely [26], or the survey about crossing numbers by Schaefer [23]." }, { "section": "Derived lower bounds", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "However, in order to prove asymptotic bounds it is also worthwhile to further investigate the quadratic programming hierarchy from De Klerk et al. [5] which we consider in this paper. One might hope to prove lower bounds \\(t_k\\) on \\(\\alpha _k\\) such that \\(8t_k /(k(k-1)) \\rightarrow 1\\) as \\(k \\rightarrow \\infty \\), thereby proving \\(\\lim _{n \\rightarrow \\infty } \\frac{\\mathop {\\textrm{cr}}\\limits (K_{n,n})}{Z(n,n)}=1\\), i.e., asymptotically proving Zarankiewicz’ conjecture. Figure 2 gives rise to the question whether \\(8\\beta _k /(k(k-1)) \\rightarrow 1\\) as \\(k \\rightarrow \\infty \\)." }, { "section": "Derived lower bounds", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "In Fig. 2, the increases are larger for odd k than for even k, a trend which was already noted in [6]. We now see that this trend continues for some larger k. As noted in [6], this is reminiscent of the fact that Zarankiewicz’s conjecture holds for \\(K_{2m,n}\\) if it holds for \\(K_{2m-1,n}\\)." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-02029-0", "title": "A polynomial time algorithm for finding a minimum 4-partition of a submodular function", "authors": [ "Tsuyoshi Hirayama", "Yuhao Liu", "Kazuhisa Makino", "Ke Shi", "Chao Xu" ], "published_online": "2023-12-06", "volume": "207", "issue": "1-2", "pages": "717-732", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-02029-0/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-02023-6", "title": "Efficient Kirszbraun extension with applications to regression", "authors": [ "Hananel Zaichyk", "Armin Biess", "Aryeh Kontorovich", "Yury Makarychev" ], "published_online": "2023-12-07", "volume": "207", "issue": "1-2", "pages": "617-642", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-02023-6/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-02033-4", "title": "Quasi-polynomial time approximation schemes for assortment optimization under Mallows-based rankings", "authors": [ "Alon Rieger", "Danny Segev" ], "published_online": "2023-12-11", "volume": "208", "issue": "1-2", "pages": "111-171", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-02033-4/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-02035-2", "title": "A single cut proximal bundle method for stochastic convex composite optimization", "authors": [ "Jiaming Liang", "Vincent Guigues", "Renato D. C. Monteiro" ], "published_online": "2023-12-11", "volume": "208", "issue": "1-2", "pages": "173-208", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-02035-2/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-02039-y", "title": "Correction to: The ancestral Benders’ cutting-plane algorithm with multi-term disjunctions for mixed-integer recourse decisions in stochastic programming", "authors": [ "Yunwei Qi", "Suvrajeet Sen" ], "published_online": "2023-12-13", "volume": "205", "issue": "1-2", "pages": "841-845", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-02039-y/fulltext.html", "article_kind": "non_research_item", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-02038-z", "title": "Hardness and approximation of submodular minimum linear ordering problems", "authors": [ "Majid Farhadi", "Swati Gupta", "Shengding Sun", "Prasad Tetali", "Michael C. Wigal" ], "published_online": "2023-12-14", "volume": "208", "issue": "1-2", "pages": "277-318", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-02038-z/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Introduction", "element": "p", "matched_phrases": [ "open_problem" ], "passage": "We summarize our hardness and approximation results on MLOP in Tables 1 and 2. The paper is structured as follows: We give an overview of our results and techniques in Sect. 2, discuss related work on MLOP variants in Sect. 3, and present preliminaries in Sect. 4. We discuss detailed proofs of our results in Sects. 5–8. We finally conclude the paper with open problems in Sect. 9." }, { "section": "Overview of results and techniques", "element": "p", "matched_phrases": [ "open_question" ], "passage": "Our results have led to multiple open questions which may be of independent interest, and are discussed in Sect. 9." }, { "section": "Approximations for minimum latency set cover (MLSC)", "element": "p", "matched_phrases": [ "we_do_not_know" ], "passage": "The integrality gap of MLSC-LP is therefore at least \\(2 - \\frac{2}{1+l}\\), but it can be more for certain families of graphs. We end this section by showing that the integrality gap of the MLSC-LP is exactly \\(2-\\frac{2}{1+\\ell }\\), for \\(\\ell \\)-uniform hypergraphs where the degree of each vertex is exactly d. We call these hypergraphs d-regular \\(\\ell \\)-uniform hypergraphs. We do not know if the integrality gap for non-regular uniform hypergraphs is strictly larger than \\(2-\\frac{2}{1+\\ell }\\)." }, { "section": "Improved approximation for monotone submodular MLOP", "element": "p", "matched_phrases": [ "unknown_whether" ], "passage": "Monotone submodular MLOP was introduced by Iwata et al. [1], where the authors also provided a factor \\((2-\\frac{2}{1+|E|})\\)-approximation algorithm using the Lovász extension of submodular functions. Fokkink et al. [16] studied the submodular search problem, which generalizes monotone submodular MLOP, and gave an approximation factor based on the total curvature of the submodular function. It was not known if a tighter approximation was possible. They considered the greedy contraction of the principal partition induced by the submodular function, an idea that has been used as early as 1992 by Pisaruk [15]. In this section, we give a different analysis to the same algorithm and improve the approximation factor to" }, { "section": "Future directions", "element": "p", "matched_phrases": [ "open_question" ], "passage": "We conclude this work by presenting a list of open questions that stem from this work." }, { "section": "Future directions", "element": "p", "matched_phrases": [ "open_question" ], "passage": "Open question 1. Given a graph G, a spanning tree T and integer k, consider the problem of whether there exists a permutation \\(\\sigma \\) of E(T) such that" }, { "section": "Future directions", "element": "p", "matched_phrases": [ "remains_open" ], "passage": "This problem is known to be NP-hard only when T is a star graph, and remains open for other simple families of trees, such as in the case where T is a path." }, { "section": "Future directions", "element": "p", "matched_phrases": [ "remains_open" ], "passage": "In Sect. 7, we showed that MLSC can be \\((2-\\frac{2}{1+\\ell })\\)-approximated using randomized scheduling techniques. Furthermore, we showed for \\(\\ell \\)-uniform regular hypergraphs, MLSC can be \\((2-\\frac{2}{1+\\ell })\\)-approximated using an LP relaxation. This question for general \\(\\ell \\)-uniform hypergraphs remains open." }, { "section": "Future directions", "element": "p", "matched_phrases": [ "open_question" ], "passage": "Open question 2. Does solving the LP relaxation provide an approximation guarantee for MLSC on \\(\\ell \\)-uniform hypergraphs by a factor of \\(2-\\frac{2}{1+\\ell }\\)?" }, { "section": "Future directions", "element": "p", "matched_phrases": [ "remains_open" ], "passage": "In Sect. 7.2, we show that the MLVC, MSVC, and MLA are all equivalent problems in decision form for regular graphs. Using techniques similar to Theorem 9, we can show that the optimal value for all three problems are related by linear shifts. It is known that MSVC on regular graphs can be 4/3-approximated (see [26]), but we have not found a formal proof that this problem is NP-hard. Thus, the following question remains open, to the best of our knowledge." }, { "section": "Future directions", "element": "p", "matched_phrases": [ "open_question" ], "passage": "Open question 3. Are MLA, MLVC, and MSVC NP-hard for the family of simple regular graphs?" }, { "section": "Future directions", "element": "p", "matched_phrases": [ "open_question" ], "passage": "In Sect. 8, we show that monotone submodular MLOP can be approximated within factor \\(2-\\frac{1+\\ell _f}{1+|E|}\\), using principal partitions. A related open question is to develop algorithms when the principal partitions are trivial, i.e., \\(f(S)|E|\\ge f(E)|S|\\) for all \\(S\\subseteq E\\). In this case, the principal partition-based algorithm studied by Fokkink et al. [16] (and by us) will simply output an arbitrary solution." }, { "section": "Future directions", "element": "p", "matched_phrases": [ "open_question" ], "passage": "Open question 4. Do there exist better polynomial time approximation algorithms for monotone submodular MLOP in the case where the function f satisfies \\(f(\\emptyset )=0\\) and \\(f(S)|E|\\ge f(E)|S|\\) for all \\(S\\subseteq E\\)?" }, { "section": "Future directions", "element": "p", "matched_phrases": [ "open_question" ], "passage": "Open question 5. Can symmetric submodular MLOP over a ground set E be approximated to a factor better than O(|E|)?" } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-02034-3", "title": "Intersecting and dense restrictions of clutters in polynomial time", "authors": [ "Martin Drees" ], "published_online": "2023-12-18", "volume": "206", "issue": "1-2", "pages": "461-477", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-02034-3/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [ { "section": "Introduction", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "A nontrivial clutter is called k-wise intersecting if it has no cover of size 1 and every k (not necessarily distinct) members have a common element. Abdi, Cornuéjols, Huynh and Lee [1] conjectured that for \\(k\\ge 4\\) these clutters are non-ideal." }, { "section": "k-wise intersecting clutters", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "As a consequence thereof, given a clutter explicitly with its members, we can decide in polynomial time whether the clutter has an intersecting minor. It is conjectured that ideal clutters with no intersecting minor also have the max-flow min-cut property [2]." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-02030-7", "title": "Beyond symmetry: best submatrix selection for the sparse truncated SVD", "authors": [ "Yongchun Li", "Weijun Xie" ], "published_online": "2023-12-21", "volume": "208", "issue": "1-2", "pages": "1-50", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-02030-7/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-02036-1", "title": "First order asymptotics of the sample average approximation method to solve risk averse stochastic programs", "authors": [ "Volker Krätschmer" ], "published_online": "2023-12-26", "volume": "208", "issue": "1-2", "pages": "209-242", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-02036-1/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-02037-0", "title": "The exact worst-case convergence rate of the alternating direction method of multipliers", "authors": [ "Moslem Zamani", "Hadi Abbaszadehpeivasti", "Etienne de Klerk" ], "published_online": "2023-12-26", "volume": "208", "issue": "1-2", "pages": "243-276", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-02037-0/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Worst-case convergence rate", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "Theorems 3 and 4 address the case that f is strongly convex relative to \\(\\Vert .\\Vert _A\\) and g is convex. Based on numerical results by solving performance estimation problems including (15) we conjecture, under the assumptions of Theorem 3, if g is \\(c_2\\)-strongly convex relative to \\(\\Vert .\\Vert _B\\), Algorithm 1 enjoys the following convergence rates" }, { "section": "Worst-case convergence rate", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "We have verified these conjectures numerically for many specific values of the parameters. Nevertheless, we could not manage to guess a closed-form formula for the residual dual in this case." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-02043-2", "title": "Asymmetry in the complexity of the multi-commodity network pricing problem", "authors": [ "Quang Minh Bui", "Margarida Carvalho", "José Neto" ], "published_online": "2024-01-03", "volume": "208", "issue": "1-2", "pages": "425-461", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-02043-2/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-02041-4", "title": "High-order methods beyond the classical complexity bounds: inexact high-order proximal-point methods", "authors": [ "Masoud Ahookhosh", "Yurii Nesterov" ], "published_online": "2024-01-04", "volume": "208", "issue": "1-2", "pages": "365-407", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-02041-4/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "BiOPT: Bi-level OPTimization framework", "element": "p", "matched_phrases": [ "future_work" ], "passage": "Let us fix \\(q\\ge 1\\). Then, the function \\(f_{y_k,p}^H(\\cdot )\\) is L-smooth relative to the scaling function \\(\\rho _k(\\cdot )\\) (3.18), which is the same for both \\(p=2q\\) and \\(p=2q+1\\). If p is even (i.e., \\(p=2q\\)), then Algorithm 4 is a 2q-order method (requiring the 2q-order oracle) and attains the complexity of \\({\\mathcal {O}}(\\varepsilon ^{-1/(2q+1)})\\), which worse than the optimal complexity \\({\\mathcal {O}}(\\varepsilon ^{-2/(6q+1)})\\). On the other hand, if p is odd (\\(p=2q+1\\)), then Algorithm 4 is again a 2q-order method (requiring the 2q-order oracle) obtaining the complexity of \\({\\mathcal {O}}(\\varepsilon ^{-1/(2q+2)})\\), which is also worse than the optimal complexity \\({\\mathcal {O}}(\\varepsilon ^{-2/(6q+1)})\\) except for \\(p=3\\) that leads to the complexity \\({\\mathcal {O}}(\\varepsilon ^{-(1/4)})\\) overpassing the optimal complexity bound of second-order methods, i.e., \\({\\mathcal {O}}(\\varepsilon ^{-(2/7)})\\), as was known from in [29]. However, in the following example, we show that the complexity of our method can overpass the classical bounds for some structured class of problems. It is arguable that one may come up with algorithms with better complexity than (3.42) for finding the acceptable solution satisfying (2.5) for general p (e.g., with a more sophisticated methods or stronger assumptions); however, this out of scope of the current study and will be further investigated in our future work." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-02044-1", "title": "A 2-approximation for the bounded treewidth sparsest cut problem in $$\\textsf{FPT}$$ Time", "authors": [ "Vincent Cohen-Addad", "Tobias Mömke", "Victor Verdugo" ], "published_online": "2024-01-04", "volume": "206", "issue": "1-2", "pages": "479-495", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-02044-1/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-02046-z", "title": "FPT algorithms for a special block-structured integer program with applications in scheduling", "authors": [ "Hua Chen", "Lin Chen", "Guochuan Zhang" ], "published_online": "2024-01-04", "volume": "208", "issue": "1-2", "pages": "463-496", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-02046-z/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-02040-5", "title": "Complementary composite minimization, small gradients in general norms, and applications", "authors": [ "Jelena Diakonikolas", "Cristóbal Guzmán" ], "published_online": "2024-01-05", "volume": "208", "issue": "1-2", "pages": "319-363", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-02040-5/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-02031-6", "title": "No dimension-free deterministic algorithm computes approximate stationarities of Lipschitzians", "authors": [ "Lai Tian", "Anthony Man-Cho So" ], "published_online": "2024-01-06", "volume": "208", "issue": "1-2", "pages": "51-74", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-02031-6/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-02042-3", "title": "Counterexample and an additional revealing poll step for a result of “analysis of direct searches for discontinuous functions”", "authors": [ "Charles Audet", "Pierre-Yves Bouchet", "Loïc Bourdin" ], "published_online": "2024-01-08", "volume": "208", "issue": "1-2", "pages": "411-424", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-02042-3/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-02045-0", "title": "A competitive algorithm for throughput maximization on identical machines", "authors": [ "Benjamin Moseley", "Kirk Pruhs", "Clifford Stein", "Rudy Zhou" ], "published_online": "2024-01-10", "volume": "206", "issue": "1-2", "pages": "497-514", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-02045-0/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [ { "section": "Conclusion", "element": "p", "matched_phrases": [ "open_question", "we_leave" ], "passage": "We considered throughput maximization on multiple machines without any high-laxity or speed augmentation assumption. Our final competitive ratio is a large unspecified constant (obtained by choosing \\(\\alpha = O(1)\\) sufficiently large.) Many interesting open questions remain. What is the right competitive ratio for throughput maximization? Can we give an improved upper bound, or any lower bound? The only known lower bounds rule out constant-competitive deterministic algorithms for \\(m = 1\\). Further, is migration necessary to achieve constant-competitiveness? Note that two out of three of our component algorithms (LMNY and Lax) are non-migratory. Naively, SRPT is migratory, and we also use migration because Lax and SRPT share the low-laxity jobs. We leave it as an open question to develop a non-migratory constant-competitive algorithm for throughput maximization on multiple machines." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-02047-y", "title": "Constrained optimization of rank-one functions with indicator variables", "authors": [ "Soroosh Shafiee", "Fatma Kılınç-Karzan" ], "published_online": "2024-01-20", "volume": "208", "issue": "1-2", "pages": "533-579", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-02047-y/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-02048-x", "title": "Characterization of matrices with bounded Graver bases and depth parameters and applications to integer programming", "authors": [ "Marcin Briański", "Martin Koutecký", "Daniel Král’", "Kristýna Pekárková", "Felix Schröder" ], "published_online": "2024-01-20", "volume": "208", "issue": "1-2", "pages": "497-531", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-02048-x/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Introduction", "element": "p", "matched_phrases": [ "open_problem" ], "passage": "One of the open problems in the area, e.g. discussed during the Dagstuhl workshop 19041 “New Horizons in Parameterized Complexity”, has been whether integer programming is fixed parameter tractable when parameterized by \\(g_1(A)\\), i.e., by the \\(\\ell _1\\)-norm of an element of the Graver basis of the constraint matrix A. Our results on the interplay of dual tree-depth, the circuit complexity and the Graver basis complexity of a matrix yield an affirmative answer. The existence of appropriate preconditioners that we establish in this paper implies that integer programming is fixed parameter tractable when parameterized by" } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02056-5", "title": "Correction: Efficient Kirszbraun extension with applications to regression", "authors": [ "Hananel Zaichyk", "Armin Biess", "Aryeh Kontorovich", "Yury Makarychev" ], "published_online": "2024-01-20", "volume": "207", "issue": "1-2", "pages": "643-643", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02056-5/fulltext.html", "article_kind": "non_research_item", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-02050-3", "title": "Mathematical Optimization for Fair Social Decisions: A Tribute to Michel Balinski", "authors": [ "Mourad Baïou", "José Correa", "Rida Laraki" ], "published_online": "2024-01-24", "volume": "203", "issue": "1-2", "pages": "1-7", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-02050-3/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [ { "section": "Game Theory", "element": "p", "matched_phrases": [ "open_problem" ], "passage": "The Price of Anarchy in Series–Parallel Network Congestion Games. Hao and Michini [REF] address an open problem in the characterization of the price of anarchy of symmetric network congestion games with affine latency functions. In this context, the authors obtain improved bounds for the case of series–parallel graphs" } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-02049-w", "title": "Adjustability in robust linear optimization", "authors": [ "Ningji Wei", "Peter Zhang" ], "published_online": "2024-01-27", "volume": "208", "issue": "1-2", "pages": "581-628", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-02049-w/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Conclusion and future extensions", "element": "p", "matched_phrases": [ "future_work" ], "passage": "For future work, it would be interesting to examine if Theorem 4 is, in general, the “optimal” way to bound adjustability ratio given an arbitrary problem setup. If the answer is affirmative, then one could examine more specialized settings to derive context-specific managerial and policy insights. In addition, the theoretical framework and new proof techniques may be of interest to researchers who work on areas mentioned in Section A." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-02054-z", "title": "A constant-factor approximation for generalized malleable scheduling under $$M ^{\\natural }$$-concave processing speeds", "authors": [ "Dimitris Fotakis", "Jannik Matuschke", "Orestis Papadigenopoulos" ], "published_online": "2024-01-29", "volume": "206", "issue": "1-2", "pages": "515-539", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-02054-z/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-02053-0", "title": "Designing tractable piecewise affine policies for multi-stage adjustable robust optimization", "authors": [ "Simon Thomä", "Grit Walther", "Maximilian Schiffer" ], "published_online": "2024-02-03", "volume": "208", "issue": "1-2", "pages": "661-716", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-02053-0/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Introduction", "element": "p", "matched_phrases": [ "future_work" ], "passage": "The rest of this paper is structured as follows. In Sect. 2, we introduce our policies and elaborate on their construction. In Sect. 3, we present our approximation bounds for the multi-stage ARO Problem (1). We present an improvement heuristic for our policies in Sect. 4. By using the results of Sects. 2 and 3, we construct tightened piecewise affine policies via lifting in Sect. 5. Finally, we provide numerical evidence for the performance of our policy compared to other state of the art policies in Sect. 6. Section 7 concludes this paper with a brief reflection of our work and avenues for future research. To keep the paper concise, we defer proofs that could possibly interrupt the reading flow to “Appendices B–O”." }, { "section": "Conclusion", "element": "p", "matched_phrases": [ "open_question", "future_work" ], "passage": "While this work extends most of the best-known approximation results for a relatively general class of ARO problems from the two-stage to the multi-stage setting, it remains an open question whether other strong two-stage ARO results can be generalized to multi-stage ARO in a similar manner. Answering this question remains an interesting area for further research. In this context, binary and uncertain recourse decisions remain particularly relevant challenges. Our analysis in Sect. 2.3 has shown that the extension of our policies to encompass these recourse decision types is not straightforward. Nevertheless, exploring the integration of our methodology into the established approaches of piecewise constant policies and k-adaptability, which have proven to be effective in these cases, appears as a promising starting point for future work. Another interesting area for future research is the extension of piecewise affine policies and the concept of domination to adjustable data-driven and distributionally robust optimization. More specifically, we believe that one can obtain tractable data-driven policies by directly fitting the polyhedral uncertainty sets used to construct our policies from data." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-02055-y", "title": "Frank–Wolfe-type methods for a class of nonconvex inequality-constrained problems", "authors": [ "Liaoyuan Zeng", "Yongle Zhang", "Guoyin Li", "Ting Kei Pong", "Xiaozhou Wang" ], "published_online": "2024-02-03", "volume": "208", "issue": "1-2", "pages": "717-761", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-02055-y/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-02032-5", "title": "A general framework for multi-marginal optimal transport", "authors": [ "Brendan Pass", "Adolfo Vargas-Jiménez" ], "published_online": "2024-02-05", "volume": "208", "issue": "1-2", "pages": "75-110", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-02032-5/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02067-2", "title": "Correction: High-order methods beyond the classical complexity bounds: inexact high-order proximal-point methods", "authors": [ "Masoud Ahookhosh", "Yurii Nesterov" ], "published_online": "2024-02-10", "volume": "208", "issue": "1-2", "pages": "409-410", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02067-2/fulltext.html", "article_kind": "non_research_item", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-02051-2", "title": "Semiglobal exponential stability of the discrete-time Arrow-Hurwicz-Uzawa primal-dual algorithm for constrained optimization", "authors": [ "Michelangelo Bin", "Ivano Notarnicola", "Thomas Parisini" ], "published_online": "2024-02-12", "volume": "208", "issue": "1-2", "pages": "629-660", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-02051-2/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Conclusions", "element": "p", "matched_phrases": [ "open_problem" ], "passage": "This article considered the long-standing open problem of nonlocal asymptotic stability of the popular discrete-time primal-dual algorithm (1) for convex, constrained optimization. In particular, under due convexity and regularity assumptions, it is proved that an optimal equilibrium exists, it is unique, and it is semiglobally asymptotically stable. Namely, for every compact set of initial conditions, there exists a sufficiently small stepsize, such that the sequences generated by the algorithm converge to the optimal solution of the optimization problem and to the optimal Lagrange multipliers. Moreover, convergence is exponential, and the optimal point is Lyapunov stable. As shown in Sect. 1.2, global asymptotic stability cannot be established for the considered algorithm, so as semiglobal guarantees are the best achievable in the general case." }, { "section": "Conclusions", "element": "p", "matched_phrases": [ "future_work" ], "passage": "Finally, it is worth remarking that the impossibility of global convergence of the algorithm complicates the development of robustness corollaries, which cannot be global in the size of the uncertainty. As a consequence, such shortfall poses new challenges in the design of distributed algorithms based on (1) and targeting, e.g., consensus optimization problems over networks [20,21,22, 24]. Future research will mainly focus on this latter extension." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02060-9", "title": "Convex hulls of monomial curves, and a sparse positivstellensatz", "authors": [ "Gennadiy Averkov", "Claus Scheiderer" ], "published_online": "2024-02-16", "volume": "209", "issue": "1-2", "pages": "113-131", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02060-9/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-023-02052-1", "title": "Deciding whether a lattice has an orthonormal basis is in co-NP", "authors": [ "Christoph Hunkenschröder" ], "published_online": "2024-02-19", "volume": "208", "issue": "1-2", "pages": "763-775", "publisher_url": "https://link.springer.com/article/10.1007/s10107-023-02052-1/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Introduction", "element": "p", "matched_phrases": [ "open_question" ], "passage": "We close this section by giving an overview of related work. The rest of the paper is structured as follows. Section 2 provides sufficient background on the topic, and shows that the rotated standard lattice problem is equivalent to the unimodular decomposition problem. We also introduce characteristic vectors and show some easy properties. Section 3 discusses the theorem of Elkies and applies it, showing that RSLP is in NP \\(\\cap \\) co-NP. Section 4 concludes with open questions." }, { "section": "Conclusion", "element": "p", "matched_phrases": [ "open_problem" ], "passage": "Another easy approach to RSLP is the following. If a lattice admits an orthogonal basis \\(\\{b_1,\\dots ,b_n\\}\\) then any HKZ-reduced basis [17, 18] is an orthogonal basis. Due to the recursive structure of those bases n calls to an oracle for the Shortest Vector Problem (SVP) are sufficient to find this basis. Hence the best known running time using this reduction is \\(2^{\\mathcal {O} (n)}\\) (see e.g. [16]) and even allows to find general orthogonal bases. But as UDP is—from a complexity point of view—supposedly easier than SVP or CVP it is of great interest to find a smarter algorithm. It seems to be a reappearing phenomenon that if a certain class of lattices allows to solve lattice problems such as SVP or CVP fast, a certain representation is needed. For instance, if \\(\\Lambda \\) is a lattice of Voronoi’s first kind, we still need to know an obtuse superbasis before using the polynomial time algorithm in [19]. Therefore, being able to find orthonormal, or even orthogonal bases remains an interesting open problem." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02059-2", "title": "Submodular maximization and its generalization through an intersection cut lens", "authors": [ "Liding Xu", "Leo Liberti" ], "published_online": "2024-02-19", "volume": "211", "issue": "1-2", "pages": "341-377", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02059-2/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02061-8", "title": "Automated tight Lyapunov analysis for first-order methods", "authors": [ "Manu Upadhyaya", "Sebastian Banert", "Adrien B. Taylor", "Pontus Giselsson" ], "published_online": "2024-02-26", "volume": "209", "issue": "1-2", "pages": "133-170", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02061-8/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Introduction", "element": "p", "matched_phrases": [ "future_work" ], "passage": "The paper is organized as follows: in Sect. 2, we introduce the problem class and the algorithm representation. Section 3 discusses interpolation results and frames them in our setting. We define the notion of a quadratic Lyapunov inequality in Sect. 4. Section 5 contains the main result on the existence of a quadratic Lyapunov inequality. Section 6 contains numerical examples and Sect. 7 contains a proof of our core result. Section 8 contains the main conclusions of this work and discusses future work." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02058-3", "title": "The effect of smooth parametrizations on nonconvex optimization landscapes", "authors": [ "Eitan Levin", "Joe Kileel", "Nicolas Boumal" ], "published_online": "2024-03-04", "volume": "209", "issue": "1-2", "pages": "63-111", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02058-3/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Characterizations of lifts", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "We conjecture that the reverse inclusion in Theorem 3.16 always holds (it does for all the lifts in Sect. 2). If this is indeed true, then \\(\\varphi \\) satisfies the “2 \\(\\Rightarrow \\!\\) 1” property at y if and only if \\((\\textrm{T}_x\\mathcal {X})^* = W_y\\), neatly echoing the condition for “1 \\(\\Rightarrow \\!\\) 1”, namely, \\((\\textrm{T}_x\\mathcal {X})^* = ({\\text {im}}\\textbf{L}_y)^\\perp \\)." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02062-7", "title": "Hessian barrier algorithms for non-convex conic optimization", "authors": [ "Pavel Dvurechensky", "Mathias Staudigl" ], "published_online": "2024-03-04", "volume": "209", "issue": "1-2", "pages": "171-229", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02062-7/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Conclusion", "element": "p", "matched_phrases": [ "open_question", "future_work" ], "passage": "We derived Hessian barrier algorithms based on first- and second-order information on the objective f. We performed a detailed analysis of their worst-case iteration complexity in order to find a suitably defined approximate KKT point. Under weak regularity assumptions and in the presence of general conic constraints, our Hessian barrier algorithms share the best known complexity rates in the literature for first- and second-order approximate KKT points. Our methods are characterized by a decomposition approach of the feasible set which leads to numerically efficient subproblems at each their iteration. Several open questions for the future remain. First, our iterations assume that the subproblems are solved exactly, and for practical reasons, this should be relaxed. Second, we mentioned that \\({{\\,\\mathrm{\\textbf{FAHBA}}\\,}}\\) can be interpreted as a discretization of the Hessian-barrier gradient system [4], but the exact relationship is not explored yet. This, however, could be an important step towards understanding acceleration techniques of \\({{\\,\\mathrm{\\textbf{FAHBA}}\\,}}\\), akin to an accelerated version of the cubic regularized Newton method. Furthermore, the cubic-regularized version has no corresponding continuous-time version yet. It will be very interesting to investigate this question further. Additionally, the question of convergence of the trajectory \\((x^{k})_{k\\ge 0}\\) generated by either scheme is open. Another interesting direction for future research would be to allow for higher-order Taylor expansions in the subproblems in order to boost convergence speed further, similar to [28]." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02057-4", "title": "Level constrained first order methods for function constrained optimization", "authors": [ "Digvijay Boob", "Qi Deng", "Guanghui Lan" ], "published_online": "2024-03-06", "volume": "209", "issue": "1-2", "pages": "1-61", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02057-4/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Inexact LCPG", "element": "p", "matched_phrases": [ "we_do_not_know" ], "passage": "According to [28, Theorem 4.5.1], the complexity of interior point methods depends not only on the time to follow the central path, but also on the time to arrive near the analytic center from an arbitrary initial point. Let us put it in the context of Algorithm 1. Despite the strict feasibility guarantee, we do not know whether \\(x^k\\) is near the analytic center of each subproblem. It remains to show how to control the complexity of approximating the analytic center." }, { "section": "Conclusion", "element": "p", "matched_phrases": [ "open_problem" ], "passage": "In this work, we presented a new LCPG method for nonconvex function constrained optimization which can achieve gradient complexity of the same order as that of unconstrained nonconvex problems. The key ingredient in our algorithm design is the use of constraint levels to ensure the subproblem feasibility, which allows us to overcome a well-known difficulty in bounding the Lagrange multipliers in the presence of nonsmooth constraints. Moreover, a merit of our convergence analysis is its striking similarity with that of gradient descent methods for unconstrained problems. Therefore, we can easily extend our method to minimizing stochastic, finite-sum, and structured nonsmooth functions with nonconvex function constraints; many of the complexity results were not known before. Another important feature of our work is that the method can deal with complex scenarios where the subproblems are not exactly solvable. To the best of our knowledge, existing work on sequential convex optimization (SQP, MBA) only assumes the subproblems to be exactly solved. We provided a detailed complexity analysis of LCPG when the subproblems are inexactly solved by customized interior point method and first-order method. Finally, we clearly distinguished the notion of gradient complexity from that of computational complexity. In terms of gradient complexity, all of our proposed methods are state-of-the-art and easy to implement. Whether the computational complexity can be further improved for composite cases remains an open problem that we leave as a future direction." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02065-4", "title": "Matroid-based TSP rounding for half-integral solutions", "authors": [ "Anupam Gupta", "Euiwoong Lee", "Jason Li", "Marcin Mucha", "Heather Newman", "Sherry Sarkar" ], "published_online": "2024-03-06", "volume": "206", "issue": "1-2", "pages": "541-576", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02065-4/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [ { "section": "Introduction", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "The (symmetric) traveling salesman problem asks: given an graph \\(G = (V,E)\\) with edge-lengths \\(c_e \\ge 0\\), find the shortest tour that visits all vertices at least once. The Christofides-Serdyukov algorithm [4, 13] gives a \\(\\nicefrac {3}{2}\\)-approximation to this APX-hard problem; this was recently improved to a \\((\\nicefrac {3}{2}-\\varepsilon )\\)-approximation by the breakthrough work of Karlin, Klein, and Oveis Gharan, where \\(\\varepsilon > 0\\) [10]. A related question is: what is the integrality gap of the subtour-elimination polytope relaxation for the TSP? Wolsey had adapted the Christofides-Serdyukov analysis to show an upper bound of \\(\\nicefrac {3}{2}\\) [16] (also [14]), and there exists a lower bound of \\(\\nicefrac {4}{3}\\). Building on their above-mentioned work, Karlin, Klein, and Oveis Gharan gave an integrality gap of \\(1.5 - \\varepsilon '\\) for another small constant \\(\\varepsilon ' > 0\\) [11], thereby making the first progress towards the conjectured optimal value of \\(\\nicefrac {4}{3}\\) in nearly half a century." }, { "section": "Introduction", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "Given the \\(\\nicefrac {3}{2}\\) barrier has been broken, we can ask: what other techniques can be effective here? How can we make further progress? These questions are interesting even for cases where the LP has additional structure. The half-integral cases (i.e., points for which \\(x_e \\in \\{0, \\nicefrac {1}{2}, 1\\}\\) for all e) are particularly interesting due to 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 [15]. The team of Karlin, Klein, and Oveis Gharan first used their max-entropy approach to get an integrality gap of 1.49993 for half-integral LP solutions [9], before they moved on to the general case in [10] and obtained an integrality gap of \\(1.5-\\varepsilon \\); the latter improvement is considerably smaller than in the half-integral case. It is natural to ask: can we do better for half-integral instances?" }, { "section": "Introduction", "element": "p", "matched_phrases": [ "open_question" ], "passage": "It remains to get rid of the two simplifying assumptions. To sample trees when |V| is odd (an open question from [7]), we add a new vertex to fix the parity, and perform local surgery on the solution to get a new TSP solution and reduce to the even case. The challenge here is to show that the losses incurred are small, and hence each edge is still even with constant probability." }, { "section": "Samplers", "element": "p", "matched_phrases": [ "open_problem" ], "passage": "Our samplers will depend on the parity of |V|: when |V| is even, the MatInt sampler is the one given by [7, Theorem 13], which we describe in Sect. 3.1. They left the case of odd |V| as an open problem, and in Appendix B we extend their procedure to the odd case." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02066-3", "title": "Multiplicative auction algorithm for approximate maximum weight bipartite matching", "authors": [ "Da Wei Zheng", "Monika Henzinger" ], "published_online": "2024-03-06", "volume": "210", "issue": "1-2", "pages": "881-894", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02066-3/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02068-1", "title": "On convergence of iterative thresholding algorithms to approximate sparse solution for composite nonconvex optimization", "authors": [ "Yaohua Hu", "Xinlin Hu", "Xiaoqi Yang" ], "published_online": "2024-03-06", "volume": "211", "issue": "1-2", "pages": "181-206", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02068-1/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02070-7", "title": "Polyhedral properties of RLT relaxations of nonconvex quadratic programs and their implications on exact relaxations", "authors": [ "Yuzhou Qiu", "E. Alper Yıldırım" ], "published_online": "2024-03-06", "volume": "209", "issue": "1-2", "pages": "397-433", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02070-7/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02064-5", "title": "Generalized minimum 0-extension problem and discrete convexity", "authors": [ "Martin Dvorak", "Vladimir Kolmogorov" ], "published_online": "2024-03-07", "volume": "209", "issue": "1-2", "pages": "279-322", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02064-5/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Introduction", "element": "p", "matched_phrases": [ "open_problem" ], "passage": "The rest of the paper is organized as follows. Section 2 reviews Hirai’s theory and defines L-convex functions on modular complexes and the SDA algorithm for minimizing them. Section 3 generalizes this to L-convex functions on extended modular complexes and presents -SDA algorithm. Both sections use the notion of submodular functions on valuated modular semilattices, which are formally defined in Sect. 4. All proofs missing in Sect. 3 are given in Sects. 5 and 6.1; this completes the proof of the tractability direction of Theorem 3. The NP-hardness direction of Theorem 3 is proven in Sect. 6.2. Section 7 concludes the paper with a list of open problems." }, { "section": "Extended modular complexes", "element": "p", "matched_phrases": [ "we_do_not_know" ], "passage": "For extended modular complex \\(\\Gamma \\) let \\(\\Phi _\\Gamma \\) be the language over domain \\(D=V_\\Gamma \\) that consists of all functions \\(f:D^n\\rightarrow \\overline{\\mathbb R}\\) such that f is L-convex on \\(\\Gamma ^n\\). The results above imply that any instance \\(\\mathcal{I}\\) of \\(\\Phi _\\Gamma \\) can be solved by the Steepest Descent Algorithm (Algorithm 1), and each subproblem in line 2 can be solved in polynomial time. However, we do not know whether the number of steps would be polynomially bounded. To get a polynomial bound, we introduce an alternative algorithm, which we denote -SDA." }, { "section": "Open questions", "element": "p", "matched_phrases": [ "open_problem" ], "passage": "In this section we state some open problems that we find interesting." }, { "section": "Open questions", "element": "p", "matched_phrases": [ "open_problem" ], "passage": "There are also some open problems concerning L-convex functions on extended modular complexes." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02063-6", "title": "Generalized Nash equilibrium problems with mixed-integer variables", "authors": [ "Tobias Harks", "Julian Schwarz" ], "published_online": "2024-03-13", "volume": "209", "issue": "1-2", "pages": "231-277", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02063-6/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02069-0", "title": "The Chvátal–Gomory procedure for integer SDPs with applications in combinatorial optimization", "authors": [ "Frank de Meijer", "Renata Sotirov" ], "published_online": "2024-03-13", "volume": "209", "issue": "1-2", "pages": "323-395", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02069-0/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Introduction", "element": "p", "matched_phrases": [ "open_problem" ], "passage": "It is known that the practical strength of CG cuts in integer linear programming is mainly due to their application in branch-and-bound methods. In this vein, we propose a generic branch-and-cut (B &C) framework for ISDPs. Our algorithm initially relaxes the PSD constraint and solves a mixed integer linear program (MILP), where the PSD constraint is imposed iteratively via CG and/or strengthened CG cuts. To derive strengthened CG cuts, we use a similar approach to the one for rational polyhedra by Dash et al. [24]. Our B &C algorithm is an extension of the algorithm of [46], in which separation is only based on positive semidefiniteness without taking into account the integrality of the variables. Our approach also builds up on the work by Çezik and Iyengar [15], in which the authors leave the separation of CG cuts for conic problems as an open problem and do not include these cuts in their computational study. We provide an example of our approach for a common class of binary SDPs that frequently appears in combinatorial optimization." }, { "section": "Conclusions", "element": "p", "matched_phrases": [ "open_problem" ], "passage": "A generic B &C algorithm for ISDPs based on strengthened CG cuts is presented in Sect. 3, see Algorithm 1. Our algorithm is a refinement of the algorithm from [46], where the authors use eigenvector based inequalities to separate infeasible integer points. Moreover, our work can be seen as an extension of [15], in which the authors introduce CG cuts for conic programs, but leave the efficient separation of CG cuts as an open problem. Our numerical results indicate the effectiveness of the use of deeper CG cuts. We also provide a separation routine for binary SDPs originating from combinatorial optimization problems, see Sect. 2." }, { "section": "Conclusions", "element": "p", "matched_phrases": [ "future_work" ], "passage": "Our work inspires several future research directions. It would be interesting to study the performance of our B &C algorithm when applied to other optimization problems that can be formulated as ISDPs. We expect the exploitation of CG cuts in the branching scheme to be effective for such ISDPs. Moreover, as for the QTSP many known classes of cuts turned out to be (strengthened) CG cuts with respect to the ISDP formulation, it would be interesting to know whether this also holds for other problems." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02075-2", "title": "Perseus: a simple and optimal high-order method for variational inequalities", "authors": [ "Tianyi Lin", "Michael I. Jordan" ], "published_online": "2024-03-13", "volume": "209", "issue": "1-2", "pages": "609-650", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02075-2/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02076-1", "title": "A slightly lifted convex relaxation for nonconvex quadratic programming with ball constraints", "authors": [ "Samuel Burer" ], "published_online": "2024-03-14", "volume": "211", "issue": "1-2", "pages": "157-179", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02076-1/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02072-5", "title": "Sum-of-squares relaxations for polynomial min–max problems over simple sets", "authors": [ "Francis Bach" ], "published_online": "2024-03-15", "volume": "209", "issue": "1-2", "pages": "475-501", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02072-5/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02073-4", "title": "Accelerated first-order methods for a class of semidefinite programs", "authors": [ "Alex L. Wang", "Fatma Kılınç-Karzan" ], "published_online": "2024-03-22", "volume": "209", "issue": "1-2", "pages": "503-556", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02073-4/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Appendices", "element": "p", "matched_phrases": [ "we_leave", "future_work" ], "passage": "We note that the Gaussian model of phase retrieval requires storing the matrix G as part of the instance. This is a matrix of size \\(O(n^2)\\) and thus limits the size of our current experiments. Nonetheless, we expect the behavior we observe with these experiments to hold in the real setting of phase retrieval where the matrix G is highly structured and can be stored implicitly. We leave this as important future work." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02071-6", "title": "Convergence rates for sums-of-squares hierarchies with correlative sparsity", "authors": [ "Milan Korda", "Victor Magron", "Rodolfo Ríos-Zertuche" ], "published_online": "2024-03-25", "volume": "209", "issue": "1-2", "pages": "435-473", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02071-6/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02083-2", "title": "Stackelberg risk preference design", "authors": [ "Shutian Liu", "Quanyan Zhu" ], "published_online": "2024-04-02", "volume": "209", "issue": "1-2", "pages": "785-823", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02083-2/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02081-4", "title": "Finding global minima via kernel approximations", "authors": [ "Alessandro Rudi", "Ulysse Marteau-Ferey", "Francis Bach" ], "published_online": "2024-04-04", "volume": "209", "issue": "1-2", "pages": "703-784", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02081-4/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02080-5", "title": "Compressing branch-and-bound trees", "authors": [ "Gonzalo Muñoz", "Joseph Paat", "Álinson S. Xavier" ], "published_online": "2024-04-06", "volume": "210", "issue": "1-2", "pages": "669-694", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02080-5/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02074-3", "title": "Non-convex scenario optimization", "authors": [ "Simone Garatti", "Marco C. Campi" ], "published_online": "2024-04-08", "volume": "209", "issue": "1-2", "pages": "557-608", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02074-3/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Non-convex robust scenario optimization", "element": "p", "matched_phrases": [ "open_problem" ], "passage": "(a digression into the convex setup) In this remark we show that even in a convex setup the thesis of Corollary 8 (that is, the property that the cumulative distribution function of the violation is lower-bounded by a \\(\\mathcal{B}(d,N \\! - \\! d \\! + \\! 1)\\) distribution) ceases to be correct if the problem is degenerate. We mention this fact explicitly to burn off a fallacious belief to the contrary that has circulated in some research environments. This digression will also allow us to introduce open problems that we feel like sharing with the community." }, { "section": "Non-convex robust scenario optimization", "element": "p", "matched_phrases": [ "open_problem" ], "passage": "An aspect we want to further discuss in relation to this example relates to the existence of a nonempty interior of the feasibility domain. Let us start by observing that the treatment of [28] for convex problems assumes that the feasibility domain of any realization of the scenario problem has a nonempty interior (Assumption 1 in [28]). Under the existence of a nonempty interior (besides existence and uniqueness of the solution), Theorem 1 in [28] claims that the cumulative distribution function of \\(V(x^*_N)\\) is always dominated (also in the degenerate case) by a Beta distribution \\(\\mathcal{B}(\\bar{d},N \\! - \\! \\bar{d} \\! + \\! 1)\\) (\\(\\bar{d}\\) is the dimension of the optimization domain). Whether this claim preserves its validity without the assumption on the existence of a nonempty interior is at present an open problem. In fact, no theoretical result confirms this claim, while no counterexample is known that confutes its validity. In contrast, when the nonempty interior assumption is dropped, the example in this remark sets a final negative word on the possibility of dominating the cumulative distribution function of \\(V(x^*_N)\\) with a Beta distribution \\(\\mathcal{B}(d,N \\! - \\! d \\! + \\! 1)\\), where d is an upper bound to the complexity strictly smaller than \\(\\bar{d}\\).Footnote 26 Whether this conclusion maintains its validity in the presence of a nonempty interior of the feasibility domain is at present another open problem.Footnote 27" }, { "section": "Notes", "element": "li", "matched_phrases": [ "we_do_not_know" ], "passage": "On the right-hand side of (50) we wrote \\(\\liminf \\) and not \\(\\lim \\) because, a-priori, we do not know if such a limit exists and, indeed, \\(\\liminf \\) suffices to close the argument. On the other hand, after proving (49) the reader may want to verify that this limit actually exists." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02082-3", "title": "Extended convergence analysis of the Scholtes-type regularization for cardinality-constrained optimization problems", "authors": [ "Sebastian Lämmel", "Vladimir Shikhman" ], "published_online": "2024-04-09", "volume": "211", "issue": "1-2", "pages": "207-243", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02082-3/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02077-0", "title": "An update-and-stabilize framework for the minimum-norm-point problem", "authors": [ "Satoru Fujishige", "Tomonari Kitahara", "László A. Végh" ], "published_online": "2024-04-18", "volume": "210", "issue": "1-2", "pages": "281-311", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02077-0/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [ { "section": "Concluding remarks", "element": "p", "matched_phrases": [ "future_work" ], "passage": "There is scope for further improvements both in the theoretical analysis and in practical implementations. In this paper, we only analyzed the running time for uncapacitated instances. Combined with existing results from [22], we expect that similar bounds can be shown for capacitated instances. We note that for the analysis, it would suffice to run minor cycles only once in a while, say after every O(n) gradient updates. From a practical perspective however, running minor cycles after every update appears to be highly beneficial in most cases. Rigorous computational experiments, using standard families of LP benchmarks, are left for future work." }, { "section": "Concluding remarks", "element": "p", "matched_phrases": [ "future_work" ], "passage": "Future work should also compare the performance of our algorithms to the gradient projection method [5, 23], using techniques from that method to our algorithm and vice versa. We note that for NNLS instances, starting from a stable point our algorithm already finds the optimal gradient update. However, a similar search as in gradient projection methods may be useful in the capacitated case. In the other direction, we note that the conjugate gradient iterations used in gradient projection do not correspond to an explicit choice of a centroid mapping. A possible enhancement of gradient projection could come from approximating a ‘local-norm’ objective as in (6) in the second stage." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02079-y", "title": "A normal fan projection algorithm for low-rank optimization", "authors": [ "James R. Scott", "Joseph Geunes" ], "published_online": "2024-04-23", "volume": "209", "issue": "1-2", "pages": "681-702", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02079-y/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02084-1", "title": "From approximate to exact integer programming", "authors": [ "Daniel Dadush", "Friedrich Eisenbrand", "Thomas Rothvoss" ], "published_online": "2024-04-24", "volume": "210", "issue": "1-2", "pages": "223-241", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02084-1/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [ { "section": "Introduction", "element": "p", "matched_phrases": [ "open_problem" ], "passage": "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 in the area of algorithms and complexity. Integer programming can be described in the following more general form. Here, a convex body is synonymous for a full-dimensional compact and convex set." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02085-0", "title": "Graph coloring and semidefinite rank", "authors": [ "Renee Mirka", "Devin Smedira", "David P. Williamson" ], "published_online": "2024-04-24", "volume": "206", "issue": "1-2", "pages": "577-605", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02085-0/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [ { "section": "Abstract", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "This paper considers the interplay between semidefinite programming, matrix rank, and graph coloring. Karger et al. (J ACM 45(2):246–265, 1998) give a vector program in which a coloring of a graph can be encoded as a semidefinite matrix of low rank. By complementary slackness conditions of semidefinite programming, if an optimal dual solution has high rank, any optimal primal solution must have low rank. We attempt to characterize graphs for which we can show that the corresponding dual optimal solution must have rank high enough that the primal solution encodes a coloring. In the case of the original Karger, Motwani, and Sudan vector program, we show that any graph which is a k-tree has sufficiently high dual rank, and we can extract the coloring from the corresponding low-rank primal solution. We can also show that if a graph is not uniquely colorable, then no sufficiently high rank dual optimal solution can exist. This allows us to completely characterize the planar graphs for which dual optimal solutions have sufficiently high dual rank, since it is known that the uniquely colorable planar graphs are precisely the planar 3-trees. We then modify the semidefinite program to have an objective function with costs, and explore when we can create an objective function such that the optimal dual solution has sufficiently high rank. We show that it is always possible to construct such an objective function given the graph coloring. The construction of the objective function gives rise to heuristics for 4-coloring planar graphs. We enumerated all maximal planar graphs with an induced \\(K_4\\) of up to 14 vertices; the heuristics successfully found a 4-coloring for 99.75% of them. Our research was motivated by trying to use semidefinite programming to prove the four-color theorem, which states that every planar graph can be colored with four colors. There is an intriguing connection of the Karger–Motwani–Sudan semidefinite program with the Colin de Verdière graph invariant (J Combin. Theory Ser B 50:11-21, 1990) (and a corresponding conjecture of Colin de Verdière), in which matrices that have some similarities to the dual feasible matrices of the semidefinite program must have high rank in the case that graphs are of a certain type; for instance, planar graphs have rank that would imply that the primal solution of the semidefinite program encodes a 4-coloring." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02078-z", "title": "Sample complexity analysis for adaptive optimization algorithms with stochastic oracles", "authors": [ "Billy Jin", "Katya Scheinberg", "Miaolan Xie" ], "published_online": "2024-04-29", "volume": "209", "issue": "1-2", "pages": "651-679", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02078-z/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02087-y", "title": "Gaining or losing perspective for convex multivariate functions on box domains", "authors": [ "Luze Xu", "Jon Lee" ], "published_online": "2024-05-08", "volume": "211", "issue": "1-2", "pages": "319-339", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02087-y/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02086-z", "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" ], "published_online": "2024-05-09", "volume": "210", "issue": "1-2", "pages": "165-188", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02086-z/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [ { "section": "Conclusion", "element": "p", "matched_phrases": [ "open_question" ], "passage": "We remark that the linear runtime \\(\\mathcal {O}(m^d)\\) of our algorithm depends on the \\(\\Omega (m^d)\\) input cost coefficients to be dense (i.e. in the case \\(d=2\\), they form a dense matrix). In the case of sparsely encoded input cost coefficients, it is an open question how to obtain a linear time algorithm." }, { "section": "Conclusion", "element": "p", "matched_phrases": [ "open_question" ], "passage": "Another open question related to the algorithm is its implementation for the practical relevant case of \\(d=2\\), and the comparison of its perfomance on linearizable instances to a generic blackbox quadratic solver, which may not be aware that the input instance is linearizable." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02092-1", "title": "A characterization of maximal homogeneous-quadratic-free sets", "authors": [ "Gonzalo Muñoz", "Joseph Paat", "Felipe Serrano" ], "published_online": "2024-05-23", "volume": "210", "issue": "1-2", "pages": "641-668", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02092-1/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02098-9", "title": "Optimal general factor problem and jump system intersection", "authors": [ "Yusuke Kobayashi" ], "published_online": "2024-05-23", "volume": "210", "issue": "1-2", "pages": "591-610", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02098-9/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02093-0", "title": "Stable set polytopes with high lift-and-project ranks for the Lovász–Schrijver SDP operator", "authors": [ "Yu Hin Au", "Levent Tunçel" ], "published_online": "2024-05-25", "volume": "212", "issue": "1-2", "pages": "79-114", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02093-0/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02094-z", "title": "A tight $$(1.5+\\epsilon )$$-approximation for unsplittable capacitated vehicle routing on trees", "authors": [ "Claire Mathieu", "Hang Zhou" ], "published_online": "2024-05-25", "volume": "212", "issue": "1-2", "pages": "115-146", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02094-z/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02096-x", "title": "ReLU neural networks of polynomial size for exact maximum flow computation", "authors": [ "Christoph Hertrich", "Leon Sering" ], "published_online": "2024-05-25", "volume": "210", "issue": "1-2", "pages": "377-406", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02096-x/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02097-w", "title": "Exploiting the polyhedral geometry of stochastic linear bilevel programming", "authors": [ "Gonzalo Muñoz", "David Salas", "Anton Svensson" ], "published_online": "2024-05-27", "volume": "210", "issue": "1-2", "pages": "695-730", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02097-w/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02099-8", "title": "Information complexity of mixed-integer convex optimization", "authors": [ "Amitabh Basu", "Hongyi Jiang", "Phillip Kerger", "Marco Molinaro" ], "published_online": "2024-05-27", "volume": "210", "issue": "1-2", "pages": "3-45", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02099-8/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02095-y", "title": "Polarized consensus-based dynamics for optimization and sampling", "authors": [ "Leon Bungert", "Tim Roith", "Philipp Wacker" ], "published_online": "2024-05-31", "volume": "211", "issue": "1-2", "pages": "125-155", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02095-y/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [ { "section": "Models", "element": "p", "matched_phrases": [ "we_leave", "future_work" ], "passage": "we expect that under reasonable Lipschitz-like assumptions on the logarithm of the kernel these arguments translate to our case, but we leave this for future work. The biggest challenge is that for standard CBO \\(t\\mapsto \\textsf{m}_{\\beta }[\\rho _t]\\) is a continuous curve in \\(\\mathbb {R}^d\\), whereas \\(t\\mapsto \\textsf{m}_{\\beta ,{{\\,\\mathrm{\\textsf{k}}\\,}}}[\\rho _t](\\cdot )\\) is not. Rather, it is a curve in a space of vector fields. This requires more sophisticated compactness arguments than the one made in [5, 7]." }, { "section": "Conclusion and outlook", "element": "p", "matched_phrases": [ "future_work" ], "passage": "There is a lot of room for future work regarding well-definedness of the Fokker–Planck equation derived above, and stability and convergence of both the mean-field and particle system to consensus using less restrictive assumptions than the ones used in this paper. We are convinced that the Lyapunov function \\(\\textsf{L}[\\rho _t]\\) which we studied in Sect. 2.4 will be very helpful in this endeavour. Future work will also focus on further numerical improvements of our method, in particular, incorporating a batching strategy similar to the one from [10] to further improve performance in high-dimensional optimization. Finally, a long-term goal will be to find multimodal sampling methods which are provably consistent for multimodal and non-Gaussian distributions. Given that even gradient-free sampling from unimodal non-Gaussian distributions is still relatively poorly understood, with [13] being a promising approach, this will be a challenging task for future work." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02088-x", "title": "An asynchronous proximal bundle method", "authors": [ "Frank Fischer" ], "published_online": "2024-06-04", "volume": "209", "issue": "1-2", "pages": "825-857", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02088-x/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Introduction", "element": "p", "matched_phrases": [ "we_do_not_know" ], "passage": "Another idea to reduce the running time required for exact oracle calls is to replace the oracles with inexact ones, see [3] for a comprehensive overview. Instead of computing the function values and subgradients exactly only approximated values are computed. The asynchronous setting is somehow similar to the inexact setting. In fact, we replace the notion of “inexact” values by “outdated” values, i.e., we do not know the exact function values in the current point but in some older (hopefully close) point. In order to get exact results from inexact oracles one typically requires controlling the inexactness in some way, e.g., to use so called “asymptotically exact” oracles, which provide increasingly better approximations during the run of the algorithm. In fact, the authors in [5] already realized that incremental bundle methods can be interpreted as inexact oracles and analyzed their algorithm within an inexact framework. Because the asynchronous setting is so closely related to incremental methods, the same could be done for our algorithm. However, there are some important differences. The algorithm developed in this paper learns the amount of inaccuracy introduced by the asynchronicity during the run and automatically adjusts the inaccuracy of future evaluations in order to guarantee convergence. We will show that this allows to interpret the guess model in our approach as an asymptotically exact oracle which has vanishing approximation errors." }, { "section": "Introduction", "element": "p", "matched_phrases": [ "future_work" ], "passage": "This paper is structured as follows. In Sect. 2 we introduce the notation and present the basic asynchronous framework and all major building blocks, which will be discussed in later sections. In Sect. 3 we start with the computation of new candidate points using the typical master problem of proximal bundle methods, and we discuss the adaptions made for descent steps in Sect. 4. We analyze the convergence of our algorithm in Sect. 5 which resembles the steps of convergence proofs for bundle methods for our asynchronous method. The basic algorithm and its analysis are presented without explicit guess models. We give some possible guess models in Sect. 6. In order to adjust the accuracy of these guess models accordingly, the Lipschitz constants must be known. However, we also show how the algorithm can be extended to the case of unknown Lipschitz constants in Sect. 6.2. Furthermore, we extend the algorithm to the case where some functions are constant along some coordinate directions in Sect. 6.4. The final algorithm with all extensions is shown in Sect. 7. Finally, we present some preliminary computational tests in Sect. 8 and give some directions for future work in Sect. 9." }, { "section": "The descent step", "element": "p", "matched_phrases": [ "we_do_not_know" ], "passage": "The problem in the asynchronous setting is that we do not know the function values \\(f(\\bar{x}^{k^-})\\) and \\(f(\\bar{x}^k)\\) exactly. Therefore, we use the best known lower bound \\(\\bar{f}^k \\le f(\\bar{x}^{k^-})\\) to approximate the function value in the center and the guess model value \\(\\tilde{f}^k\\) to approximate the function value in the candidate. Denoting the predicted decrease by" } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02091-2", "title": "The complexity of first-order optimization methods from a metric perspective", "authors": [ "A. S. Lewis", "Tonghua Tian" ], "published_online": "2024-06-04", "volume": "212", "issue": "1-2", "pages": "49-78", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02091-2/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02100-4", "title": "Universal heavy-ball method for nonconvex optimization under Hölder continuous Hessians", "authors": [ "Naoki Marumo", "Akiko Takeda" ], "published_online": "2024-06-04", "volume": "212", "issue": "1-2", "pages": "147-175", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02100-4/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02102-2", "title": "A unified framework for symmetry handling", "authors": [ "Jasper van Doornmalen", "Christopher Hojny" ], "published_online": "2024-06-04", "volume": "212", "issue": "1-2", "pages": "217-271", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02102-2/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Computational study", "element": "p", "matched_phrases": [ "we_leave", "future_work" ], "passage": "Comparing the two parameterizations of orbitopal fixing, we see that the median variant performs \\({4.7}\\,\\%\\) better than the first variant. This shows that the choice for the column reordering by symmetry \\(\\varphi _\\beta \\) has a measurable impact on the running time for dynamic orbitopal reduction. A possible explanation for the median rule to perform better than the first rule is that median creates more balanced branch-and-bound trees, cf. Sect. 3.2. Consequently, the right choice of \\(\\varphi _\\beta \\) might significantly change the performance of orbitopal reduction. We leave it as future research to find a good rule for the selection of \\(\\varphi _\\beta \\) as this rule might be based on the structure of the underlying problem, e.g., size of the variable domains, number of rows, and number of columns." }, { "section": "Computational study", "element": "p", "matched_phrases": [ "future_work" ], "passage": "Table 4 presents the computational results for the unbounded and reformulated instances. The figures for the other instance sets are similar, and are tabulated in Appendix D. Without symmetry handling, no instance with \\(n \\ge 7\\) can be solved, whereas instances up to \\(n = 14\\) are solved with symmetry handling. Comparing static and dynamic orbitopal reduction, the static version is more performant than the dynamic variant for small values of n. For \\(n \\ge 11\\), however, the dynamic version is faster and can solve one more instance. Handling symmetries via SST cuts is, in general, the method performing best. One possible explanation is that the explicit addition of symmetry handling inequalities in the problem formulation allows to strengthen the relaxation of the non-linear kissing number problem, cf. [44]. Combining static dynamic orbitopal reduction and SST cuts allows to improve the running time, in particular, this combination is faster than dynamic orbitopal reduction. Except for \\(n = 12\\), however, the combined setting is not faster than exclusively using SST cuts to handle symmetries. For future research, it would be interesting to see whether one can gain synergies between SST cuts and static orbitopal reduction. In principle, one can expect that such synergies exist since SST cuts are implied by the symmetry handling constraints that static orbitopal reduction is also using. But this requires a more in-depth study of the symmetries of the kissing number problem and how they interact with the solution techniques for non-linear problems, which is out of scope of this article." }, { "section": "Conclusions and future research", "element": "p", "matched_phrases": [ "future_work" ], "passage": "Due to the flexibility of our framework, it is not only applicable for the methods discussed in this article, but also allows to apply methods that will be developed in the future (provided they are compatible with SHC s (2)). This opens, among others, the following directions for future research." }, { "section": "Conclusions and future research", "element": "p", "matched_phrases": [ "future_work" ], "passage": "Last, in the computational results we describe decision rules for enabling/disabling certain symmetry handling methods. If new symmetry handling methods are cast into our framework, however, these rules need to be updated. Future research could thus encompass the derivation of good rules for how to handle symmetries in our framework." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02106-y", "title": "A PTAS for the horizontal rectangle stabbing problem", "authors": [ "Arindam Khan", "Aditya Subramanian", "Andreas Wiese" ], "published_online": "2024-06-13", "volume": "206", "issue": "1-2", "pages": "607-630", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02106-y/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02101-3", "title": "Generalized scaling for the constrained maximum-entropy sampling problem", "authors": [ "Zhongzhu Chen", "Marcia Fampa", "Jon Lee" ], "published_online": "2024-06-20", "volume": "212", "issue": "1-2", "pages": "177-216", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02101-3/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02108-w", "title": "A slope generalization of Attouch theorem", "authors": [ "Aris Daniilidis", "David Salas", "Sebastián Tapia-García" ], "published_online": "2024-06-20", "volume": "212", "issue": "1-2", "pages": "319-348", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02108-w/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Main result, final comments and perspectives", "element": "p", "matched_phrases": [ "open_problem" ], "passage": "Open problems: This work is motivated by the celebrated Attouch theorem (Theorem 1.4), the determination result of slopes [27], and the sensitivity result of [14]. All of these results are valid in Hilbert spaces, while the first two are also valid in Banach spaces (see [4, 12] and [33]). Therefore, a natural question is whether Theorem 1.6 (our main result) is true in Hilbert spaces, or more generally, in reflexive Banach spaces (or even in general Banach spaces). While there is no obvious obstruction for this extension, the present work relies heavily on local compactness of the space, for many of its intermediate results and consequently any potential extension should rather rely in a completely different approach." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02105-z", "title": "On the tightness of an SDP relaxation for homogeneous QCQP with three real or four complex homogeneous constraints", "authors": [ "Wenbao Ai", "Wei Liang", "Jianhua Yuan" ], "published_online": "2024-06-21", "volume": "211", "issue": "1-2", "pages": "5-48", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02105-z/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02115-x", "title": "Special Issue: Integer Programming and Combinatorial Optimization (IPCO) 2022", "authors": [ "Karen Aardal", "Laura Sanità" ], "published_online": "2024-06-28", "volume": "206", "issue": "1-2", "pages": "1-2", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02115-x/fulltext.html", "article_kind": "non_research_item", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02103-1", "title": "On solving a rank regularized minimization problem via equivalent factorized column-sparse regularized models", "authors": [ "Wenjing Li", "Wei Bian", "Kim-Chuan Toh" ], "published_online": "2024-07-03", "volume": "212", "issue": "1-2", "pages": "273-318", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02103-1/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02089-w", "title": "On the directional asymptotic approach in optimization theory", "authors": [ "Matúš Benko", "Patrick Mehlitz" ], "published_online": "2024-07-05", "volume": "209", "issue": "1-2", "pages": "859-937", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02089-w/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Directional asymptotic stationarity in nonsmooth optimization", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "We conjecture that the sufficient condition (2.12) can be weakened to just pseudo-subregularity of \\(\\varPhi \\) in Corollary 4.2 (b). However, it would require a different proof to show this, so we will not explore this option. For \\(\\gamma :=1\\), such a result is known to hold, see [36, Theorem 7]." }, { "section": "Concluding remarks", "element": "p", "matched_phrases": [ "future_work" ], "passage": "In this paper, we completely neglected to study the potential value of directional asymptotic regularity in numerical optimization which might be a promising topic of future research. Furthermore, it has been shown in [63] that nondirectional asymptotic regularity can be applied nicely as a qualification condition in the limiting variational calculus. Most likely, directional asymptotic regularity may play a similar role in the directional limiting calculus. Finally, it seems desirable to further develop the calculus for pseudo-coderivatives for mappings which possess a more difficult structure than constraint mappings." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02090-3", "title": "Optimal methods for convex nested stochastic composite optimization", "authors": [ "Zhe Zhang", "Guanghui Lan" ], "published_online": "2024-07-05", "volume": "212", "issue": "1-2", "pages": "1-48", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02090-3/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02117-9", "title": "Matrix discrepancy and the log-rank conjecture", "authors": [ "Benny Sudakov", "István Tomon" ], "published_online": "2024-07-05", "volume": "212", "issue": "1-2", "pages": "567-579", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02117-9/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Abstract", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "We use this result to obtain a modest improvement of Lovett’s best known upper bound on the log-rank conjecture. We prove that any \\(m\\times n\\) binary matrix M of rank at most r contains an \\((m\\cdot 2^{-O(\\sqrt{r})})\\times (n\\cdot 2^{-O(\\sqrt{r})})\\) sized all-1 or all-0 submatrix, which implies that the deterministic communication complexity of any Boolean function of rank r is at most \\(O(\\sqrt{r})\\)." }, { "section": "Introduction", "element": "p", "matched_phrases": [ "open_problem", "conjecture" ], "passage": "The log-rank conjecture, proposed by Lovász and Saks [12] in 1988, is one of the fundamental open problems in communication complexity. It states that for any Boolean function \\(f:X\\times Y\\rightarrow \\{-1,1\\}\\) of rank r, its deterministic communication complexity \\(\\text{ CC}^{det}(f)\\) is bounded by \\({{\\,\\textrm{polylog}\\,}}(r)\\). We refer the reader to Kushilevitz and Nisan [7] for exact definitions, or the recent survey of Lee and Shraibman [10] for a detailed overview of the problem. The log-rank conjecture has a number of combinatorial interpretations, which will be the main focus of our paper." }, { "section": "Introduction", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "Lovász and Saks [12] observed (see also [13]) that this problem is closely related to bounding the chromatic number of graphs, whose adjacency matrix has bounded rank. Indeed, if c(r) denotes the maximum of \\(\\text{ CC}^{det}(f)\\) among every f of rank r, then \\(\\chi (G)\\le 2^{c(r+1)}\\) for every graph G, whose adjacency matrix has rank at most r. Furthermore, the log-rank conjecture is related to finding monochromatic submatrices in low-rank binary matrices. To this end, let \\(\\alpha (r)>0\\) be such that any binary matrix \\(M\\in \\{0,1\\}^{m\\times n}\\) of rank at most r contains an all-0 or all-1 submatrix of area at least \\(mn/2^{\\alpha (r)}\\) (here, the area of a matrix refers to the product of the number of rows and columns). It was proved by Nisan and Wigderson [15] that \\(c(f)=O((\\log r)^2+\\sum _{i=0}^{\\log _2 r}\\alpha (r/2^i))\\)." }, { "section": "Concluding remarks", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "Finally, let us mention the following interesting conjecture from [13], which states that a relaxation of Theorem 4.2 should also hold for matrices that are not necessarily binary. We believe the methods presented in our paper could be useful in making progress on this conjecture. See [14, 17] for some recent developments." }, { "section": "Concluding remarks", "element": "h3", "matched_phrases": [ "conjecture" ], "passage": "Conjecture 5.1" }, { "section": "About this article", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "Sudakov, B., Tomon, I. Matrix discrepancy and the log-rank conjecture. Math. Program. 212, 567–579 (2025). https://doi.org/10.1007/s10107-024-02117-9" }, { "section": "About this article", "element": "li", "matched_phrases": [ "conjecture" ], "passage": "Log-rank conjecture" } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02107-x", "title": "On circuit diameter bounds via circuit imbalances", "authors": [ "Daniel Dadush", "Zhuan Khye Koh", "Bento Natura", "László A. Végh" ], "published_online": "2024-07-08", "volume": "206", "issue": "1-2", "pages": "631-662", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02107-x/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [ { "section": "Introduction", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "The combinatorial diameter of a polyhedron P is the diameter of the vertex-edge graph associated with P. Hirsch’s famous conjecture from 1957 asserted that the combinatorial diameter of a d-dimensional polytope (bounded polyhedron) with f facets is at most \\(f-d\\). This was disproved by Santos in 2012 [30]. The polynomial Hirsch conjecture, i.e., finding a poly(f) bound on the combinatorial diameter remains a central question in the theory of linear programming." }, { "section": "Introduction", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "The first quasipolynomial bound was given by Kalai and Kleitman [24, 25], see [32] for the best current bound and an overview of the literature. Dyer and Frieze [11] proved the polynomial Hirsch conjecture for totally unimodular (TU) matrices. For a system \\(\\{x\\in \\mathbb {R}^d:\\, Mx\\le b\\}\\) with integer constraint matrix M, polynomial diameter bounds were given in terms of the maximum subdeterminant \\(\\Delta _M\\) [4, 7, 12, 20]. These arguments can be strengthened to using a parametrization by a ‘discrete curvature measure’ \\(\\delta _M\\ge 1/(d\\Delta ^2_M)\\). The best such bound was given by Dadush and Hähnle [12] as \\(O(d^3\\log (d/\\delta _M)/\\delta _M)\\), using a shadow vertex simplex algorithm." }, { "section": "Introduction", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "A circuit walk is a sequence of points \\(x^{(0)},x^{(1)},\\ldots ,x^{(k)}\\) in P such that for each \\(i=0,\\ldots ,k-1\\), \\(x^{(i+1)}=x^{(i)}+\\alpha ^{(i)} g^{(i)}\\) for some \\(g^{(i)}\\in \\mathcal {E}(A)\\) and \\(\\alpha ^{(i)} > 0\\), and further, \\(x^{(i)}+\\alpha g^{(i)}\\notin P\\) for any \\(\\alpha >\\alpha ^{(i)}\\), i.e., each consecutive circuit step is maximal. The circuit diameter of P is the maximum length (number of steps) of a shortest circuit walk between any two vertices \\(x,y\\in P\\). Note that, in contrast to walks in the vertex-edge graph, circuit walks are non-reversible and the minimum length from x to y may be different from the one from y to x; this is due to the maximal step requirement. The circuit-analogue of Hirsch conjecture, formulated in [5], asserts that the circuit diameter of d-dimensional polyhedron with f facets is at most \\(f-d\\); this may be true even for unbounded polyhedra, see [8]. For P in the form (P), \\(d=n-m\\) and the number of facets is at most n; hence, the conjectured bound is m." }, { "section": "Introduction", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "In light of these results, the next important step towards the polynomial Hirsch conjecture might be to show a poly\\((n,\\log \\kappa _A)\\) bound on the combinatorial diameter of (P). Note that—in contrast with the circuit diameter—not even a poly\\((n,{\\text {size}}(A,b))\\) bound is known. In this context, the best known general bound is \\(O((n-m)^{3} m \\kappa _{A}\\log (\\kappa _{A}+n))\\) implied by [12]." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02112-0", "title": "Monoidal strengthening and unique lifting in MIQCPs", "authors": [ "Antonia Chmiela", "Gonzalo Muñoz", "Felipe Serrano" ], "published_online": "2024-07-08", "volume": "210", "issue": "1-2", "pages": "189-222", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02112-0/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02119-7", "title": "Structural iterative rounding for generalized k-median problems", "authors": [ "Anupam Gupta", "Benjamin Moseley", "Rudy Zhou" ], "published_online": "2024-07-10", "volume": "212", "issue": "1-2", "pages": "581-634", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02119-7/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Conclusion", "element": "p", "matched_phrases": [ "open_problem" ], "passage": "To conclude, we highlight some extensions and open problems." }, { "section": "Conclusion", "element": "li", "matched_phrases": [ "open_question" ], "passage": "It is an interesting open question to obtain a true approximation for GKM. In our paper, we obtain a 6.387-approximation fractional solution for GKM with \\(O(r_1 + r_2)\\) fractional facilities, where \\(r_1\\) and \\(r_2\\) are the number of packing and covering constraint, respectively. It is likely that the pre-processing of [2]—summarized in Theorem 27—extends to GKM and gives analogous Extra Invariants (Definition 9). The main challenge seems to be rounding the final \\(O(r_1 + r_2)\\) fractional facilities such that there is at least one open facility in each \\(C^*\\)-ball and all packing and covering constraints are satisfied. In both special cases we consider, knapsack median and k-median with outliers, there is only one non-trivial constraint to satisfy (the knapsack constraint and outlier constraint, respectively). With our current techniques, it seems difficult to handle multiple constraints simultaneously." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02104-0", "title": "Efficient separation of RLT cuts for implicit and explicit bilinear terms", "authors": [ "Ksenia Bestuzheva", "Ambros Gleixner", "Tobias Achterberg" ], "published_online": "2024-07-16", "volume": "210", "issue": "1-2", "pages": "47-74", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02104-0/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02109-9", "title": "On the geometry and refined rate of primal–dual hybrid gradient for linear programming", "authors": [ "Haihao Lu", "Jinwen Yang" ], "published_online": "2024-07-17", "volume": "212", "issue": "1-2", "pages": "349-387", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02109-9/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Introduction", "element": "p", "matched_phrases": [ "future_work" ], "passage": "We discuss the sharpness of homogeneous linear inequality system and collect necessary convergence results of PDHG in Section 2. The analysis of identification stage (Stage I) and local convergence (Stage II) are presented respectively in Section 3 and 4. Section 5 illustrates and verifies the theoretical results through numerical experiments. We conclude the paper and propose several future research directions in Section 6." }, { "section": "Conclusion and future directions", "element": "p", "matched_phrases": [ "open_question" ], "passage": "We end the paper by presenting a few interesting open questions for further investigation:" }, { "section": "Conclusion and future directions", "element": "li", "matched_phrases": [ "we_leave", "future_work" ], "passage": "Extensions to the restarted algorithm. PDLP utilizes a new technique, restarting, to obtain the optimal linear convergence rate with global sharpness condition. We believe our theoretical results can be extended to the restarted algorithms, and we leave it as future work." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02114-y", "title": "Distributional utility preference robust optimization models in multi-attribute decision making", "authors": [ "Jian Hu", "Dali Zhang", "Huifu Xu", "Sainan Zhang" ], "published_online": "2024-07-17", "volume": "212", "issue": "1-2", "pages": "519-565", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02114-y/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02120-0", "title": "New notions of simultaneous diagonalizability of quadratic forms with applications to QCQPs", "authors": [ "Alex L. Wang", "Rujun Jiang" ], "published_online": "2024-07-17", "volume": "212", "issue": "1-2", "pages": "635-682", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02120-0/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02110-2", "title": "A trust region-type normal map-based semismooth Newton method for nonsmooth nonconvex composite optimization", "authors": [ "Wenqing Ouyang", "Andre Milzarek" ], "published_online": "2024-07-22", "volume": "212", "issue": "1-2", "pages": "389-435", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02110-2/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02118-8", "title": "Competitive kill-and-restart and preemptive strategies for non-clairvoyant scheduling", "authors": [ "Sven Jäger", "Guillaume Sagnol", "Daniel Schmidt genannt Waldschmidt", "Philipp Warode" ], "published_online": "2024-07-22", "volume": "210", "issue": "1-2", "pages": "457-509", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02118-8/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02122-y", "title": "The pseudo-Boolean polytope and polynomial-size extended formulations for binary polynomial optimization", "authors": [ "Alberto Del Pia", "Aida Khajavirad" ], "published_online": "2024-07-22", "volume": "212", "issue": "1-2", "pages": "717-761", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02122-y/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02111-1", "title": "On supervalid inequalities for binary interdiction games", "authors": [ "Ningji Wei", "Jose L. Walteros" ], "published_online": "2024-07-27", "volume": "212", "issue": "1-2", "pages": "437-478", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02111-1/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02124-w", "title": "Convergence in distribution of randomized algorithms: the case of partially separable optimization", "authors": [ "D. Russell Luke" ], "published_online": "2024-07-27", "volume": "212", "issue": "1-2", "pages": "763-798", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02124-w/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Introduction", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "A noteworthy feature of blockwise methods, and what distinguishes the present study from [18,19,20] is that, even when the objective in (1) is convex, blockwise algorithms do not satisfy the usual regularity properties enjoyed by convex optimization algorithms that lead generically to global convergence. This is demonstrated in Example 2. The stochastic implementations for convex problems, however, do enjoy nice properties in expectation (see Theorem 1), and this is enough to guarantee generic global convergence (Theorem 7, Proposition 17). While this fact lies implicitly behind the convergence analysis of, for instance, [25] and many others, it was recognized in [40] as the important property of descent in expectation. We place these observations in the context of Markov operators with update functions that satisfy desirable properties in expectation (see Theorem 3). These notions, at the level of the Markov operator, have already been defined in [18,19,20]; the convergence results presented in those works, however, are based on the assumption that each of the update functions that generate the Markov operator have the same class of regularity that they have in expectation. Blockwise algorithms for partially separable optimization do not enjoy this structure, and therefore many of the results of [18,19,20] do not immediately apply; indeed, we conjecture that some of the stronger convergence results of [18, 19] are not true without additional compactness assumptions, hence our analogous global convergence statement for the convex case Proposition 6, is weaker than its counterparts [18, Theorem 3.6] or [19, Theorem 2.9]." }, { "section": "Notation and random function iterations", "element": "p", "matched_phrases": [ "future_work" ], "passage": "The stochastic blockwise Douglas-Rachford Algorithm 3 therefore can be understood as a stochastic blockwise ADMM algorithm for solving (21). The discussion above about the separability of f is yet another way of understanding the observed computational difficulty of implementing this algorithm; it is quite unlikely that f given by (21) will be separable in the standard basis and therefore the resolvent (16c) will have to be computed numerically. Alternative primal-dual methods that circumvent this are the topic of future research." }, { "section": "Final remarks", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "There a several open technicalities lurking between the lines above, and one rather obvious challenge hiding in plain sight. To the hidden technicalities belong the question of whether metric subregularity is necessary for quantitative convergence in some appropriate metric of Markov operators that are not paracontractions in measure. We conjecture that this is true. Another open technical issue concerns the statement of asymptotic regularity in Proposition 6. This result is incomplete without some extension to a weak type of convergence in distribution. For consistent stochastic fixed point problems, if each of the update functions \\(T_i\\) were \\(\\alpha \\)-fne, then almost sure weak convergence of the iterates is guaranteed [18, Theorem 3.9]; at issue here is whether this holds when \\(T_i\\) is pointwise \\(\\alpha \\)-fne in expectation at invariant measures of the corresponding Markov operator. We expect that there should be a counterexample to this claim. Characterization of the supports of invariant measures in the inconsistent case is quite challenging and essential for meaningfully connecting the limiting distributions of the algorithms to solutions to the underlying optimization problem. Finally, the restriction of the study to single-valued mappings does not allow one to capture the full extent of behavior one sees with nonconvex problems. Projection methods for sparse affine feasibility, for instance, have the property that the projection onto a sparsity constraint can be multi-valued on all neighborhoods of a solution (see [21, Lemma III.2]). An extension of the analysis presented here to multi-valued mappings, is required." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02128-6", "title": "Optimizing distortion riskmetrics with distributional uncertainty", "authors": [ "Silvana M. Pesenti", "Qiuqi Wang", "Ruodu Wang" ], "published_online": "2024-07-29", "volume": "213", "issue": "1-2", "pages": "51-106", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02128-6/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02113-z", "title": "Unified smoothing approach for best hyperparameter selection problem using a bilevel optimization strategy", "authors": [ "Jan Harold Alcantara", "Chieu Thanh Nguyen", "Takayuki Okuno", "Akiko Takeda", "Jein-Shan Chen" ], "published_online": "2024-08-08", "volume": "212", "issue": "1-2", "pages": "479-518", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02113-z/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02116-w", "title": "On the correlation gap of matroids", "authors": [ "Edin Husić", "Zhuan Khye Koh", "Georg Loho", "László A. Végh" ], "published_online": "2024-08-08", "volume": "210", "issue": "1-2", "pages": "407-456", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02116-w/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [ { "section": "Conclusion", "element": "p", "matched_phrases": [ "future_work" ], "passage": "where \\(\\rho \\) is the rank and \\(\\gamma \\) is the girth of the matroid. The difference between this upper bound and the lower bound given in Theorem 1.5 motivates a further analysis to reduce this gap. Another direction for future work is to explore other matroid parameters for quantifying the the correlation gap. This could be motivated by tighter bounds on the correlation gap for special matroid classes. Finally, it remains to open to which extent our analysis can be modified to give new insights also for other (classes of) submodular functions." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02123-x", "title": "Nonsmooth convex–concave saddle point problems with cardinality penalties", "authors": [ "Wei Bian", "Xiaojun Chen" ], "published_online": "2024-08-08", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02123-x/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02125-9", "title": "A radial basis function method for noisy global optimisation", "authors": [ "Dirk Banholzer", "Jörg Fliege", "Ralf Werner" ], "published_online": "2024-08-08", "volume": "211", "issue": "1-2", "pages": "49-92", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02125-9/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [ { "section": "Convergence of method", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "One possibility for establishing density of the iterates \\(x_1, \\ldots , x_n\\) is to resort to the available convergence results of Gutmann’s method (see the supplementary Section A for a brief summary of the main results), and show that these pertain if the exact function values \\(f(x_i)\\) are replaced by the noisy observations \\(\\hat{f}^{(n)} (x_i)\\) (\\(1 \\le i \\le n\\)). An indispensable assumption is thus that the involved level of noise decreases to zero over the course of the optimisation. As one may already conjecture from the construction of regularised least-squares approximants through (20), this will be required to adopt Gutmann’s proof of convergence for noisy function values. Nevertheless, in the vanishing iterative noise model, for a natural choice of \\(\\gamma _n\\), the sequence \\(\\{ n \\gamma _n\\}\\) converges to zero if we require that \\(\\epsilon _i^{(n)} \\rightarrow 0\\) fast enough as \\(n \\rightarrow \\infty \\), as the following Theorem 3 shows." }, { "section": "Conclusions", "element": "p", "matched_phrases": [ "future_work" ], "passage": "In this paper, we have addressed the global optimisation of an expensive and noisy objective function where observed function values are assumed to lie within error bounds. Based on Gutmann’s original RBF method for minimising a deterministic objective function, relying on radial basis function interpolation, we have first discussed common approaches of radial basis function approximations for integration into a response surface method. Arguing in favour of regularised least-squares approximants, we then have presented a noisy RBF method that constructs the smoothest possible response surfaces that stay within the given error bounds at the evaluated points, and determines new evaluation points by minimising a regularised least-squares criterion in terms of a target value. Further on, we have established convergence of the noisy RBF method to the global minimum of any continuous function, under some additional assumptions on the error bounds, and provided relevant convergence results. Finally, we have provided a numerical illustration of our RBF method for noisy objective functions by considering a simple test problem. Future work will include the assessment of the proposed method on various academic and real-world test functions." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02126-8", "title": "Optimizing for strategy diversity in the design of video games", "authors": [ "Oussama Hanguir", "Will Ma", "Jiangze Han", "Christopher Thomas Ryan" ], "published_online": "2024-08-08", "volume": "210", "issue": "1-2", "pages": "335-376", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02126-8/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02132-w", "title": "Configuration balancing for stochastic requests", "authors": [ "Franziska Eberle", "Anupam Gupta", "Nicole Megow", "Benjamin Moseley", "Rudy Zhou" ], "published_online": "2024-08-08", "volume": "210", "issue": "1-2", "pages": "243-279", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02132-w/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [ { "section": "Introduction", "element": "p", "matched_phrases": [ "open_question" ], "passage": "We note that we have not tried to optimize the constant for our offline algorithm. It remains an interesting open question whether the online setting admits an O(1)-competitive algorithm." }, { "section": "Conclusion", "element": "p", "matched_phrases": [ "open_problem" ], "passage": "We considered the configuration balancing problem under uncertainty. In contrast to the (often overly optimistic) clairvoyant settings and the (often overly pessimistic) non-clairvoyant settings, we consider the stochastic setting where each request j presents a set of random vectors, and we need to (adaptively) pick one of these vectors, to minimize the expected maximum load over the m resources. We give logarithmic bounds for several general settings (which are existentially tight), and a much better O(1) offline and \\(O(\\log \\log m)\\) online bound for the related machines setting. Closing the gap for online related-machines load balancing remains an intriguing open problem. More generally, getting a better understanding of both adaptive and non-adaptive algorithms for stochastic packing and scheduling problems remains an exciting direction for research." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02131-x", "title": "Convexification techniques for fractional programs", "authors": [ "Taotao He", "Siyue Liu", "Mohit Tawarmalani" ], "published_online": "2024-08-16", "volume": "213", "issue": "1-2", "pages": "107-149", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02131-x/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02121-z", "title": "On the strength of Lagrangian duality in multiobjective integer programming", "authors": [ "Matthew Brun", "Tyler Perini", "Saumya Sinha", "Andrew J. Schaefer" ], "published_online": "2024-08-20", "volume": "212", "issue": "1-2", "pages": "683-715", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02121-z/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02127-7", "title": "Nonlinear conjugate gradient methods: worst-case convergence rates via computer-assisted analyses", "authors": [ "Shuvomoy Das Gupta", "Robert M. Freund", "Xu Andy Sun", "Adrien Taylor" ], "published_online": "2024-08-22", "volume": "213", "issue": "1-2", "pages": "1-49", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02127-7/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02130-y", "title": "Machine learning augmented branch and bound for mixed integer linear programming", "authors": [ "Lara Scavuzzo", "Karen Aardal", "Andrea Lodi", "Neil Yorke-Smith" ], "published_online": "2024-08-22", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02130-y/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [ { "section": "Learning tasks", "element": "p", "matched_phrases": [ "open_question" ], "passage": "New directions The work we surveyed showcases the potential in mixing the extensive body of domain knowledge in variable selection with new learning techniques. Still a lot of open questions remain. For example, little attention has been directed towards highlighting important subsets of variables, as opposed to choosing a single one at each node. Khalil et al. [66] propose an approach to finding such important subsets, in this case the so-called backdoors [33], and show promise in using them as prioritized branching candidates. Another relevant gap is the absence of expert knowledge for certain classes of MILPs. In any case, it is clear that new ML-based methods need to build upon the pre-existing knowledge on variable selection to achieve a fruitful combination." }, { "section": "Learning tasks", "element": "p", "matched_phrases": [ "open_question", "future_work" ], "passage": "Many open questions in this vast research area still exist, making this a promising area of future work." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02133-9", "title": "A fast combinatorial algorithm for the bilevel knapsack problem with interdiction constraints", "authors": [ "Noah Weninger", "Ricardo Fukasawa" ], "published_online": "2024-08-22", "volume": "210", "issue": "1-2", "pages": "847-879", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02133-9/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02138-4", "title": "Accelerated stochastic approximation with state-dependent noise", "authors": [ "Sasila Ilandarideva", "Anatoli Juditsky", "Guanghui Lan", "Tianjiao Li" ], "published_online": "2024-08-27", "volume": "213", "issue": "1-2", "pages": "239-280", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02138-4/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02135-7", "title": "Recycling valid inequalities for robust combinatorial optimization with budgeted uncertainty", "authors": [ "Christina Büsing", "Timo Gersing", "Arie M. C. A. Koster" ], "published_online": "2024-08-29", "volume": "210", "issue": "1-2", "pages": "97-146", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02135-7/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [ { "section": "Conclusions", "element": "p", "matched_phrases": [ "future_work" ], "passage": "For future research, it would be interesting to further analyze the recycling of non-knapsack inequalities and evaluate whether one can obtain facet-defining robust inequalities from specific classes of nominal inequalities. Furthermore, recycling should be tested in combination with other tailored approaches and the effect of recycling should be evaluated for robust problems with uncertain constraints. Another interesting line of research is to investigate whether the recycling approach can be generalized to other uncertainty sets that have similar structures as the budgeted uncertainty set." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02137-5", "title": "Complexity of chordal conversion for sparse semidefinite programs with small treewidth", "authors": [ "Richard Y. Zhang" ], "published_online": "2024-09-17", "volume": "213", "issue": "1-2", "pages": "201-237", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02137-5/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02136-6", "title": "Fast convergence to non-isolated minima: four equivalent conditions for $${\\textrm{C}^{2}}$$ functions", "authors": [ "Quentin Rebjock", "Nicolas Boumal" ], "published_online": "2024-09-19", "volume": "213", "issue": "1-2", "pages": "151-199", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02136-6/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02141-9", "title": "A continuous approximation model for the electric vehicle fleet sizing problem", "authors": [ "Brais González-Rodríguez", "Aurélien Froger", "Ola Jabali", "Joe Naoum-Sawaya" ], "published_online": "2024-09-21", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02141-9/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [ { "section": "Appendices", "element": "p", "matched_phrases": [ "future_work" ], "passage": "In summary, this section shows that the framework that we present in this paper is fairly generic and can accommodate different approximation models for the optimal tour length. The computational results in terms of computational performance, number of partitions, and the associated charging costs are however dependent on the approximation. As such, future research may potentially investigate approximations that are best suited for the framework that is presented in this paper." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02139-3", "title": "The computational complexity of finding stationary points in non-convex optimization", "authors": [ "Alexandros Hollender", "Manolis Zampetakis" ], "published_online": "2024-09-27", "volume": "213", "issue": "1-2", "pages": "281-341", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02139-3/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Introduction", "element": "p", "matched_phrases": [ "remains_open" ], "passage": "Although the above results draw an almost complete picture for answering Questions 1 and 2 in the constrained optimization setting the following two questions still remain open." }, { "section": "Introduction", "element": "li", "matched_phrases": [ "open_problem" ], "passage": "2. Unconstrained algorithm for \\(d = 2\\) (Theorem4.1). For \\(d = 2\\) we provide a zero-order algorithm that finds \\(\\varepsilon \\)-stationary points of smooth and bounded functions in unconstrained optimization using at most \\(O(1/\\varepsilon )\\) value queries to the objective. This result resolves a recent open problem of [8]." }, { "section": "Introduction", "element": "p", "matched_phrases": [ "open_problem" ], "passage": "Unconstrained algorithm for \\(d = 2\\) (Theorem4.1). For \\(d = 2\\) we provide a zero-order algorithm that finds \\(\\varepsilon \\)-stationary points of smooth and bounded functions in unconstrained optimization using at most \\(O(1/\\varepsilon )\\) value queries to the objective. This result resolves a recent open problem of [8]." }, { "section": "Introduction", "element": "h3", "matched_phrases": [ "open_question" ], "passage": "1.2 Open questions" }, { "section": "Introduction", "element": "p", "matched_phrases": [ "open_question" ], "passage": "The following couple of open questions arise from our work:" } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02144-6", "title": "Decomposition of probability marginals for security games in max-flow/min-cut systems", "authors": [ "Jannik Matuschke" ], "published_online": "2024-10-03", "volume": "210", "issue": "1-2", "pages": "611-640", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02144-6/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [ { "section": "Abstract", "element": "p", "matched_phrases": [ "open_question" ], "passage": "As a subroutine of our algorithm, we provide a combinatorial algorithm for computing shortest paths in abstract networks, partially answering an open question by McCormick [20]. We further show that a conservation law proposed in [6] for the requirement vector \\(\\pi \\) in DAGs can be reduced to the setting of affine requirements described above." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02146-4", "title": "Dyadic linear programming and extensions", "authors": [ "Ahmad Abdi", "Gérard Cornuéjols", "Bertrand Guenin", "Levent Tunçel" ], "published_online": "2024-10-03", "volume": "213", "issue": "1-2", "pages": "473-516", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02146-4/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Introduction", "element": "p", "matched_phrases": [ "remains_open", "conjecture" ], "passage": "The interest in dyadic linear programming stems not only from the computer science perspective mentioned above, but also from mathematics and from optimization. Take a mathematical point of view: Given a prime integer \\(p \\ge 2\\), we say that a rational number is finitely p-adic if it is of the form \\(\\frac{r}{p^k}\\) for some integer r and nonnegative integer k. This concept is closely related to the notion of the p-adic numbers introduced by Hensel, formally defined as the set of “finite-tailed\" infinite series \\(\\sum _{i=N}^{+\\infty } a_i p^i\\) where \\(N\\in \\mathbb {Z}\\), and \\(a_i \\in \\mathbb {Z}\\) and \\(0 \\le a_i < p\\) for each \\(i\\ge N\\).Footnote 1 The study of p-adic numbers gives rise to beautiful and powerful mathematics; see the excellent book by Gouvêa for more [17]. It can be readily checked that the set of finitely p-adic numbers is the set of finite series of the form \\(\\sum _{i=N}^{M} a_i p^i\\), where \\(M,N\\in \\mathbb {Z}\\), \\(M\\ge N\\), and \\(0 \\le a_i < p, a_i \\in \\mathbb {Z}\\) for all \\(N\\le i\\le M\\), justifying our terminology.Footnote 2 In this paper, we will only deal with finitely p-adic numbers; for simplicity we refer to them as p-adic numbers throughout the paper. Also we refer to “2-adic\" as “dyadic\". More generally, we say that a rational number is [p]-adic if it is of the form \\(\\frac{r}{s}\\) where r is an integer and s is a product of powers of primes between 2 and p. These numbers appear naturally in some of the theorems in this paper. For the optimization point of view, let us mention an intriguing conjecture of Seymour dating back to 1975; see Schrijver [28] 79.3e. Let A be a 0,1 matrix such that the set covering polyhedron \\(Ax \\ge \\textbf{1}, x \\ge 0\\) has only integral vertices, where \\(\\textbf{1}\\) denotes the vector of all 1 s. Thus the linear program \\(\\min \\left\\{ c^\\top x: Ax\\ge \\textbf{1}, x \\ge 0\\right\\} \\) has an optimal 0,1 solution for any objective function \\(c \\in \\mathbb {Z}^n_+\\). Seymour conjectured that the dual linear program \\(\\max \\left\\{ \\textbf{1}^\\top y: A^\\top y\\le c, y \\ge 0 \\right\\} \\) always has a dyadic optimal solution y. This conjecture is still open but is known to hold in a few important special cases. For example, when A is the T-cut versus edge incidence matrix of a graph, an optimal dual solution y is \\(\\frac{1}{2}\\)-integral (Lovász [22]). Seymour’s conjecture was proved recently in a couple of other special cases [2, 5]." }, { "section": "Introduction", "element": "p", "matched_phrases": [ "future_work" ], "passage": "Section 6 provides conclusions and possible directions for future research." }, { "section": "Concluding remarks and future research", "element": "p", "matched_phrases": [ "remains_open", "conjecture" ], "passage": "has an optimal solution that is dyadic, i.e., \\(\\frac{1}{2^k}\\)-integral for some integer \\(k\\ge 0\\) [5]. The proof provides no upper bound guarantee on k. That said, it has been conjectured by Seymour that \\(k\\le 2\\) ([12], Conjecture 2.15, also see Schrijver [28] 79.3e). An upper bound of \\(k\\le c\\log (|E|)\\) for some universal constant c, also remains open. The conjecture of Seymour combined with our approach in the current manuscript, suggests a study of classes of linear programs with integral data such that for every integral objective function vector, the primal has a \\(\\frac{1}{2^{k_1}}\\)-integral optimal solution (whenever it has an optimal solution) and the dual has a \\(\\frac{1}{2^{k_2}}\\)-integral optimal solution, for some fixed pair of nonegative integers \\(k_1\\) and \\(k_2\\). Seymour’s conjecture above corresponds to the special case \\(k_1:=0\\), \\(k_2:=2\\)." }, { "section": "Concluding remarks and future research", "element": "p", "matched_phrases": [ "open_question" ], "passage": "In the special case of \\(\\Vert A\\Vert _\\infty =1\\), the best lower bound on the support size of an optimal solution to a [p]-adic linear program that we can show is at most O(m), while our upper bound is \\(O(m\\ln m)\\). Closing the gap in this case remains an intriguing open question. An important special case comes in the dyadic (\\(p=2\\)) case from the fractional T-join packing problem mentioned above. By developing a column generation technique for solving dyadic linear programs, and by leveraging tools from matching theory, we achieve a matching upper bound of O(m) (note \\(m=|E|\\) in this case) [1]." }, { "section": "Concluding remarks and future research", "element": "p", "matched_phrases": [ "future_work" ], "passage": "We have used the size of the input, in particular \\(\\ln \\Vert A\\Vert _{\\infty }\\) to state our results (for bounds on computational complexity as well as support size bounds etc.). However, for specially structured instances, there are better complexity measures, capturing more intrinsic properties of the instance. This typically yields tighter and more insightful bounds. Thus, it would be fruitful to pursue this direction in future research." }, { "section": "Concluding remarks and future research", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "Let \\(a^1,\\ldots ,a^n\\in \\mathbb {Z}^m\\). The set \\(\\{a^1,\\ldots ,a^n\\}\\) is a dyadic generating set for a cone (DGSC) if every integral vector in the conic hull of the vectors can be expressed as a dyadic conic combination of the vectors. This notion was coined and studied in our first work on dyadic linear programming [4]. Given a DGSC \\(\\{a^1,\\ldots ,a^n\\}\\) and an integral vector b in the conic hull, we know that b can be expressed as a dyadic conic combination of the vectors. What is the fewest number k of nonzero coefficients in such a representation? While Corollary 5.12 gives an upper bound of \\(O(m\\ln (m\\Vert A\\Vert ^2_{\\infty }))\\) on k, we conjecture that there is a O(m) upper bound on k. The rationale behind this comes from the observation that a DGSC may be viewed as the dyadic analogue of Hilbert bases for integer linear programming [16] for which the analogous upper bound guarantee is \\(2m-2\\) [29]." }, { "section": "Concluding remarks and future research", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "Finally, we propose a weakening of a known conjecture. A matrix \\(A\\in \\{0,1\\}^{m\\times n}\\) is ideal if the set covering linear program \\(\\min \\{c^\\top x:Ax\\ge \\textbf{1},x\\ge \\textbf{0}\\}\\) has an integral optimal solution for all \\(c\\in \\mathbb {Z}^n_{\\ge 0}\\). Seymour conjectures that the dual linear program \\(\\max \\{\\textbf{1}^\\top y:A^\\top y\\le c,y\\ge \\textbf{0}\\}\\) has a dyadic optimal solution for all \\(c\\in \\mathbb {Z}^n_{\\ge 0}\\) (see Schrijver [28] 79.3e). We propose the following weakening of this conjecture." }, { "section": "Concluding remarks and future research", "element": "h3", "matched_phrases": [ "conjecture" ], "passage": "Conjecture 6.1" }, { "section": "Concluding remarks and future research", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "This conjecture has been verified for the clutter of dijoins of a digraph ([20], Theorem 2.13)." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02147-3", "title": "Cuts and semidefinite liftings for the complex cut polytope", "authors": [ "Lennart Sinjorgo", "Renata Sotirov", "Miguel F. Anjos" ], "published_online": "2024-10-09", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02147-3/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [ { "section": "Introduction", "element": "p", "matched_phrases": [ "future_work" ], "passage": "This paper is organized as follows. Notation is given in Sect. 1.1. We provide the definitions of \\(\\textrm{CUT}^n_m\\) and \\(\\mathcal {E}^n_m\\) in Sect. 2. In Sect. 3, we introduce a framework for finding valid inequalities for \\(\\textrm{CUT}^n_m\\) and provide some valid cuts. In Sect. 4, we provide an exact description of \\(\\textrm{CUT}^3_3\\), and use the derived facets of \\(\\textrm{CUT}^3_3\\) to strengthen \\(\\mathcal {E}^n_3\\) for general \\(n \\ge 3\\). In Sect. 5 we provide an efficient reformulation of CSDPs whose convex feasible sets are closed under complex conjugation. In Sect. 6 we investigate the sets \\(\\textrm{CUT}^n_\\infty \\) and second semidefinite liftings of \\(\\textrm{CUT}^n_\\infty \\). In Sect. 7, we study rank 2 extreme points of \\(\\mathcal {E}^n_m\\) for integer \\(m > 2\\). In Sect. 8, we numerically investigate the effect of adding cuts to \\(\\mathcal {E}^n_m\\) for various optimization problems from the literature. Lastly, in Sect. 9 we draw conclusions, and propose future research directions." }, { "section": "Second semidefinite lifting of \\(\\textrm{CUT}^n_\\infty \\)", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "Unfortunately, we are not able to prove or disprove the existence of such rank 3 points. Numerical tests, see also [21], lead us to the following conjecture:" }, { "section": "Second semidefinite lifting of \\(\\textrm{CUT}^n_\\infty \\)", "element": "h3", "matched_phrases": [ "conjecture" ], "passage": "Conjecture 1" }, { "section": "Second semidefinite lifting of \\(\\textrm{CUT}^n_\\infty \\)", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "Conjecture 1 can be connected to the notion of a flat extension from the literature, see e.g., [7]. The term extension is related to the definition of completion as given in Definition 1. In that definition, the larger matrix X is considered an extension of the smaller matrix \\(\\widetilde{X}\\). Matrix X is considered a flat extension of \\(\\widetilde{X}\\) if, along with the properties outlined in Definition 1, we also have \\(\\textrm{rk}(X) = \\textrm{rk}(\\widetilde{X})\\). In [23], there are several results related to such flat extensions, specifically for the bases \\(\\mathbb {N}^p_r := \\left\\{ \\alpha \\in \\mathbb {N}^p \\, | \\, \\sum _{i=1}^p \\alpha _i \\le r \\right\\} \\). Now [23, Theorem 5.2] leads to an equivalent reformulation of Conjecture 1. Namely, Conjecture 1 is true if and only if all \\(\\widetilde{X} \\in \\textbf{L}(\\mathscr {B}_{1}) \\subseteq \\mathcal {E}^n_\\infty \\) admit a flat extension \\(X \\in \\mathcal {F}(\\mathbb {N}^3_{3})\\), for \\(\\mathcal {F}(\\cdot )\\) as in (36)." }, { "section": "Conclusions and future work", "element": "p", "matched_phrases": [ "future_work", "conjecture" ], "passage": "For future work, it would be interesting to have Conjecture 1 resolved. We are also interested in finding faster methods for solving large CSDPs arising from \\(\\textbf{L}^n(\\mathscr {C}_{p})\\). Table 6 shows clearly that larger values of p greatly improve the strength of relaxations, although the required computational effort (both time and memory) to solve them with off-the-shelf interior point method solvers quickly becomes prohibitive. A tailored solver might be able to handle much larger values of n than 25. Many approaches for improving the scalability of real SDP have been proposed in the literature, most of them via exploiting some form of sparsity in the objective function and/or constraints, see e.g., [47, 52,53,54] (note that our numerical experiments involved only dense matrices). The authors of [51] remark that some of those approaches can be also applied to CSDPs with only real coefficients. Future research is to investigate how these methods translate for the case of CSDPs over relaxations of \\(\\textrm{CUT}^n_m\\)." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02142-8", "title": "Coderivative-based semi-Newton method in nonsmooth difference programming", "authors": [ "Francisco J. Aragón-Artacho", "Boris S. Mordukhovich", "Pedro Pérez-Aros" ], "published_online": "2024-10-15", "volume": "213", "issue": "1-2", "pages": "385-432", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02142-8/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02150-8", "title": "ALSO-X#: better convex approximations for distributionally robust chance constrained programs", "authors": [ "Nan Jiang", "Weijun Xie" ], "published_online": "2024-10-15", "volume": "213", "issue": "1-2", "pages": "575-638", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02150-8/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02140-w", "title": "Fast convergence of trust-regions for non-isolated minima via analysis of CG on indefinite matrices", "authors": [ "Quentin Rebjock", "Nicolas Boumal" ], "published_online": "2024-10-18", "volume": "213", "issue": "1-2", "pages": "343-384", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02140-w/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02152-6", "title": "The mixed integer trust region problem", "authors": [ "Alberto Del Pia" ], "published_online": "2024-10-23", "volume": "213", "issue": "1-2", "pages": "699-736", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02152-6/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02143-7", "title": "Adaptive proximal algorithms for convex optimization under local Lipschitz continuity of the gradient", "authors": [ "Puya Latafat", "Andreas Themelis", "Lorenzo Stella", "Panagiotis Patrinos" ], "published_online": "2024-10-28", "volume": "213", "issue": "1-2", "pages": "433-471", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02143-7/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02148-2", "title": "Advances on strictly $$\\Delta $$-modular IPs", "authors": [ "Martin Nägele", "Christian Nöbel", "Richard Santiago", "Rico Zenklusen" ], "published_online": "2024-10-30", "volume": "210", "issue": "1-2", "pages": "731-760", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02148-2/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [ { "section": "Abstract", "element": "p", "matched_phrases": [ "remains_open", "conjecture" ], "passage": "There has been significant work recently on integer programs (IPs) \\(\\min \\{c^\\top x :Ax\\le b,\\,x\\in \\mathbb {Z}^n\\}\\) with a constraint marix A with bounded subdeterminants. This is motivated by a well-known conjecture claiming that, for any constant \\(\\Delta \\in \\mathbb {Z}_{>0}\\), \\(\\Delta \\)-modular IPs are efficiently solvable, which are IPs where the constraint matrix \\(A\\in \\mathbb {Z}^{m\\times n}\\) has full column rank and all \\(n\\times n\\) minors of A are within \\(\\{-\\Delta , \\dots , \\Delta \\}\\). Previous progress on this question, in particular for \\(\\Delta =2\\), relies on algorithms that solve an important special case, namely strictly \\(\\Delta \\)-modular IPs, which further restrict the \\(n\\times n\\) minors of A to be within \\(\\{-\\Delta , 0, \\Delta \\}\\). Even for \\(\\Delta =2\\), such problems include well-known combinatorial optimization problems like the minimum odd/even cut problem. The conjecture remains open even for strictly \\(\\Delta \\)-modular IPs. Prior advances were restricted to prime \\(\\Delta \\), which allows for employing strong number-theoretic results. In this work, we make first progress beyond the prime case by presenting techniques not relying on such strong number-theoretic prime results. In particular, our approach implies that there is a randomized algorithm to check feasibility of strictly \\(\\Delta \\)-modular IPs in strongly polynomial time if \\(\\Delta \\le 4\\)." }, { "section": "Introduction", "element": "p", "matched_phrases": [ "we_do_not_know", "conjecture" ], "passage": "Integer Programs (IPs) \\(\\min \\{c^\\top x:Ax\\le b,\\, x\\in \\mathbb {Z}^n\\}\\) are a central NP-hard problem class in Combinatorial Optimization. There is substantial prior work and interest in identifying special classes of polynomial-time solvable IPs while remaining as general as possible. One of the best-known such classes are IPs with a constraint matrix that is totally unimodular (TU), i.e., the determinant of any of its square submatrices is within \\(\\{-1,0,1\\}\\). A long-standing open conjecture in the field is whether this result can be generalized to \\(\\Delta \\)-modular constraint matrices for constant \\(\\Delta \\). Here, we say that a matrix \\(A\\in \\mathbb {Z}^{k\\times n}\\) is \\(\\Delta \\)-modular if it has full column rank and all \\(n\\times n\\) submatrices have determinants in \\(\\{-\\Delta ,\\ldots ,\\Delta \\}\\).Footnote 1 For brevity, we call an IP with \\(\\Delta \\)-modular constraint matrix a \\(\\Delta \\)-modular IP. We recap the above-mentioned conjecture below. Unfortunately, we do not know its precise origin; it may be considered folklore in the field." }, { "section": "Introduction", "element": "h3", "matched_phrases": [ "conjecture" ], "passage": "Conjecture 1" }, { "section": "Introduction", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "First progress on Conjecture 1 was made by [1], who showed that it holds for \\(\\Delta =2\\) (the bimodular case). [2] show that the conjecture is true for an arbitrary constant \\(\\Delta \\) under the extra condition that the constraint matrix has at most two non-zero entries per row or column. Through a non-trivial extension of the techniques in [1], it was shown by [3] that there is a randomized algorithm to check feasibility of an IP with a strictly 3-modular constraint matrix in polynomial time. Here, a matrix \\(A\\in \\mathbb {Z}^{k\\times n}\\) is called strictly \\(\\Delta \\)-modular if it has full column rank and all its \\(n\\times n\\) submatrices have determinants in \\(\\{-\\Delta , 0, \\Delta \\}\\)." }, { "section": "Introduction", "element": "p", "matched_phrases": [ "open_question", "conjecture" ], "passage": "Combinatorial optimization problems with congruency constraints are highly non-trivial and many open questions remain. As they are already captured by strictly \\(\\Delta \\)-modular IPs, this motivates the following weakening of Conjecture 1." }, { "section": "Introduction", "element": "h3", "matched_phrases": [ "conjecture" ], "passage": "Conjecture 2" }, { "section": "Introduction", "element": "p", "matched_phrases": [ "open_problem", "conjecture" ], "passage": "Even resolving this weaker conjecture would settle several open problems, including congruency-constrained min cuts (in both directed and undirected graphs), or the problem of efficiently and deterministically finding a perfect matching in a red/blue edge-colored bipartite graph such that the number of red matching edges is \\(r \\, (\\text{ mod } m)\\). (This is a simplified version of the famous red-blue matching problem, where the task is to find a perfect matching with a specified number of red edges; for both versions, randomized algorithms are known.) Interestingly, for the bimodular case (\\(\\Delta =2\\)), a result by [10] implies that Conjecture 1 and Conjecture 2 are equivalent (see [1])." }, { "section": "Introduction", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "Our goal is to shed further light on Conjecture 2 and overcome some important hurdles of prior approaches. In a first step, we note that a positive resolution of Conjecture 2 does not only imply efficient solvability of MCCTU problems, but also vice versa, and this backward reduction works in strongly polynomial time." }, { "section": "Introduction", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "Further, we are interested in making progress regarding the feasibility version of Conjecture 2, i.e., efficiently deciding whether a strictly \\(\\Delta \\)-modular IP is feasible. Prior approaches settle this question for \\(\\Delta =2\\) [1] and—using a randomized algorithm— for \\(\\Delta =3\\) [3]. A main hurdle to extend these is that they crucially rely on \\(\\Delta \\) being prime, for example through the use of the Cauchy-Davenport Theorem. Our main contribution here is to address this. In particular, we can check feasibility for \\(\\Delta =4\\) with a randomized algorithm, which is the first result in this context for non-prime \\(\\Delta \\). More importantly, our techniques will hopefully prove useful for future advances on this challenging question." }, { "section": "Introduction", "element": "p", "matched_phrases": [ "remains_open" ], "passage": "Besides being a key part of our approach, Theorem 6 underlines that base block GCTUF problems are not merely special cases, but play a key role in progress on general GCTUF problems. There are only two non-trivial types of such base block GCTUF problems, namely when the constraint matrix is a so-called network matrix or a transpose thereof. Both cases cover combinatorial problems that are interesting on their own, and their complexity status remains open to date. If the constraint matrix is a network matrix, GCTUF can be cast as a circulation problem with a group constraint. By reducing to and exploiting results of [17] on exact perfect matching problems, a randomized algorithm for the congruency-constrained case has been presented in [3]. We observe that these results extend to the group-constrained setting. The other base block case, where the constraint matrix is the transpose of a network matrix, can be cast as a group-constrained directed minimum cut problem by leveraging a result in [3]. Prior work combined this reduction with results on congruency-constrained submodular minimization [8] to solve the optimization version of the problem for congruency-constraints of prime power modulus. We show that the feasibility question on this base block can be solved efficiently on any finite abelian group of constant order, thus circumventing the prime power restriction that is intrinsic in prior approaches." }, { "section": "GCTUF with transposed network constraint matrices", "element": "p", "matched_phrases": [ "remains_open" ], "passage": "where \\(\\mathcal {L}\\subseteq 2^N\\) is a lattice on some finite ground set N, \\(\\gamma :N\\rightarrow \\mathbb {Z}\\), \\(r\\in \\mathbb {Z}\\), \\(m\\in \\mathbb {Z}_{>0}\\), \\(w:N\\rightarrow \\mathbb {R}\\), and we use \\(\\gamma (S):==\\sum _{v\\in S}\\gamma (v)\\) as well as \\(w(S):==\\sum _{v\\in S}w(v)\\).Footnote 5 Being a special case of congruency-constrained submodular minimization, it is known that such problems, and thus the corresponding congruency-constrained TU problems with a transposed network constraint matrix, can be solved in strongly polynomial time for constant prime power moduli m, while the case of general constant composite moduli remains open [8]. The progress on GCTUF , particularly the reduction to base block feasibility problems through Theorem 6 and its generalization (Theorem 16 in Sect. 4), motivates studying these reductions and results in the feasibility setting and with a group constraint instead of a congruency constraint, giving rise to the following problem." }, { "section": "Notes", "element": "li", "matched_phrases": [ "conjecture" ], "passage": "A weaker variant of the conjecture claims efficient solvability of IPs with totally \\(\\Delta \\)-modular constraint matrices, where all subdeterminants are bounded by \\(\\Delta \\) in absolute value. The conjecture involving \\(\\Delta \\)-modular matrices implies the weaker variant. Indeed, an IP \\(\\min \\{c^\\top x :Ax \\le b, x\\in \\mathbb {Z}^n\\}\\) with a totally \\(\\Delta \\)-modular constraint matrix can be reformulated as \\(\\min \\{c^\\top (x^+-x^-) :A(x^+-x^-)\\le b, x^+, x^- \\in \\mathbb {Z}_{\\ge 0}^n\\}\\). It is not hard to see that the constraint matrix of the new LP remains totally \\(\\Delta \\)-modular; moreover, it has full column rank because of the non-negativity constraints." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02158-0", "title": "Algebraic combinatorial optimization on the degree of determinants of noncommutative symbolic matrices", "authors": [ "Hiroshi Hirai", "Yuni Iwamasa", "Taihei Oki", "Tasuku Soma" ], "published_online": "2024-11-02", "volume": "213", "issue": "1-2", "pages": "941-984", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02158-0/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Introduction", "element": "p", "matched_phrases": [ "open_problem" ], "passage": "where \\(A_k\\) are matrices over a field \\(\\mathbb {K}\\), \\(x_k\\) are variables, and the rank of A is considered in the rational function field \\(\\mathbb {K}(x_1,x_2,\\ldots ,x_m)\\). This problem is sometimes called Edmonds’ problem. A deterministic polynomial-time algorithm for Edmonds’ problem is not known, and is one of the major open problems in theoretical computer science. The maximum cardinality bipartite matching problem is just the rank computation of (1.1) for the case where each \\(A_k\\) has exactly one nonzero entry. Other representatives of polynomially-solvable combinatorial optimization problems, such as linear matroid intersection, nonbipartite matching, and linear matroid matching, are also formulated as Edmonds’ problem for special matrices; see e.g., [29, 51]. Although min–max theorems in some of these problems were found/can be deduced via such algebraic formulation (see e.g., [59, Section 16.2b]), it is not as well understood as the polyhedral methods." }, { "section": "Introduction", "element": "p", "matched_phrases": [ "open_problem" ], "passage": "In Sect. 4, we present applications. For the linear matroid matching case, the rank and nc-rank are not equal. However, Oki and Soma [56] showed that the nc-rank is equal to twice the maximum size of fractional matroid matchings. We show its weighted extension: \\(\\varDelta _\\textrm{max}(A[c])\\) is equal to the maximum weight of fractional matroid matchings, with respect to the weight vector c. As a consequence, the nc-independent set polytope \\(\\mathcal{Q}(A)\\) is the twice expansion of the fractional linear matroid matching polytope. Also, our Hungarian Deg-Det algorithm brings a new strongly polynomial-time algorithm, which is a variation of Gijswijt and Pap [30], and can care about the bit complexity. This result can be applied to the Brascamp–Lieb inequality [7, 49]. It is known [4] that the parameter space of nontrivial BL-inequality forms an interesting rational polytope (BL-polytope). This polytope is defined by exponentially many inequalities, and coincides with a matroid base polytope if the defining matrices are all rank-one [3]. The computational complexity on BL-polytopes is an intriguing open problem in theoretical computer science [25]. Franks, Soma, and Goemans [23] found that the BL-polytope defined by rank-2 matrices is equal to the perfect fractional matroid matching polytope associated with these matrices, and showed that the membership problem for the corresponding BL-polytope is in NP \\(\\cap \\) co-NP. Our result implies that it is indeed in P." }, { "section": "Introduction", "element": "p", "matched_phrases": [ "open_problem" ], "passage": "Related work and literature. Edmonds’ problem is at a crossroad of various fields in mathematical science. In theoretical computer science, it is at the core of deterministic vs. randomization issue, particularly, polynomial identity testing, which has been an active research field in theoretical computer science for decades. If the underlying field is sufficiently large, then Edmonds’ problem admits a simple randomized polynomial-time algorithm (Lovász [50]). However, any deterministic polynomial-time algorithm is not known. Such an algorithm would imply a nontrivial circuit lower bound [45], which is another long-standing open problem in computational complexity theory. Edmonds’ problem can also be solved by matrix completion of finding a substitution of \\(x_k\\) that attains the maximum rank. The above randomized polynomial-time algorithm is based on random substitution. Deterministic polynomial-time matrix completions are known for a few classes of matrices; they are related to well-solved combinatorial optimization problems mentioned above; see e.g., [28]. The rank computation of symbolic matrices also plays important roles in structural analysis on engineering systems, where variables \\(x_k\\) are physical parameters. Mixed matrices (Murota [53]) are a class of symbolic matrices, providing a numeric-symbolic integrated analysis on such systems by means of matroid algorithms. Rigidity of bar-joint frameworks in generic positions is also a typical problem of the symbolic rank computation; see [51, 57]. Several results (e.g., Laman’s theorem) as well as long-standing open problems (e.g., 3-dimensional generic rigidity) in rigidity theory are about combinatorial rank characterizations of symbolic matrices arising as rigidity matrices." }, { "section": "Applications", "element": "p", "matched_phrases": [ "future_work" ], "passage": "It is an interesting future research to incorporate the CLV-algorithm into a primal-dual framework in Sect. 3.2.3, which will improve the time complexity greatly. It is expected that an extreme fractional matching will work as a primal solution and its characterization in [11] will help." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02154-4", "title": "Accelerated-gradient-based generalized Levenberg–Marquardt method with oracle complexity bound and local quadratic convergence", "authors": [ "Naoki Marumo", "Takayuki Okuno", "Akiko Takeda" ], "published_online": "2024-11-04", "volume": "213", "issue": "1-2", "pages": "771-822", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02154-4/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Local convergence under Hölderian growth condition", "element": "p", "matched_phrases": [ "future_work" ], "passage": "Because \\(\\left\\| {c(x_k) - c(x^*)}\\right\\| ^2 \\le \\left\\| {c(x_k)}\\right\\| ^2 - \\Vert c(x^*)\\Vert ^2\\) in some cases and not in others, we cannot conclude which condition is stronger. However, using Assumption 3, which extends QG, facilitates comparison with problem settings other than least squares. We leave the investigation of whether our algorithm exhibits favorable local convergence under (5.4), in place of Assumption 3, for future work." }, { "section": "Numerical experiments", "element": "p", "matched_phrases": [ "we_do_not_know" ], "passage": "Sensitivity to \\(g^* + h^*\\). The proposed algorithm has other input parameters, \\(g^*\\) and \\(h^*\\), that should be set as in (3.5). So far, we have set \\(g^* = h^* = 0\\) since we know that \\(\\inf _{x \\in {\\mathbb {R}}^d} g(x) = \\inf _{y \\in {\\mathbb {R}}^n} h(y) = 0\\) in all four experimental settings. Here, let us hypothetically consider the case where we do not know the infimum of g or h. A practical choice in such cases is to compute numerically the minimum of the convex functions g and h. Numerical solutions often give an upper bound, but subtracting a small value from the upper bound is expected to result in a lower bound. Assuming such a case, we tested the proposed algorithm with different values of \\(g^* + h^*\\), and the results are presented in Fig. 4.Footnote 12 The results suggest that the algorithm’s performance is not significantly affected as long as \\(g^* + h^*\\) is close to 0 (i.e., \\(\\inf _{x \\in {\\mathbb {R}}^d} g(x) + \\inf _{y \\in {\\mathbb {R}}^n} h(y)\\)). However, if their differences become too large, the performance deteriorates. Note that we should consider the scale of the function value to compare the magnitude of the value of \\(g^* + h^*\\). For example, in the NMF setting, the function h has the form" }, { "section": "Conclusion", "element": "p", "matched_phrases": [ "future_work" ], "passage": "Such a finite-sum setting often appears in machine learning, and stochastic methods [29] effectively solve the problem when N is large. An interesting direction for future work would be to investigate whether oracle complexity bound and local convergence can still be achieved with stochastic LM methods." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02145-5", "title": "A nearly optimal randomized algorithm for explorable heap selection", "authors": [ "Sander Borst", "Daniel Dadush", "Sophie Huiberts", "Danish Kashaev" ], "published_online": "2024-11-05", "volume": "210", "issue": "1-2", "pages": "75-96", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02145-5/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [ { "section": "Introduction", "element": "p", "matched_phrases": [ "open_problem" ], "passage": "Improving on the best-first strategy, Karp, Saks and Wigderson [2] gave a randomized algorithm with expected cost \\(n\\cdot \\exp (O(\\sqrt{\\log (n)}))\\) using \\(O(\\sqrt{\\log (n)})\\) working space. They also showed how to make the algorithm deterministic using \\(O(\\log (n)^{2.5})\\) space. In this work, our main contribution is an improved randomized algorithm with expected cost \\(O(n\\log (n)^3)\\) using \\(O(\\log (n))\\) space. Given the \\(\\Omega (n)\\) lower bound, our travel cost is optimal up to logarithmic factors. Furthermore we show that any algorithm for explorable heap selection that uses only s units of memory, must take at least \\(n\\cdot \\log _s (n)\\) time in expectation. An interesting open problem is the question whether a superlinear lower bound also holds without any restriction on the memory usage." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02149-1", "title": "Optimization of trigonometric polynomials with crystallographic symmetry and spectral bounds for set avoiding graphs", "authors": [ "Evelyne Hubert", "Tobias Metzlaff", "Philippe Moustrou", "Cordian Riener" ], "published_online": "2024-11-05", "volume": "213", "issue": "1-2", "pages": "517-573", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02149-1/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Introduction", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "In this article, \\(\\Omega \\) is the weight lattice of a crystallographic root system in \\(\\mathbb {R}^n\\). Root and weight lattices provide optimal configurations for a variety of problems in geometry and information theory, with incidence in physics and chemistry. The \\(\\textrm{A}_{2}\\) lattice (the hexagonal lattice) is classically known to be optimal for sampling, packing, covering, and quantization in the plane [17, 37], but also proved, or conjectured, to be optimal for energy minimization problems [6, 55]. More recently, the \\(\\textrm{E}_{8}\\) lattice was proven to give an optimal solution for the sphere packing problem and a large class of energy minimization problems in dimension 8 [15, 58]. From an approximation point of view, weight lattices of root systems describe Gaussian cubatures [46, 50], a rare occurence on multidimensional domains. In a different direction, the triangulations associated with infinite families of root systems are relevant in graphics and computational geometry, see for instance [16] and references therein." }, { "section": "Spectral bounds for set avoiding graphs", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "Finally we consider the case of the Euclidean space \\(V=\\mathbb {R}^n\\) as a set of vertices, where the avoided set \\(S=\\partial \\mathcal {P}\\) is the boundary of a convex centrally-symmetric polytope \\(\\mathcal {P}\\). This setting was studied in [3], giving bounds on \\(\\chi _m(\\mathbb {R}^n,\\partial \\mathcal {P})\\) without using spectral bounds. There it was proven that \\(\\chi _m(\\mathbb {R}^n,\\partial \\mathcal {P}) \\le 2^n\\) whenever \\(\\mathcal {P}\\) tiles \\(\\mathbb {R}^n\\) and equality is conjectured. We now investigate the strength of the spectral bound for certain instances of this graph (Fig. 12)." }, { "section": "Spectral bounds for set avoiding graphs", "element": "p", "matched_phrases": [ "we_do_not_know", "conjecture" ], "passage": "Since we do not know a priori how large is the gap between the spectral bound and the actual chromatic number, it is interesting to understand better the behavior of the spectral bound for such graphs in itself. In this direction, in addition to provide bounds on the chromatic number of the graphs that we consider, our method gives information on the discrete measures supported on lattice points up to scaling. For example, in the case of the hexagon, even by increasing the number of support points, we did not get a discrete measure providing a better bound, see Table 5. Our experiments then suggest that the optimal measure supported on rational points is the one supported by two orbits: the vertices of the hexagon, with weight 1/3, and the middle of the edges, with weight 2/3. In the case of the cross-polytope from Sect. 4.3, we observe a different phenomenon: when increasing the number of possible support points, the optimal measure distribution does not appear to stabilize. It seems then reasonable to expect the bound to get better when increasing the number of points, even though it is hard to conjecture for an optimal discrete measure after our experiments, see Fig. 11. Moreover, we note that the larger the set of possible support points is, the higher we need to go in the order of the hierarchy to get a good bound. This can be explained by the fact that the weighted degrees of the involved Chebyshev polynomials get higher, making the semi-definite programs harder to solve." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02155-3", "title": "From coordinate subspaces over finite fields to ideal multipartite uniform clutters", "authors": [ "Ahmad Abdi", "Dabeen Lee" ], "published_online": "2024-11-05", "volume": "213", "issue": "1-2", "pages": "823-861", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02155-3/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Abstract", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "Take a prime power q, an integer \\(n\\ge 2\\), and a coordinate subspace \\(S\\subseteq GF(q)^n\\) over the Galois field GF(q). One can associate with S an n-partite n-uniform clutter \\(\\mathcal {C}\\), where every part has size q and there is a bijection between the vectors in S and the members of \\(\\mathcal {C}\\). In this paper, we determine when the clutter \\(\\mathcal {C}\\) is ideal, a property developed in connection to Packing and Covering problems in the areas of Integer Programming and Combinatorial Optimization. Interestingly, the characterization differs depending on whether q is 2, 4, a higher power of 2, or otherwise. Each characterization uses crucially that idealness is a minor-closed property: first the list of excluded minors is identified, and only then is the global structure determined. A key insight is that idealness of \\(\\mathcal {C}\\) depends solely on the underlying matroid of S. Our theorems also extend from idealness to the stronger max-flow min-cut property. As a consequence, we prove the Replication and \\(\\tau =2\\) Conjectures for this class of clutters." }, { "section": "Introduction", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "Every clutter whose members are pairwise disjoint is obviously ideal. Many non-trivial examples of ideal clutters can be found in Combinatorial Optimization – let us mention a few here: the clutter of st-paths of a graph [27], (inclusionwise) minimal st-cuts of a graph [14], minimal T-joins of a graph [17], minimal T-cuts of a graph [17], and odd circuits of a signed graph that has no odd-\\(K_5\\) minor [18]. Each of these examples has as ground set the edge set of the associated graph. In general, it is co-NP-complete to decide whether a clutter is ideal [13], and understanding the various aspects of the theory of ideal clutters is one of the long-standing open research directions in the area: 11 of the 18 conjectures in the book Combinatorial Optimization. Packing and Covering [10] are directly about general or special instances of ideal clutters." }, { "section": "Introduction", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "As a corollary, idealness and the MFMC property coincide when q is an odd prime power or q is a power of 2 greater than 4. In contrast, there is an example of a vector space over GF(4) whose multipartite uniform clutter is ideal but does not have the MFMC property. We demonstrate this example in Sect. 8. Theorem 1.5 also has a consequence on the Replication Conjecture, proposed by Conforti and Cornuéjols [9]. In particular, the Replication Conjecture is a set covering analogue of the Duplication Lemma for perfect graphs [25]." }, { "section": "Introduction", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "(proved in Sect. 8). The Replication Conjecture holds true for the class of multipartite uniform clutters from coordinate subspaces." }, { "section": "Introduction", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "Another corollary of Theorem 1.5 is on the \\(\\tau =2\\) Conjecture, proposed by Cornuéjols, Guenin, and Margot [11]. They showed that if the \\(\\tau =2\\) Conjecture holds, then so does the Replication Conjecture [11], providing a way of tackling the Replication Conjecture." }, { "section": "Introduction", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "(proved in Sect. 8). The \\(\\tau =2\\) Conjecture holds true for the class of multipartite uniform clutters from coordinate subspaces." }, { "section": "Introduction", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "We will formally state the Replication Conjecture and the \\(\\tau =2\\) Conjecture along with the proofs of Corollaries 1.6 and 1.7 in Sect. 8." }, { "section": "Introduction", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "We conclude the paper by proving Corollaries 1.6 and 1.7 on the Replication Conjecture and the \\(\\tau =2\\) Conjecture, respectively, for the class of multipartite clutters from coordinate subspaces in Sect. 8. Section 2 provides some basics of multipartite uniform clutters and matroid theory. More advanced concepts in matroid theory and the theory of ideal clutters are defined and explained whenever necessary." }, { "section": "The replication and \\(\\tau =2\\) conjectures", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "We say that clutter \\(\\mathcal {C}\\) packs if \\(\\tau (\\mathcal {C},\\textbf{1})=\\nu (\\mathcal {C},\\textbf{1})\\). We say that \\(\\mathcal {C}\\) has the packing property if every minor of \\(\\mathcal {C}\\) packs. It was observed in [11] that minimally non-ideal clutters do not pack due to Lehman’s theorem [22] and that if a clutter has the packing property, then it is ideal. Moreover, notice that the packing property is a relaxed notion of the max-flow min-cut property. Here, the Replication Conjecture predicts that the packing property implies the max-flow min-cut property. We answer the conjecture in the affirmative for the class of multipartite uniform clutters from coordinate subspaces." }, { "section": "The replication and \\(\\tau =2\\) conjectures", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "Next we consider the \\(\\tau =2\\) Conjecture [11] which predicts that a stronger statement than the Replication Conjecture holds true. We call a clutter minimally non-packing if it does not have the packing property but every proper minor of it does. It is known that a minimally non-packing clutter is either ideal or minimally non-ideal [11]. Here, the \\(\\tau =2\\) Conjecture is that if a clutter \\(\\mathcal {C}\\) is ideal and minimally non-packing, then its covering number, defined as \\(\\tau (\\mathcal {C},\\textbf{1})\\), is two. We show that if the multipartite uniform clutter of a coordinate subspace is ideal and minimally non-packing, then its covering number is two." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02157-1", "title": "A projection-free method for solving convex bilevel optimization problems", "authors": [ "Khanh-Hung Giang-Tran", "Nam Ho-Nguyen", "Dabeen Lee" ], "published_online": "2024-11-05", "volume": "213", "issue": "1-2", "pages": "907-940", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02157-1/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Iteratively regularized conditional gradient method", "element": "p", "matched_phrases": [ "we_do_not_know" ], "passage": "While strong duality holds, the dual is not always solvable in general. Hence, we do not know whether there exists a finite \\(\\lambda _{{{\\,\\textrm{opt}\\,}}} \\ge 0\\) such that \\(\\min _{x \\in X} L(x,\\lambda _{{{\\,\\textrm{opt}\\,}}}) = f_{{{\\,\\textrm{opt}\\,}}}\\) and \\(X_{{{\\,\\textrm{opt}\\,}}} \\subseteq \\mathop {\\mathrm {arg\\,min}}\\limits _{x \\in X} L(x,\\lambda _{{{\\,\\textrm{opt}\\,}}})\\) or not. However, since \\(L(x,\\lambda )\\) is non-decreasing in \\(\\lambda \\) for each \\(x \\in X\\), regardless of the existence of \\(\\lambda _{{{\\,\\textrm{opt}\\,}}}\\), we must have that" }, { "section": "Conclusion", "element": "p", "matched_phrases": [ "future_work" ], "passage": "One possible future research direction is to explore projection-free bilevel methods that mitigate the well-known zig-zag behavior of objective values under conditional gradient-type methods, which often results in slow local convergence. Another direction is to consider stochastic algorithms, which may be useful in large-scale problems in which exact gradient computation may be expensive." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02153-5", "title": "Neural spectrahedra and semidefinite lifts: global convex optimization of degree-two polynomial activation neural networks in polynomial-time", "authors": [ "Burak Bartan", "Mert Pilanci" ], "published_online": "2024-11-08", "volume": "213", "issue": "1-2", "pages": "737-769", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02153-5/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02156-2", "title": "Riemannian trust-region methods for strict saddle functions with complexity guarantees", "authors": [ "Florentin Goyens", "Clément W. Royer" ], "published_online": "2024-11-08", "volume": "213", "issue": "1-2", "pages": "863-905", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02156-2/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02159-z", "title": "Exact and approximation algorithms for routing a convoy through a graph", "authors": [ "Martijn van Ee", "Tim Oosterwijk", "René Sitters", "Andreas Wiese" ], "published_online": "2024-11-09", "volume": "213", "issue": "1-2", "pages": "985-1008", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02159-z/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02162-4", "title": "Extended formulations for binary optimal control problems", "authors": [ "Christoph Buchheim" ], "published_online": "2024-11-12", "volume": "213", "issue": "1-2", "pages": "1039-1062", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02162-4/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02160-6", "title": "Improved global guarantees for the nonconvex Burer–Monteiro factorization via rank overparameterization", "authors": [ "Richard Y. Zhang" ], "published_online": "2024-11-20", "volume": "213", "issue": "1-2", "pages": "1009-1038", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02160-6/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02161-5", "title": "Sparsity penalized mean–variance portfolio selection: analysis and computation", "authors": [ "Buse Şen", "Deniz Akkaya", "Mustafa Ç. Pınar" ], "published_online": "2024-11-25", "volume": "211", "issue": "1-2", "pages": "281-318", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02161-5/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02164-2", "title": "Acceleration by stepsize hedging: Silver Stepsize Schedule for smooth convex optimization", "authors": [ "Jason M. Altschuler", "Pablo A. Parrilo" ], "published_online": "2024-11-25", "volume": "213", "issue": "1-2", "pages": "1105-1118", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02164-2/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Introduction", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "We make several remarks. (1) This rate \\(n^{-\\log _2 \\rho } \\approx n^{-1.2716}\\) is intermediate between the textbook unaccelerated rate \\(n^{-1}\\) and the accelerated rate \\(n^{-2}\\) due to Nesterov in 1983 [9]; see Fig. 2 for a visualization. (2) We conjecture that, among all possible stepsize schedules, the exponent \\(\\log _{2} \\rho \\) in the rate is exactly optimal, and the multiplicative constant \\(\\tfrac{1}{2}\\) in the rightmost-expression of (4) is within a modest constant factor of optimality. (3) For the Silver Stepsize Schedule, Theorem 1 shows that \\(r_k\\) is an upper bound on its convergence rate; this is essentially tight since there exist problem instancesFootnote 1 for which the Silver Stepsize Schedule has rate \\(\\tfrac{1}{4\\rho ^k - 2} \\approx \\tfrac{1}{4 n^{1.2716}}\\), which is within a factor of two of \\(r_k \\approx \\tfrac{1}{2n^{1.2716}}\\). (4) The Silver Stepsize Schedule is independent of the horizon n, see Sect. 2. (5) The result holds for the final iterate \\(x_n\\), which is more desirable than the averaged iterate—the standard alternative in optimization results. (6) Due to standard black-box reductionsFootnote 2, this accelerated rate for smooth convex optimization immediately implies an analogously accelerated rate for smooth strongly-convex optimization by repeating, for large enough n, the n-step Silver Stepsize Schedule (3) for the present (non-strongly) convex setting. This asymptotically recovers the convergence rate of [8, Theorem 1.1], albeit with slightly worse constants non-asymptotically." }, { "section": "Introduction", "element": "p", "matched_phrases": [ "future_work" ], "passage": "More broadly, this alternative mechanism for accelerated optimization opens up many directions for future work, such as robustness and generality to other settings and other algorithms; see the discussion in [8, Sect. 6]. We believe that the paradigm of stepsize hedging will be useful for the design and analysis of optimization algorithms in many contexts, and that the present setting of gradient descent is just scratching the surface of this algorithmic opportunity." }, { "section": "Introduction", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "In July 2023, Grimmer [12] showed how to prove asymptotic rates for the (non-strongly) convex setting by periodically cycling through finite schedules. While in the strongly convex setting composing progress from different cycles just amounts to multiplying contraction rates, in the non-strongly convex setting this can be more subtle depending on the approach.Footnote 3 To deal with this, he introduced the notion of straightforward stepsize patterns and obtained a constant-factor improvement over the textbook unaccelerated rate by cycling through approximate schedules of length \\(n=127\\). From these numerics, he conjectured that further optimizing stepsizes might lead to an asymptotic rate of \\(O((n \\log n)^{-1})\\), a milder improvement than that conjectured by [11]." }, { "section": "Introduction", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "In September 2023, two concurrent lines of work appeared. One line of work was the two companion papers of the present authors—Altschuler and Parrilo [8] and the present paper. The other was the paper Grimmer et al. [13]. Altschuler and Parrilo [8] proposed the Silver Stepsize Schedule of arbitrary size to prove asymptotic acceleration for the strongly convex setting. This improved the dependence on the condition number \\(\\kappa \\) in the textbook unaccelerated rate \\(O(\\kappa )\\) to \\(O(\\kappa ^{\\log _\\rho 2}) \\approx O(\\kappa ^{0.7864})\\). (All these bounds scale in the precision as \\(\\log 1/\\varepsilon \\).) We conjectured and provided partial evidence that these rates are optimal among all possible stepsize schedules. This result was achieved by introducing the technique of recursive gluing to establish multi-step descent, which we make use of here. The other paper, Grimmer et al. [13], utilized a certain non-periodic sequence of increasingly large stepsizes and built upon the “straightforwardness” machinery in [12], to prove the rate \\(O(n^{-1.0245})\\) for the convex setting in v1, later improved to \\(O(n^{-1.0564})\\) in v2." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02165-1", "title": "Quantifying low rank approximations of third order symmetric tensors", "authors": [ "Shenglong Hu", "Defeng Sun", "Kim-Chuan Toh" ], "published_online": "2024-11-28", "volume": "213", "issue": "1-2", "pages": "1119-1168", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02165-1/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Conclusions", "element": "p", "matched_phrases": [ "future_work" ], "passage": "The emphasis of this paper is on the global optimality and quantified quasi-optimality certification of the best low rank approximation. Numerical illustration is presented for the validation of the theory as well. However, more carefully and wisely designed numerical methods should be investigated in our future research for solving the hard optimization problems involved in the theory. In particular, the method employed in this paper for the problem (35) is a first order method, which typically has a slow convergence and it is difficult to get a high accuracy solution for large scale problems. In order to facilitate the global optimality certification, high accuracy solution for the dual problem (12) is necessary (cf. Table 4). While problem (12) has a complicated nonsmooth convex objective function, it is a challenging problem to be solved, especially at degenerate solutions. On the other hand, note that we used the approximate solution (the multipler for (35)) for the dual problem of (35) to generate a candidate for the solution of (11). The polynomial optimization ingredients in our reformulation (11) require a high accuracy solution; otherwise, the flatness condition is impossible to be satisfied. Hence, methods and theory for solving (35) and its dual problem targeted with high accuracy and fast convergent properties should be developed. In particular, properties of the conjugate function and the proximal mapping of the squared low rank projection function involved in (36) should be investigated." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02168-y", "title": "The polyhedral geometry of truthful auctions", "authors": [ "Michael Joswig", "Max Klimm", "Sylvain Spitz" ], "published_online": "2024-11-28", "volume": "210", "issue": "1-2", "pages": "539-566", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02168-y/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [ { "section": "Introduction", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "Characterizing the combinatorial types of mechanisms is interesting for several reasons. First, such geometric arguments are often used to characterize the set of allocation functions that are implementable by a DSIC mechanism. For instance, the difference sets whose closures have nonempty intersection correspond exactly to two-cycles in the allocation network used by Rochet [48] to characterize the allocation functions that are implementable by DSIC mechanisms. More recently, Edelman and Weymark [23] showed that under certain conditions it is necessary that the length of all 2-cycles is zero. As we show, the 2-cycles in the allocation network correspond exactly to those allocations A and \\(A'\\) such that the intersections of \\(Q_{A}\\) and \\(Q_{A'}\\) are nonempty. Examining the mechanisms illustrated in Fig. 1 shows that different combinatorial types give rise to different nonempty intersections and, hence, different necessary conditions on implementability. Second, the combinatorial types can be used to study the sensitivity of mechanisms to deviations in the reported types. As an example consider the mechanism in Fig. 1a. For any \\(\\epsilon > 0\\), reporting the type \\((1-\\epsilon , 1-\\epsilon )\\) yields no item for player 1 while the report of the type \\((1+\\epsilon , 1+\\epsilon )\\) grants them both items. Put differently, a small change in the reported type may change the outcome from no items being allocated to player 1 to all items being allocated to the same player. This is in contrast with the mechanism shown in Fig. 1c where a small change in the reported type may change the cardinality of the set of allocated items only by 1.Footnote 1 Third, the combinatorial types of the mechanism are relevant for the efficiency of the mechanism; see, e.g., [16, 54]. For example, one of the most influential models in the algorithmic game theory literature is the unrelated machine scheduling problem introduced by Nisan and Ronen [42]. Here, players correspond to machines. There are m jobs and each machine is requires \\(t_{i,j}\\) time units to process job j. The values \\(t_{i,j}\\) for \\(j \\in [m]\\) are private information of machine i. If a subset \\(S \\subseteq [m]\\) of jobs gets assigned to machine i it obtains a (negative) utility of \\(-\\sum _{j \\in S} t_{i,j}\\). Thus, the setting is equivalent to a combinatorial auction with \\(\\theta _{i,j} = -t_{i,j}\\) for all \\(i \\in [n]\\) and \\(j \\in [m]\\). Nisan and Ronen [42] conjectured that no DSIC mechanism approximates the optimal makespan (i.e., the minimum of the machines’ utilities) by a factor always strictly better than n. Very recently, Christodoulou et al. [14] proved this conjecture using the geometric characterization of two-job mechanisms shown in Fig. 1." }, { "section": "Introduction", "element": "p", "matched_phrases": [ "open_question" ], "passage": "In Sect. 3 we give a complete characterization of the combinatorial types of all DSIC combinatorial auctions with m items for any value of m. This answers an open question of Vidali [54] about how to generalize her results for \\(m =2\\) and \\(m=3\\) to larger values of m. We employ methods from polyhedral geometry [20] and tropical combinatorics [34, 46]. Our results rest on the observation that a multi-player mechanism is DSIC if and only if its single-player components are DSIC; see [49]. We show that for m items, the combinatorial types of those single-player components are in bijection to equivalence classes of regular subdivisions of the m-dimensional unit cube. We identify the relevant symmetries for exchangeable items and conclude that there are exactly 23 nondegenerate combinatorial types for \\(m=3\\) and 3,706,261 such types for \\(m=4\\) (Corollary 10). In Sect. 4 we then use this characterization to study the optimal sensitivity of mechanisms to slight changes in the reported types. Specifically, we show that for any number of items m, there is a one-player combinatorial auction so that the cardinality of the set of items received by the player changes by at most 1 when the reported type is slightly perturbed (Proposition 13). We also give bounds on a similar measure involving the Hamming distance of the set of received items (Proposition 14). In Sect. 5 we further show how to apply the same methodology in order to classify the combinatorial types of affine maximizers, a special class of mechanisms. More precisely, we show that, for m items and n players, where \\(n\\ge 2\\), the combinatorial types of these affine maximizers are in bijection to the regular subdivisions of the m-fold product of the \\((n{-}1)\\)-dimensional simplex. Here the one-player case studied in Sect. 3 arises as the two-player case, but now all goods are distributed between the two players. In this way, the second player is automatically assigned those goods which are not taken by the first player. Geometrically, this amounts to a homogenization, and so the m-dimensional cube re-appears as the m-fold product of the one-dimensional simplex. It turns out that there are exactly 21 combinatorial types of nondegenerate affine maximizers for the case of \\(m=2\\) items and \\(n=3\\) players; the corresponding count for \\((m,n)=(2,4)\\) reads 96,722 (Theorem 20). Products of simplices are prominent in polytope theory ([20, Chapter 6]) and tropical combinatorics ([34, §4.7])." }, { "section": "Sensitivity of mechanisms", "element": "p", "matched_phrases": [ "we_do_not_know" ], "passage": "There is a large gap between the lower and upper bound on \\(M_h(m)\\), which we do not know how to close. At least for 4 items, we can use the computations we made earlier to show that in this case, the upper bound is tight." }, { "section": "Conclusion", "element": "p", "matched_phrases": [ "open_problem", "we_leave", "future_work" ], "passage": "Our focus lied on single-unit allocation mechanisms. Though, our methods can also be applied to multi-unit auctions. The implementability of indifference complexes are then tied to the existence of corresponding regular subdivisions of the appropriate subset of the lattice \\({\\mathbb N}^m\\). For example, an implementable indifference complex for a single-buyer multi-unit auction where there are two units of the first item and three units of the second item corresponds to a regular subdivision of the integer points in the rectangle \\([0,2]\\times [0,3]\\); see Fig. 6. We leave the details of this generalization as an interesting open problem for future research." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02170-4", "title": "Mean robust optimization", "authors": [ "Irina Wang", "Cole Becker", "Bart Van Parys", "Bartolomeo Stellato" ], "published_online": "2024-11-28", "volume": "213", "issue": "1-2", "pages": "1235-1277", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02170-4/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02169-x", "title": "Stabilization of capacitated matching games", "authors": [ "Matthew Gerstbrein", "Laura Sanità", "Lucy Verberk" ], "published_online": "2024-11-29", "volume": "210", "issue": "1-2", "pages": "313-334", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02169-x/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02166-0", "title": "On rank-monotone graph operations and minimal obstruction graphs for the Lovász–Schrijver SDP hierarchy", "authors": [ "Yu Hin Au", "Levent Tunçel" ], "published_online": "2024-12-10", "volume": "213", "issue": "1-2", "pages": "1169-1209", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02166-0/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02151-7", "title": "Nonlinear distributionally robust optimization", "authors": [ "Mohammed Rayyan Sheriff", "Peyman Mohajerin Esfahani" ], "published_online": "2024-12-12", "volume": "213", "issue": "1-2", "pages": "639-698", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02151-7/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02172-2", "title": "Optimization over convex polyhedra via Hadamard parametrizations", "authors": [ "Tianyun Tang", "Kim-Chuan Toh" ], "published_online": "2024-12-12", "volume": "214", "issue": "1-2", "pages": "91-131", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02172-2/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02167-z", "title": "Regular packing of rooted hyperforests with root constraints in hypergraphs", "authors": [ "Pierre Hoppenot", "Mathis Martin", "Zoltán Szigeti" ], "published_online": "2024-12-16", "volume": "213", "issue": "1-2", "pages": "1211-1233", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02167-z/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Inline Recommendations", "element": "h3", "matched_phrases": [ "conjecture" ], "passage": "The Strong Nine Dragon Tree Conjecture is True for \\(d \\le k + 1\\)" } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02174-0", "title": "Neighborhood persistency of the linear optimization relaxation of integer linear optimization", "authors": [ "Kei Kimura", "Kotaro Nakayama" ], "published_online": "2024-12-18", "volume": "214", "issue": "1-2", "pages": "133-151", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02174-0/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02173-1", "title": "Explicit convex hull description of bivariate quadratic sets with indicator variables", "authors": [ "Antonio De Rosa", "Aida Khajavirad" ], "published_online": "2024-12-19", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02173-1/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [ { "section": "Introduction", "element": "p", "matched_phrases": [ "open_question" ], "passage": "In [1], the authors consider the set \\({{\\mathcal {S}}}_2\\) with the additional constraints \\(x_1, x_2 \\le 1\\), and obtain an extended formulation, containing three auxiliary variables, for the convex hull of this set. Their representation consists of second-order cone constraints and PSD conditions, and hence can be embedded in conic solvers. To this date, obtaining an explicit characterization for the convex hull of \\({{\\mathcal {S}}}_2\\) in the original space (x, X, z) remains an open question. The main obstruction in obtaining such an explicit description is that there exists no general methodology to perform projection when nonlinear expressions are involved. This is in contrast with the polyhedral setting in which standard techniques such as Fourier-Motzkin elimination can be applied to obtain the projection in the original space. We should remark that in the case of semi-algebraic sets, cylindrical algebraic decomposition provides a scheme to perform projection via the use of quantifier elimination [11]. One of the goals of this paper is to build necessary tools to perform projections of convex sets defined by nonlinear inequalities. We then utilize these tools to obtain an explicit characterization for the convex hull of \\({{\\mathcal {S}}}_2\\) in the original space." }, { "section": "Preliminary numerical experiments", "element": "p", "matched_phrases": [ "future_work" ], "passage": "In this section, we conduct a preliminary computational study to demonstrate the potential benefits of our proposed relaxations for quadratic optimization problems with indicator variables. A comprehensive computational study that includes various problem types and real data sets from the literature is a topic of future research." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02175-z", "title": "Optimization meets machine learning: an exact algorithm for semi-supervised support vector machines", "authors": [ "Veronica Piccialli", "Jan Schwiddessen", "Antonio M. Sudoso" ], "published_online": "2024-12-19", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02175-z/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02171-3", "title": "Homogenization of SGD in high-dimensions: exact dynamics and generalization properties", "authors": [ "Courtney Paquette", "Elliot Paquette", "Ben Adlam", "Jeffrey Pennington" ], "published_online": "2024-12-24", "volume": "214", "issue": "1-2", "pages": "1-90", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02171-3/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Notes", "element": "li", "matched_phrases": [ "conjecture" ], "passage": "In particular, the authors of [57, Theorem 12] showed that \\(\\beta \\le n^{1/5-\\delta }\\) for any \\( 1/5> \\delta > 0\\) has the same dynamical behavior of SGD when the data \\({\\varvec{A}}\\) satisfied a left-orthogonal invariance condition. It is conjectured that this holds up to \\(\\beta = o(n)\\)." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02163-3", "title": "A first-order augmented Lagrangian method for constrained minimax optimization", "authors": [ "Zhaosong Lu", "Sanyou Mei" ], "published_online": "2024-12-27", "volume": "213", "issue": "1-2", "pages": "1063-1104", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02163-3/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02184-y", "title": "Online bipartite matching in the probe-commit model", "authors": [ "Allan Borodin", "Calum MacRury" ], "published_online": "2025-01-03", "volume": "214", "issue": "1-2", "pages": "303-356", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02184-y/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02179-9", "title": "Polyhedral aspects of feedback vertex set and pseudoforest deletion set", "authors": [ "Karthekeyan Chandrasekaran", "Chandra Chekuri", "Samuel Fiorini", "Shubhang Kulkarni", "Stefan Weltge" ], "published_online": "2025-01-04", "volume": "214", "issue": "1-2", "pages": "153-200", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02179-9/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Abstract", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "We consider the feedback vertex set problem in undirected graphs (FVS). The input to FVS is an undirected graph \\(G=(V,E)\\) with non-negative vertex costs. The goal is to find a minimum cost subset of vertices \\(S \\subseteq V\\) such that \\(G-S\\) is acyclic. FVS is a well-known NP-hard problem and does not admit a \\((2-\\epsilon )\\)-approximation for any fixed \\(\\epsilon > 0\\) assuming the Unique Games Conjecture. There are combinatorial 2-approximation algorithms (Bafna et al., in: Algorithms and computations, pp 142–151, 1995; Becker and Geiger in Artif Intell 83:167–188, 1996) and also primal-dual based 2-approximations (Chudak et al. in Oper Res Lett 22(4):111–118, 1998; Fujito in J Algorithms 31(1):211–227, 1999). Despite the existence of these algorithms for several decades, there is no known polynomial-time solvable LP relaxation for FVS with a provable integrality gap of at most 2. More recent work (Chekuri and Madan, in: Proceedings of the twenty-seventh annual ACM-SIAM symposium on discrete algorithms, pp 808–820, SODA, 2016) developed a polynomial-sized LP relaxation for a more general problem, namely Subset FVS, and showed that its integrality gap is at most 13 for Subset FVS, and hence also for FVS. Motivated by this gap in our knowledge, we undertake a polyhedral study of FVS and related problems. In this work, we formulate new integer linear programs (ILPs) for FVS whose LP-relaxation can be solved in polynomial time, and whose integrality gap is at most 2. The new insights in this process also enable us to prove that the formulation in Chekuri and Madan (in: Proceedings of the twenty-seventh annual ACM-SIAM symposium on discrete algorithms, pp 808–820, SODA, 2016) has an integrality gap of at most 2 for FVS. Our results for FVS are inspired by new formulations and polyhedral results for the closely-related pseudoforest deletion set problem (PFDS). Our formulations for PFDS are in turn inspired by a connection to the densest subgraph problem. We also conjecture an extreme point property for a LP-relaxation for FVS, and give evidence for the conjecture via a corresponding result for PFDS." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02187-9", "title": "Subdifferentials at infinity and applications in optimization", "authors": [ "Do Sang Kim", "Minh Tùng Nguyen", "Tien-Son Pham" ], "published_online": "2025-01-04", "volume": "214", "issue": "1-2", "pages": "409-440", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02187-9/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02189-7", "title": "Maximum entropy on the mean and the Cramér rate function in statistical estimation and inverse problems: properties, models, and algorithms", "authors": [ "Yakov Vaisbourd", "Rustum Choksi", "Ariel Goodwin", "Tim Hoheisel", "Carola-Bibiane Schönlieb" ], "published_online": "2025-01-04", "volume": "214", "issue": "1-2", "pages": "441-490", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02189-7/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02176-y", "title": "Cubic-quartic regularization models for solving polynomial subproblems in third-order tensor methods", "authors": [ "Coralia Cartis", "Wenqi Zhu" ], "published_online": "2025-01-06", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02176-y/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [ { "section": "Introduction and problem set-up", "element": "p", "matched_phrases": [ "open_question" ], "passage": "Finding ways to efficiently minimize \\(m_3\\) remains an open question and is the main focus of this paper. While there are previous algorithms related to minimizing the AR3 subproblems, they often lack explicit handling of the tensor term or the fourth-order regularization. Schnabel et al. [17, 38] considered solving unconstrained optimization using third-order tensor local models without the regularization term. Birgin et al. [5] introduced a variant of the AR3 algorithm, comparing its performance with AR2/ARC. In the case of a convex \\(m_3\\), Nesterov proposed a series of second-order methods for its minimization [30,31,32,33,34,35]. These methods use various convex optimization tools, including Bregman gradient methods, high-order proximal-point operators, and iterative minimization of convex quadratic models with quartic regularization. Recently, we introduced the Quadratic Quartic Regularisation (QQR) method [16, 39]. QQR approximates the third-order tensor term of the AR3 model by a linear combination of quadratic and quartic terms, resulting in (possibly nonconvex) local models that can be solved to global optimality. In this paper, we aim to enhance the previous approach by explicitly incorporating (simple) third-order terms. Note that for local minimization of \\(m_3\\), standard first and second-order methods designed for minimizing a general objective f(x) can be applied directly; for instance, cubic regularization (ARC/AR2) will be applied as baseline in our numerical experiments." }, { "section": "Characterising global minima of the CQR polynomial", "element": "p", "matched_phrases": [ "future_work" ], "passage": "As suggested by Theorem 2.1 and Theorem 2.5, we have transformed the task of identifying a global minimizer of \\(M_c\\) into the problem of solving the nonlinear equations (27) for \\(\\lambda \\) in the corresponding branch. To numerically determine the root of (27), we use the global optimality characterisations outlined in Sects. 2.1 to 2.2. Additionally, we extend well-established Cholesky factorization-based root-finding techniques, as detailed in [14, Ch 8.2.3]. The primary objective of this subsection is to formulate an algorithm for locating the root of the univariate (nonlinear) secular equation. We employ univariate Newton iteration to find the necessary root of (27). Further convergence analysis for the root-finding method is deferred to future work." }, { "section": "Characterising global minima of the CQR polynomial", "element": "p", "matched_phrases": [ "future_work" ], "passage": "Regardless of the case, we ensure that \\(H_i + \\lambda ^{(0)} I_n\\) is positive definite, making the Cholesky factorization possible in the first iteration. While it is worth noting that this does not entirely avoid the hard case, as it is only the first iteration, this is a measure we take to ensure that we do not start the algorithm with an indefinite \\(H_i + \\lambda ^{(0)} I_n\\). Throughout the iterations of Algorithm 1, in all the numerical examples we examined, the positive definiteness of \\(H_i + \\lambda ^{(k)} I_n\\) is preserved, and we did not encounter the hard case. The Cholesky factorization also remains valid in the cases we tested. However, note that Algorithm 1 is a preliminary implementation of Newton root-finding for the nonlinear equation (27). Addressing and implementing the hard case, as well as the convergence analysis of Algorithm 1, is dedicated to future work." }, { "section": "Characterising global minima of the CQR polynomial", "element": "p", "matched_phrases": [ "future_work" ], "passage": "Our current approach here, for solving the system (21)–(23) and (27), uses Cholesky factorization and Newton’s method to address the secular equation (27), and is just one of many possibilities. Similarly to existing approaches for trust region and regularization subproblems ([9, Ch. 6] and [15, Ch. 8, 10]), scalable iterative algorithms could be designed here, based on Krylov methods, eigenvalue formulations, or subspace optimization. While we recognize the potential of these tools for solving (21)–(23) and finding the global minimum of the CQR polynomial, our primary focus in this paper is on using the CQR polynomial to minimize the \\(m_3\\) subproblem. As a result, we defer a detailed analysis of these alternative approaches to future work." }, { "section": "Conclusions and future work", "element": "p", "matched_phrases": [ "future_work" ], "passage": "The proposed CQR methods offer a way to approximate the third-order tensor term in the AR3 subproblem by a simple cubic term. This is a first step towards developing local models and their optimality conditions that incorporate cubic tensor information. In the future, we plan to incorporate more detailed tensor information from the AR3 subproblem into local quartic models, to capture third-order directions more accurately. By proposing efficient AR3 minimization algorithms, this paper brings high-order tensor methods for general objectives closer to their practical implementation. In terms of further uses, polynomial optimization problems for which only (approximate) local minimizers are required, would be good test cases for CQR algorithms. Finding specific real-life applications of high-order optimization methods is the subject of future work. We believe that problems on which first- and second-order algorithms are inefficient, due to ill-conditioning, or near stationary (in both first- and second-order derivatives), would provide reasonable examples for which computational uses of third-order information may bring some benefit." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02180-2", "title": "Accelerated affine-invariant convergence rates of the Frank–Wolfe algorithm with open-loop step-sizes", "authors": [ "Elias Wirth", "Javier Peña", "Sebastian Pokutta" ], "published_online": "2025-01-06", "volume": "214", "issue": "1-2", "pages": "201-245", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02180-2/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Introduction", "element": "p", "matched_phrases": [ "future_work" ], "passage": "Section 6 reports computational experiments involving problem instances from data science applications. Therein, FW with open-loop steps attains accelerated convergence rates even in some cases that are not yet covered by our theoretical developments. We hope the developments in this paper inspire future research to investigate new classes of problems that satisfy growth properties." }, { "section": "Strong (M, r)-growth", "element": "p", "matched_phrases": [ "open_question" ], "passage": "This section presents affine-invariant convergence results for the Frank–Wolfe algorithm (FW) with open-loop step-size of the form \\(\\eta _t = \\frac{\\ell }{t+\\ell }\\) for \\(\\ell \\in \\mathbb {N}_{\\ge 2}\\) when \\((\\mathcal {C}, f)\\) satisfies the strong (M, r)-growth property for some \\(r\\in [0,1]\\). As we detail in Sect. 3.2 below, the developments in this section answer several open questions raised in [45] and also improve on some of the main convergence results therein." }, { "section": "Strong (M, r)-growth", "element": "p", "matched_phrases": [ "open_question" ], "passage": "The results established in this section address several open questions raised in [45] and significantly advance the understanding of FW with \\(\\eta _t = \\frac{\\ell }{t+\\ell }\\) for \\(\\ell \\in \\mathbb {N}_{\\ge 1}\\). In particular, our rates are affine-invariant, faster than those in [45], and in \\({\\textsf{primaldual}}_t\\), which bounds \\({\\textsf{subopt}}_t\\) from above. Furthermore, we derive bounds on the dual gap." }, { "section": "Other numerical experiments", "element": "p", "matched_phrases": [ "we_do_not_know" ], "passage": "where \\({\\textbf{a}}_1,\\ldots , {\\textbf{a}}_m \\in \\mathbb {R}^d\\) are feature vectors and \\(b_1, \\ldots , b_m \\in \\{-1,+1\\}\\) are the corresponding labels. The \\(\\ell _1\\)-constraints induce sparsity in the iterates of FW. In Fig. 9, for FW with open-loop step-size \\(\\eta _t = \\frac{\\ell }{t+\\ell }\\) for \\(\\ell = 4\\) and all \\(t\\in \\mathbb {N}\\), we compare \\({\\textsf{gap}}_t\\), \\({\\textsf{primaldual}}_t\\), and \\({\\textsf{subopt}}_t\\) on the Z-score normalized Gisette datasetFootnote 3 [19] with \\(m=2000\\) and \\(n=5000\\). We also plot \\(\\mathcal {O}(t^{-\\ell })\\) and \\(\\mathcal {O}(t^{-2})\\) for better visualization. We cannot plot \\({S}\\) as in Theorem 5.2 predicting when acceleration kicks in as we do not know the value of \\({Q}\\) for which Assumption 4 is satisfied." }, { "section": "Other numerical experiments", "element": "p", "matched_phrases": [ "open_question" ], "passage": "In Fig. 10a, all optimality measures decay at a rate of \\(\\mathcal {O}(t^{-\\ell })\\), in Fig. 10b, all optimality measures decay at a rate of order \\(\\mathcal {O}(t^{-2})\\), and in Fig. 10c, \\({\\textsf{gap}}_t\\) and \\({\\textsf{primaldual}}_t\\) appear to decay at a slower rate than \\({\\textsf{subopt}}_t\\), which again appears to converge at a rate of order \\(\\mathcal {O}(t^{-\\ell })\\). To the best of our knowledge, none of these accelerated rates are explained by the current literature. This raises the open question to characterize the acceleration FW enjoys when optimizing over the nuclear norm ball by, for example, exploiting the findings from [11]." }, { "section": "Discussion", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "Throughout, we derived affine-invariant accelerated convergence rates for FW with open-loop step-sizes of the form \\(\\eta _t = \\frac{\\ell }{t+\\ell }\\), \\(\\ell \\in \\mathbb {N}_{\\ge 1}\\). Notably, our results subsume the ones in [45] up to constant factors. In particular, when the feasible region is strongly convex and the gradient of the objective is bounded away from zero, we confirm the conjecture of [45] and extend prior rates of order \\(\\mathcal {O}(t^{-\\ell /2})\\) to \\(\\mathcal {O}(t^{-\\ell })\\). Furthermore, we establish the first affine-invariant rates of order \\(\\mathcal {O}(t^{-2})\\) in the setting of Wolfe’s lower bound, which demonstrates that FW with line-search and short-step can be outperformed by FW with open-loop step-sizes in one of the most important settings of optimization." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02181-1", "title": "McCormick envelopes in mixed-integer PDE-constrained optimization", "authors": [ "Sven Leyffer", "Paul Manns" ], "published_online": "2025-01-06", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02181-1/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [ { "section": "McCormick envelopes for optimal control problems with state equations that have bilinear terms", "element": "p", "matched_phrases": [ "we_do_not_know" ], "passage": "Without any additional constraints, we do not know whether we can actually bound z uniformly in H, however, so we add these bound constraints on u, which are again satisfied by all feasible points for (OCP)." }, { "section": "Conclusion", "element": "p", "matched_phrases": [ "future_work" ], "passage": "As mentioned before, we have used a priori error estimates in our analysis so far. Because the actual optimal objective values for \\((\\textrm{McC}_{hh})\\) were very close to a true lower bound on (OCP) for relatively large values of h (coarser local averaging) both in 1D and 2D, we are convinced that the analysis and integration of a posteriori error estimates into the procedure is key for future research to be able to use relatively coarse grids for \\((\\textrm{McC}_{hh})\\) and scale this methodology." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02183-z", "title": "Quantum computing inspired iterative refinement for semidefinite optimization", "authors": [ "Mohammadhossein Mohammadisiahroudi", "Brandon Augustino", "Pouya Sampourmahani", "Tamás Terlaky" ], "published_online": "2025-01-11", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02183-z/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [ { "section": "Introduction", "element": "p", "matched_phrases": [ "open_question" ], "passage": "Very recently, Apers and Gribling [3] proposed a QIPM for LO that avoids dependence on a condition number. Under some assumptionsFootnote 2 and access to QRAM, their framework achieves a quantum speedup for “tall” LOPs, in which all constraints are inequalities and the number of constraints is significantly larger than the number of variables. Rather than using QLSAs to solve the Newton system, the Newton steps are computed via spectral approximations of the Hessian. While it is an interesting open question whether the techniques from [3] can be generalized to SDO, it has been posited in the classical literature (see, e.g., a discussion in the conclusion of [29]) that sampling is unlikely to speed up Hessian computation as it does in IPMs for LO. Therefore, we seek another avenue for alleviating the condition of number dependence." }, { "section": "Application to quantum interior point methods", "element": "p", "matched_phrases": [ "open_question", "future_work" ], "passage": "In this framework, one can set the target precision of the IF-QIPM to e.g., \\(\\epsilon =10^{-2}\\), and the final precision of the IR method as needed. It is straightforward to verify that the refining optimization problems in IR have the same coefficient matrix, thereby rendering associated characteristics (such as condition number) fixed over the course of the refining scheme. Consequently, the matrix T and its condition number \\(\\kappa _T\\) also remain fixed through the iterations of the IR algorithm. That said, the right-hand side vectors do change from one refining problem to the next, and as a consequence, the norm bound \\(\\omega ^{(k)}\\) for the solution varies over the refinement iterates. We remark that providing an upper bound on the bound for the norm of solutions \\(\\omega \\) for general SDO problems is complicated, and some pathological examples highlight that the solution to a given SDOP can be exponentially large [54]. Thus, we assume that there exist some \\({\\bar{\\omega }}\\ge \\omega ^{(k)}\\) for any integer \\(k>0\\). Although \\({\\bar{\\omega }}\\) can be exponentially large, a detailed analysis of this upper bound remains an open question for future research." }, { "section": "Conclusion", "element": "p", "matched_phrases": [ "future_work" ], "passage": "Further, we introduce two alternate variants of infeasible IR methods adaptable to infeasible oracles. Although these methods exhibit a quadratic reduction of the optimality gap, they concurrently reduce infeasibility at a linear rate. An intriguing avenue for future research is to find the initialization of IPMs in the refining problem so that the number of IPM iterations remains uniformly bounded during the IR iterations." }, { "section": "Conclusion", "element": "p", "matched_phrases": [ "open_question" ], "passage": "To further enhance the complexity of QIPMs, techniques such as preconditioning [44] and regularization [43, 60] can be deployed, addressing constants like \\({\\bar{\\omega }}\\) and \\(\\kappa _T\\). In addition, an interesting question raised by one of the referees is how \\(\\omega ^{(k)}\\) evolves through iterations of the IR methods. Although such analysis remains an open question, we showed the condition number of linear systems will be bounded by a constant even for degenerate SDOPs as the condition number series of the sequence of linear systems start from a fixed number and have similar growth. Thus an upper bound for \\(\\omega ^{(k)}\\) or a tighter bound for the condition number on Newton systems arising in IR-IF-QIPM can further improve the total complexity analysis of IR-IF-QIPM. In another direction, additional empirical and theoretical analyses are essential to explore and realize the full potential of employing IR with other inexact algorithms, such as ADMM-based IPMs [15], Spectral Bundle Methods [25], and various others." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02177-x", "title": "Constant-competitiveness for random assignment Matroid secretary without knowing the Matroid", "authors": [ "Richard Santiago", "Ivan Sergeev", "Rico Zenklusen" ], "published_online": "2025-01-15", "volume": "210", "issue": "1-2", "pages": "815-846", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02177-x/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [ { "section": "Abstract", "element": "p", "matched_phrases": [ "open_problem", "open_question", "conjecture" ], "passage": "The Matroid Secretary Conjecture is a notorious open problem in online optimization. It claims the existence of an O(1)-competitive algorithm for the Matroid Secretary Problem (MSP). Here, the elements of a weighted matroid appear one-by-one, revealing their weight at appearance, and the task is to select elements online with the goal to get an independent set of largest possible weight. O(1)-competitive MSP algorithms have so far only been obtained for restricted matroid classes and for MSP variations, including Random-Assignment MSP (RA-MSP), where an adversary fixes a number of weights equal to the ground set size of the matroid, which then get assigned randomly to the elements of the ground set. Unfortunately, these approaches heavily rely on knowing the full matroid upfront. This is an arguably undesirable requirement, and there are good reasons to believe that an approach towards resolving the MSP Conjecture should not rely on it. Thus, both Soto (SIAM Journal on Computing 42(1): 178-211, 2013.) and Oveis Gharan and Vondrák (Algorithmica 67(4): 472-497, 2013.) raised as an open question whether RA-MSP admits an O(1)-competitive algorithm even without knowing the matroid upfront. In this work, we answer this question affirmatively. Our result makes RA-MSP the first well-known MSP variant with an O(1)-competitive algorithm that does not need to know the underlying matroid upfront and without any restriction on the underlying matroid. Our approach is based on first approximately learning the rank-density curve of the matroid, which we then exploit algorithmically." }, { "section": "Introduction", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "The Matroid Secretary Problem (MSP), introduced by Babaioff et al. [1], is a natural and well-known generalization of the classical Secretary Problem [2], motivated by strong connections and applications in mechanism design. Formally, MSP is an online selection problem where we are given a matroid \\(\\mathcal {M}= (N, \\mathcal {I})\\),Footnote 1 with elements of unknown weights \\(w :N \\rightarrow \\mathbb {R}_{\\ge 0}\\) that appear one-by-one in uniformly random order. Whenever an element appears, it reveals its weight and one has to immediately and irrevocably decide whether to select it. The goal is to select a set of elements \\(I \\subseteq N\\) that (i) is independent, i.e., \\(I \\in \\mathcal {I}\\), and (ii) has weight \\(w(I) = \\sum _{e \\in I} w(e)\\) as large as possible. The key challenge in the area is to settle the notorious Matroid Secretary Problem (MSP) Conjecture:" }, { "section": "Introduction", "element": "h3", "matched_phrases": [ "conjecture" ], "passage": "Conjecture 1.1" }, { "section": "Introduction", "element": "p", "matched_phrases": [ "remains_open", "conjecture" ], "passage": "Whereas the MSP Conjecture remains open, extensive work in the field has led to constant-competitive algorithms for variants of the problem and restricted settings. This includes constant-competitive algorithms for specific classes of matroids [5,6,7,8,9,10,11,12,13]. Moreover, in terms of natural variations of the problem, Soto [8] showed that constant-competitiveness is achievable in the so-called Random-Assignment MSP, or RA-MSP for short. Here, an adversary chooses \\(|N|\\) weights, which are then assigned uniformly at random to the ground set elements N of the matroid. (Soto’s result was later extended by Oveis Gharan and Vondrák [14] to the setting where the arrival order of the elements is adversarial instead of uniformly random.) Constant-competitive algorithms also exist for the Free Order Model, where the algorithm can choose the order in which elements appear [9]." }, { "section": "Introduction", "element": "p", "matched_phrases": [ "open_question", "conjecture" ], "passage": "A high reliance on knowing the matroid \\(\\mathcal {M}= (N, \\mathcal {I})\\) upfront (except for its size \\(|N|\\)) is undesirable when trying to approach the MSP Conjecture, because it is easy to obstruct an MSP instance by adding zero-weight elements. Not surprisingly, all prior advances on the general MSP conjecture, like the above-mentioned \\(O(\\log \\log ({{\\,\\textrm{rank}\\,}}(\\mathcal {M})))\\)-competitive algorithms [3, 4] and also earlier procedures [1, 15], only need to know \\(|N|\\) upfront and make calls to an independence oracle on elements revealed so far. Thus, for RA-MSP, it was raised as an open question, both in [8] and [14], whether a constant-competitive algorithm exists without knowing the matroid upfront. The key contribution of this work is to affirmatively answer this question, making the random assignment setting the first MSP variant for which a constant-competitive algorithm is known without knowing the matroid and without any restriction on the underlying matroid." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02188-8", "title": "Integer points in arbitrary convex cones: the case of the PSD and SOC cones", "authors": [ "Jesús A. De Loera", "Brittney Marsters", "Luze Xu", "Shixuan Zhang" ], "published_online": "2025-01-16", "volume": "216", "issue": "1-2", "pages": "3-27", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02188-8/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02178-w", "title": "Towards an optimal contention resolution scheme for matchings", "authors": [ "Pranav Nuti", "Jan Vondrák" ], "published_online": "2025-01-22", "volume": "210", "issue": "1-2", "pages": "761-792", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02178-w/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02186-w", "title": "Low-rank optimization on Tucker tensor varieties", "authors": [ "Bin Gao", "Renfeng Peng", "Ya-xiang Yuan" ], "published_online": "2025-01-27", "volume": "214", "issue": "1-2", "pages": "357-407", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02186-w/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02185-x", "title": "Monoidal strengthening of simple $$\\mathcal {V}$$-polyhedral disjunctive cuts", "authors": [ "Aleksandr M. Kazachkov", "Egon Balas" ], "published_online": "2025-01-29", "volume": "210", "issue": "1-2", "pages": "567-590", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02185-x/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02191-z", "title": "Sparsity and proximity transference in integer programming", "authors": [ "Iskander Aliev", "Marcel Celaya", "Martin Henk" ], "published_online": "2025-01-31", "volume": "216", "issue": "1-2", "pages": "29-48", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02191-z/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [ { "section": "Introduction and main results", "element": "p", "matched_phrases": [ "future_work" ], "passage": "In this paper, we establish the first transference bounds that involve the integrality gap of integer linear programs and hold in the general case, addressing a future research question posed in [3]. The proofs explore a new geometric approach that combines Minkowski’s geometry of numbers and box slicing inequalities." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02195-3", "title": "Cut-sufficient directed 2-commodity multiflow topologies", "authors": [ "Joseph Poremba", "F. Bruce Shepherd" ], "published_online": "2025-02-03", "volume": "210", "issue": "1-2", "pages": "793-814", "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02195-3/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02182-0", "title": "Convergence of the preconditioned proximal point method and Douglas–Rachford splitting in the absence of monotonicity", "authors": [ "Brecht Evens", "Pieter Pas", "Puya Latafat", "Panagiotis Patrinos" ], "published_online": "2025-02-04", "volume": "214", "issue": "1-2", "pages": "247-301", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02182-0/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02194-4", "title": "On penalty-based bilevel gradient descent method", "authors": [ "Han Shen", "Quan Xiao", "Tianyi Chen" ], "published_online": "2025-02-05", "volume": "214", "issue": "1-2", "pages": "539-589", "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02194-4/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02198-0", "title": "A quadratically convergent semismooth Newton method for nonlinear semidefinite programming without generalized Jacobian regularity", "authors": [ "Fuxiaoyue Feng", "Chao Ding", "Xudong Li" ], "published_online": "2025-02-05", "volume": "214", "issue": "1-2", "pages": "643-683", "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02198-0/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02200-9", "title": "Partitioned matching games for international kidney exchange", "authors": [ "Márton Benedek", "Péter Biró", "Walter Kern", "Dömötör Pálvölgyi", "Daniel Paulusma" ], "published_online": "2025-02-11", "volume": "214", "issue": "1-2", "pages": "723-758", "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02200-9/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Introduction", "element": "p", "matched_phrases": [ "future_work" ], "passage": "Table 1 summarizes the known results for problems P1–P3 for matching games and their variants (in this table we have also included our new results, which we explain below in detail). In Sect. 5 we discuss future work. There, we also mention some results for other solution concepts. For an in-depth discussion on complexity aspects of solution concepts for matching games and their variants, we refer to the recent survey of Benedek et al. [5]." }, { "section": "Introduction", "element": "p", "matched_phrases": [ "left_open" ], "passage": "We start with the known generalization of matching games, namely the b-matching games. In Sect. 2 we prove the following result, solving the only case left open in [11] (see also Table 1)." }, { "section": "Introduction", "element": "p", "matched_phrases": [ "open_problem" ], "passage": "We now make a connection to the Exact Perfect Matching problem introduced by Papadimitriou and Yannakakis [28] forty years ago. This problem has as input an undirected graph G whose edge set is partitioned into a set R of red edges and a set B of blue edges. The question is whether G has a perfect matching with exactly k red edges for some given integer k. The complexity status of Exact Perfect Matching is a longstanding open problem, and so far only partial results were shown (see, for example, [20])." }, { "section": "Introduction", "element": "p", "matched_phrases": [ "future_work" ], "passage": "In Sect. 5 we finish our paper with a discussion on future work." }, { "section": "Conclusions", "element": "p", "matched_phrases": [ "open_problem" ], "passage": "We found two sets of results. One set of results was about ensuring stability of the international collaboration. The aim was to choose in each round of the international programme a kidney transplant distribution as close as possible to some prescribed fair distribution for that round. Roughly speaking, we proved that this problem can be solved efficiently when transplant weights are equal, but otherwise the problem becomes quickly computationally hard. We pose the following two open problems; recall that we showed some partial results for directed partitioned matching games (N, v) with \\(|N|=2\\) in Theorems 8 and 9Footnote 4:" }, { "section": "Conclusions", "element": "p", "matched_phrases": [ "we_do_not_know", "we_leave", "future_work" ], "passage": "In our other set of results, we linked the core of partitioned matching games to the core of b-matching games. We resolved a complexity gap for computing core allocations of b-matching games. As a consequence we can settle the computational complexity for the three core-related problems P1–P3 for partitioned matching games as well; see also Table 1. We note that Table 1 contains two co-NP-hardness results for P2. We do not know if P2 for b-matching games with \\(b\\not \\le 2\\) and for partitioned matching games of width \\(c\\ge 3\\) is co-NP-complete. Given that P1 is co-NP-complete for these cases (see Table 1), P2 is likely to be computationally harder. We leave this question for future research." }, { "section": "Conclusions", "element": "p", "matched_phrases": [ "remains_open" ], "passage": "It is also interesting to consider other solution concepts for b-matching games and partitioned matching games, such as the nucleolus. Könemann, Pashkovich and Toth [23] proved that the nucleolus of a matching game can be computed in polynomial time. In contrast, Könemann, Toth and Zhou [25] proved that computing the nucleolus is NP-hard even for uniform b-assignment games with \\(b\\le 3\\). We refer to [4, 24, 25] for some positive results (see also [5]), but determining the complexity of computing the nucleolus is still open for the following games (see also [25]):" }, { "section": "Notes", "element": "li", "matched_phrases": [ "future_work" ], "passage": "As future work we plan to perform simulations for finding weakly and strongly close maximum weight matchings using integer linear programming (ILP) and Theorems 8 and 9 justify using an ILP model." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02203-6", "title": "Correction: Regular packing of rooted hyperforests with root constraints in hypergraphs", "authors": [ "Pierre Hoppenot", "Mathis Martin", "Zoltán Szigeti" ], "published_online": "2025-02-11", "volume": "213", "issue": "1-2", "pages": "1279-1279", "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02203-6/fulltext.html", "article_kind": "non_research_item", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02205-4", "title": "Special Issue: Integer Programming and Combinatorial Optimization (IPCO) 2023", "authors": [ "Alberto Del Pia", "Volker Kaibel" ], "published_online": "2025-02-11", "volume": "210", "issue": "1-2", "pages": "1-2", "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02205-4/fulltext.html", "article_kind": "non_research_item", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02199-z", "title": "Finite convergence of the Moment-SOS hierarchy for polynomial matrix optimization", "authors": [ "Lei Huang", "Jiawang Nie" ], "published_online": "2025-02-15", "volume": "214", "issue": "1-2", "pages": "685-722", "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02199-z/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02193-5", "title": "A $$\\frac{4}{3}$$-approximation algorithm for half-integral cycle cut instances of the TSP", "authors": [ "Billy Jin", "Nathan Klein", "David P. Williamson" ], "published_online": "2025-02-17", "volume": "210", "issue": "1-2", "pages": "511-538", "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02193-5/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [ { "section": "Abstract", "element": "p", "matched_phrases": [ "remains_open", "conjecture" ], "passage": "A long-standing conjecture for the traveling salesman problem (TSP) states that the integrality gap of the standard linear programming relaxation of the TSP (sometimes called the Subtour LP or the Held-Karp bound) is at most 4/3 for symmetric instances of the TSP obeying the triangle inequality; that is, the cost of an optimal tour is at most 4/3 times the value of the value of the corresponding linear program. There is a variety of evidence in support of the conjecture (see, for instance, Goemans in Math Program 69:335–349, 1995; Benoit and Boyd in Math Oper Res 33:921–931, 2008). It has long been known that the integrality gap is at most 3/2 (Wolsey in Math Program Study 13:121–134, 1980; Shmoys and Williamson in Inf Process Lett 35:281–285, 1990). Despite significant efforts by the community, the conjecture remains open. In this paper we consider the half-integral case, in which a feasible solution to the LP has solution values in \\(\\{0, 1/2, 1\\}\\). Such instances have been conjectured to be the most difficult instances for the overall four-thirds conjecture (Schalekamp et al. in Math Oper Res 39(2):403–417, 2014). Karlin et al. (in: Proceedings of the 52nd Annual ACM Symposium on the the Theory of Computing, ACM, New York, 2020), in a breakthrough result, were able to show that in the half-integral case, the integrality gap is at most 1.49993; Gupta et al. (in: Integer Programming and Combinatorial Optimization. Lecture Notes in Computer Science, 2022. https://arxiv.org/abs/2111.09290) showed a slight improvement of this result to 1.4983. Additionally, this result led to the first significant progress on the overall conjecture in decades; the same authors showed the integrality gap of the Subtour LP is at most \\(1.5-\\epsilon \\) for some \\(\\epsilon >10^{-36}\\) Karlin et al. in 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS). https://doi.org/10.1109/FOCS54457.2022.00084. With the improvements on the 3/2 bound remaining very incremental, even in the half-integral case, we turn the question around and look for a large class of half-integral instances for which we can prove that the 4/3 conjecture is correct, preferably one containing the known worst-case instances. In Karlin et al.’s work on the half-integral case, they perform induction on a hierarchy of critical tight sets in the support graph of the LP solution, in which some of the sets correspond to cycle cuts and the others to degree cuts. Here we show that if all the sets in the hierarchy correspond to cycle cuts, then we can find a distribution of tours whose expected cost is at most 4/3 times the value of the half-integral LP solution; sampling from the distribution gives us a randomized 4/3-approximation algorithm. We note that two important bad cases with an integrality gap of 4/3 have a half-integral LP solution in which all the critical tight sets in the hierarchy are cycle cuts; thus our result is tight. Our overall approach is novel. Most recent work has focused on showing that some variation of the Christofides-Serdyukov algorithm (Christofides in “Worst case analysis of a new heuristic for the traveling salesman problem”. Report 388, Graduate School of Industrial Administration, Carnegie Mellon University, Pittsburgh, 1976; Serdyukov in Upravlyaemye Sistemy 17:76–79, 1978) that combines a randomly sampled spanning tree plus a T-join (or a matching) can be shown to give a bound better than 1.5. Here we show that for any point in a region of “patterns” of edges incident to each cycle cut, we can give a distribution of patterns connecting all the child cycle cuts such that the distribution of patterns for each child also falls in the region. This region gives rise to a distribution on Eulerian tours in which each edge in the support of the LP is used at most four-thirds of its LP value of the time, which then gives the result." }, { "section": "Notes", "element": "li", "matched_phrases": [ "conjecture" ], "passage": "The first place that the authors are aware of a published statement of the conjecture is in a 1995 paper of Goemans [1], but the conjecture was in circulation earlier than that." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02202-7", "title": "On the smallest support size of integer solutions to linear equations", "authors": [ "Yatharth Dubey", "Siyue Liu" ], "published_online": "2025-02-18", "volume": "214", "issue": "1-2", "pages": "789-800", "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02202-7/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02201-8", "title": "Some primal-dual theory for subgradient methods for strongly convex optimization", "authors": [ "Benjamin Grimmer", "Danlin Li" ], "published_online": "2025-02-22", "volume": "214", "issue": "1-2", "pages": "759-788", "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02201-8/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02190-0", "title": "The Boosted Double-proximal Subgradient Algorithm for nonconvex optimization", "authors": [ "Francisco J. Aragón-Artacho", "Pedro Pérez-Aros", "David Torregrosa-Belén" ], "published_online": "2025-02-25", "volume": "214", "issue": "1-2", "pages": "491-537", "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02190-0/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Conclusions and future work", "element": "p", "matched_phrases": [ "future_work" ], "passage": "To conclude, we point out two possible directions for future research." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02197-1", "title": "Nonsmooth nonconvex–nonconcave minimax optimization: Primal–dual balancing and iteration complexity analysis", "authors": [ "Jiajin Li", "Linglingzhi Zhu", "Anthony Man-Cho So" ], "published_online": "2025-03-05", "volume": "214", "issue": "1-2", "pages": "591-641", "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02197-1/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Notes", "element": "li", "matched_phrases": [ "open_question" ], "passage": "As far as we know, the lowest complexity required for finding approximate stationary points of nonconvex–PŁ minimax/pointwise maximum problems remains an open question. Nevertheless, as we mentioned, for smooth nonconvex–strongly concave problems, the \\(\\mathcal {O}(\\epsilon ^{-2})\\) complexity is already optimal." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02210-7", "title": "Solving nonconvex optimization problems using outer approximations of the set-copositive cone", "authors": [ "Markus Gabl", "Kurt M. Anstreicher" ], "published_online": "2025-03-10", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02210-7/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02196-2", "title": "Branch-and-bound for integer D-optimality with fast local search and variable-bound tightening", "authors": [ "Gabriel Ponte", "Marcia Fampa", "Jon Lee" ], "published_online": "2025-03-13", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02196-2/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02211-6", "title": "A prophet inequality based approach to the adaptive ProbeTop$$\\varvec{K}$$ problem", "authors": [ "Guillermo Gallego", "Danny Segev" ], "published_online": "2025-03-13", "volume": "215", "issue": "1-2", "pages": "57-93", "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02211-6/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Introduction", "element": "p", "matched_phrases": [ "open_question" ], "passage": "In this paper, we revisit one of the most eye-opening paradigms in the subfield of stochastic probing, commonly known as the adaptive ProbeTop\\(K\\) problem. Indeed, in spite of its stylized formulation, this setting still captures numerous technical hurdles inherent to stochastic optimization, related to both information structure and efficient computation. For these reasons, special cases and variants of this problem have repeatedly been serving as a test bed for a multitude of algorithmic methods, which will be surveyed in Sect. 1.2, and concurrently as a popular teaching tool in courses and tutorials dedicated to recent trends in optimization under uncertainty; see, e.g., [3, 15, 21, 22, 27, 34]. In order to rigorously discuss existing work in this context, to highlight pending open questions, and to present our main contributions, we proceed by providing a complete mathematical description of the problem in question." }, { "section": "Introduction", "element": "h3", "matched_phrases": [ "open_question" ], "passage": "1.2 Existing work and open questions" }, { "section": "Introduction", "element": "p", "matched_phrases": [ "open_question" ], "passage": "Motivating questions. In light of the preceding discussion, the primary open questions that motivate our work aim to fill several interrelated voids in the current literature, along the following axes:" }, { "section": "Concluding remarks", "element": "p", "matched_phrases": [ "open_question", "future_work" ], "passage": "We conclude this paper by highlighting several open questions, potential directions for future research, and interesting observations. The next few points are intended to investigate whether our upper bounding methods can be used in broader settings, as well as to discuss very recent developments for adaptive ProbeTop\\(K\\) in the discrete and finite support case." }, { "section": "Concluding remarks", "element": "p", "matched_phrases": [ "future_work" ], "passage": "Extensions to additional feasibility constraints? Taking a broader perspective, the adaptive ProbeTop\\(K\\) problem falls within the framework of stochastic probing subject to downward-closed constraints, capturing much of its analytical and computational challenges. As reported in Sect. 1.2, Gupta and Nagarajan [16] proposed an LP-based randomized rounding approach, creating a distribution over feasible sets whose expected objective value approximates the adaptive optimum within factor 3. These results are applicable in the discrete case, with finite support, assuming that the specific constraints in question allow us to efficiently solve this LP-relaxation, which is indeed doable for matroid, knapsack, and k-system constraints, to mention a few. As part of future research, it would be interesting to study whether suitable adaptations of our min-max upper bound can be carried over to the downward-closed setting. Particularly relevant questions in this context are those of obtaining improved adaptivity gaps, deterministic constructions of static policies, and simple threshold-based policies." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02213-4", "title": "Online combinatorial assignment in independence systems", "authors": [ "Javier Marinkovic", "José A. Soto", "Victor Verdugo" ], "published_online": "2025-03-13", "volume": "216", "issue": "1-2", "pages": "49-86", "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02213-4/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02214-3", "title": "Correction: Accelerated affine-invariant convergence rates of the Frank–Wolfe algorithm with open-loop step-sizes", "authors": [ "Elias Wirth", "Javier Peña", "Sebastian Pokutta" ], "published_online": "2025-03-13", "volume": "214", "issue": "1-2", "pages": "941-942", "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02214-3/fulltext.html", "article_kind": "non_research_item", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02212-5", "title": "Convex envelopes of bounded monomials on two-variable cones", "authors": [ "Pietro Belotti" ], "published_online": "2025-03-17", "volume": "211", "issue": "1-2", "pages": "93-123", "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02212-5/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [ { "section": "Introduction", "element": "p", "matched_phrases": [ "open_question" ], "passage": "In Sect. 2 we discuss the convex hull of \\(F(\\mathbb {R}_+^n)\\). Then we obtain the lower envelope of f over \\(X \\cap W_{ij}\\) for \\(n=2\\) (Sect. 3) and the upper envelope of f over \\(X \\cap W_{ij}\\) for \\(n\\ge 2\\) (Sect. 4). In Sect. 5 we present branching rules that retain the convex envelope structure in the subproblems, while in Sect. 6 we compute the volume of the convex hull of \\(F(W_{12})\\) for \\(n=2\\). Concluding remarks and open questions are in Sect. 7." }, { "section": "Upper envelope over \\(X \\cap W_{ij}\\)", "element": "p", "matched_phrases": [ "open_problem" ], "passage": "Table 2 summarizes the results from this and the previous two sections. The lower envelope for \\(W_{ij}\\) is an open problem for \\(n>2\\)." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-024-02192-y", "title": "Distributionally Risk-Receptive and Robust Multistage Stochastic Integer Programs and Interdiction Models", "authors": [ "Sumin Kang", "Manish Bansal" ], "published_online": "2025-03-20", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-024-02192-y/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02206-3", "title": "Primal methods for variational inequality problems with functional constraints", "authors": [ "Liang Zhang", "Niao He", "Michael Muehlebach" ], "published_online": "2025-03-20", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02206-3/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [ { "section": "Convergence analysis", "element": "p", "matched_phrases": [ "we_leave", "future_work" ], "passage": "In the strongly-monotone setting, the convergence rate of CGM in terms of the optimality gap is \\(\\mathcal {O}(1/T)\\), which matches the rate of projection-based methods [84]. However, it is worth noting that the rate in terms of the constraint violation approaches \\(\\mathcal {O}(1/T)\\) only as \\(\\gamma \\) grows large enough. We leave the task of closing this gap to future work. As a comparison, the primal-dual method in Yang et al. [42] achieves \\(\\mathcal {O}(1/\\sqrt{T})\\) rate measured by the distance \\(\\Vert \\bar{x}_T - x^*\\Vert \\). If the operator is additionally Lipschitz, the same \\(\\mathcal {O}(1/\\sqrt{T})\\) rate is attained for a strong approximate solution." }, { "section": "Conclusion", "element": "p", "matched_phrases": [ "unknown_whether" ], "passage": "We present a primal method for solving monotone variational inequality problems with general functional constraints. Our algorithm achieves the same complexity on querying F as projection-based methods and does not rely on any information about the optimal Lagrange multipliers unlike existing primal-dual methods. Several interesting questions have yet to be explored. The current analysis of CGM (Algorithm 1) for the strongly-monotone setting achieves a rate slightly worse than \\(\\mathcal {O}(1/T)\\) on the constraint violation, where T is the number of iterations. It remains unknown whether such guarantees can be improved. Recently, Zamani and Glineur [94] presented a refined analysis of the projected subgradient method, demonstrating last iterate convergence for nonsmooth convex optimization. Extending our analysis in this direction could be interesting. An exploration into the monotone and Lipschitz case is also valuable, particularly to determine if the improved complexity \\(\\mathcal {O}(1/\\epsilon )\\) can be achieved, as observed in the context of projection-based methods, such as extragradient and optimistic gradient [28, 29]. Furthermore, extensions to stochastic settings [23, 47, 62] and non-monotone settings [74, 76] are left for future investigations." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02216-1", "title": "Duality in convex stochastic optimization", "authors": [ "Teemu Pennanen", "Ari-Pekka Perkkiö" ], "published_online": "2025-03-21", "volume": "215", "issue": "1-2", "pages": "145-191", "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02216-1/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02218-z", "title": "Mean–semideviation–based distributionally robust learning with weakly convex losses: convergence rates and finite-sample guarantees", "authors": [ "Landi Zhu", "Mert Gürbüzbalaban", "Andrzej Ruszczyński" ], "published_online": "2025-03-21", "volume": "215", "issue": "1-2", "pages": "237-267", "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02218-z/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Convergence rate for continuously differentiable losses", "element": "p", "matched_phrases": [ "we_leave", "future_work" ], "passage": "Our non-asymptotic analysis is facilitated by the constant stepsize, which allows for the simplification of many terms in Lemma 2 and Theorem 1. For a decaying stepsize sequence, the non-asymptotic analysis appears feasible as well, but it would be significantly more complicated; even for convex single-level stochastic optimization problems, the non-asymptotic analysis with decaying stepsize may require tedious computations, e.g. [32]. For such an extension, the first step would be to generalize Lemma 2 (which provides a bound on the inner level estimate \\(\\mathbb {E}|u_k - h(x_k)|\\)) from constant to decaying stepsizes \\(\\{\\tau _k\\}_{k\\ge 0}\\); this would yield terms involving \\(\\tau _1, \\tau _2,.., \\tau _{k-1}\\) and their various products in the upper bound (20) given in Lemma 2. Then, the proof of Theorem 1 would need to be modified where the bounds on the Moreau envelope \\(\\mathbb {E}[\\varphi _{1/\\bar{\\rho }}(x^{N})]\\) after N iterations would also depend on more complicated terms involving \\(\\tau _1, \\tau _2,.., \\tau _{N}\\) and their various products. Since this requires lengthy and tedious calculations, we leave it for future work to be reported in a specialized outlet." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02217-0", "title": "An accelerated distributed stochastic gradient method with momentum", "authors": [ "Kun Huang", "Shi Pu", "Angelia Nedić" ], "published_online": "2025-03-26", "volume": "215", "issue": "1-2", "pages": "193-236", "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02217-0/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02204-5", "title": "Anisotropic proximal gradient", "authors": [ "Emanuel Laude", "Panagiotis Patrinos" ], "published_online": "2025-04-07", "volume": "214", "issue": "1-2", "pages": "801-845", "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02204-5/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02225-0", "title": "Special Issue: Global Optimization", "authors": [ "Tibor Csendes", "Boglárka G.-Tóth", "Marco Locatelli" ], "published_online": "2025-04-09", "volume": "211", "issue": "1-2", "pages": "1-3", "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02225-0/fulltext.html", "article_kind": "non_research_item", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02220-5", "title": "On the split closure of the periodic timetabling polytope", "authors": [ "Niels Lindner", "Berenike Masing" ], "published_online": "2025-04-15", "volume": "215", "issue": "1-2", "pages": "325-367", "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02220-5/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02208-1", "title": "Preference ambiguity and robustness in multistage decision making", "authors": [ "Jia Liu", "Zhiping Chen", "Huifu Xu" ], "published_online": "2025-04-16", "volume": "214", "issue": "1-2", "pages": "847-939", "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02208-1/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Appendices", "element": "p", "matched_phrases": [ "unknown_whether" ], "passage": "States are not known If we do not have complete information about states, then we may divide the data points into several groups, each of which will be used to construct a piecewise linear utility function. Alternatively, we can sort out the state-dependent data points with all available data by solving a single optimization problem and construct the state-dependent utility functions accordingly with the optimal solutions. We can do so by the central estimation approach with some minor modifications." }, { "section": "Appendices", "element": "p", "matched_phrases": [ "we_do_not_know" ], "passage": "Here, we generate K Kantorovich balls simultaneously, each of which corresponds to a state, with center \\(u_k^C\\), upper boundary \\(u^U_k\\) and lower boundary \\(u^L_k\\) for the k-th ball at each breakpoint. In the case that \\(K=1\\), (E45) coincides with (E43). The objective is to minimize the sum of the Kantorovich distance from the center of each ball to the upper or the lower boundary whichever is greater. Constraints (E45b)–(E45d) are imposed to ensure monotonicity and concavity of the utility function underlying the values \\(u_k^C\\), \\(u^U_k\\) and \\(u^L_k\\) in each state k. (E45e) is to guarantee that \\(u^U_k\\) and \\(u^L_k\\) cover all the consumption-utility pairs under state k. Since in this case we do not know which the state each of the consumption-utility pairs belongs to, the index set \\(S_k\\) is a decision variable. To get rid of the index set, we may introduce variables \\(z_{i,k}\\) which takes a value of 0 or 1. Consequently we can reformulate (E45) as" } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02215-2", "title": "The two-stripe symmetric circulant TSP is in P", "authors": [ "Samuel C. Gutekunst", "Billy Jin", "David P. Williamson" ], "published_online": "2025-04-17", "volume": "215", "issue": "1-2", "pages": "95-143", "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02215-2/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Abstract", "element": "p", "matched_phrases": [ "open_question" ], "passage": "The symmetric circulant TSP is a special case of the traveling salesman problem in which edge costs are symmetric and obey circulant symmetry. Despite the substantial symmetry of the input, remarkably little is known about the symmetric circulant TSP, and the complexity of the problem has been an often-cited open question. Considerable effort has been made to understand the case in which only edges of two lengths are allowed to have finite cost: the two-stripe symmetric circulant TSP. In this paper, we resolve the complexity of the two-stripe symmetric circulant TSP. To do so, we reduce two-stripe symmetric circulant TSP to the problem of finding certain minimum-cost Hamiltonian paths on cylindrical graphs. We then solve this Hamiltonian path problem. Our results show that the two-stripe symmetric circulant TSP is in P. Note that a two-stripe symmetric circulant TSP instance consists of a constant number of inputs (including n, the number of cities), so that a polynomial-time algorithm for the decision problem must run in time polylogarithmic in n, and a polynomial-time algorithm for the optimization problem cannot output the tour. We address this latter difficulty by showing that the optimal tour must fall into one of two parameterized classes of tours, and that we can output the class and the parameters in polynomial time. Thus we make a substantial contribution to the set of polynomial-time solvable special cases of the TSP, and we take an important step towards resolving the complexity of the general symmetric circulant TSP." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02207-2", "title": "Adjustable robust nonlinear network design without controllable elements under load scenario uncertainties", "authors": [ "Johannes Thürauf", "Julia Grübel", "Martin Schmidt" ], "published_online": "2025-04-21", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02207-2/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [ { "section": "Introduction", "element": "p", "matched_phrases": [ "future_work" ], "passage": "The remainder of the paper is organized as follows. In Sect. 2, we introduce potential-based flows and state the considered adjustable robust mixed-integer nonlinear network design problem under load scenario uncertainties. In Sect. 3, we derive an characterization of adjustable robust feasibility of a given network expansion based on finitely many worst-case load scenarios. Subsequently, we embed this result in an exact adversarial approach that solves the uncertain network design problem. We present different solution techniques that speed up the performance of the developed approach in Sect. 4. Using an academic example, we then discuss that the number of necessary worst-case load scenarios in the algorithm can significantly vary depending on the capacity of the sources; see Sect. 5. We finally demonstrate the applicability of the developed approach using the example of gas networks in Sect. 6, followed by a discussion of possible future research directions in Sect. 7." }, { "section": "Conclusion", "element": "p", "matched_phrases": [ "future_work" ], "passage": "One promising direction for future research consists of developing valid inequalities for network expansion problems with nonlinear potential-based flows. In contrast to the large literature on valid inequalities for network design problems with capacitated linear flows, the corresponding literature on potential-based flows is rather scarce. Moreover, including controllable elements such as compressors in gas networks or pumps in water networks is a challenging but important future research direction that leads to solving challenging nonconvex bilevel problems; see Remarks 2 and 3. Certain ideas to address controllable elements in specific nonconvex bilevel problems can already be found in the literature [32]. However, these techniques themselves require rather strong assumptions such as that the controllable elements are not placed on cycle arcs of the network. Nevertheless, these ideas might be a good starting point to generalize the model discussed in this paper." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02224-1", "title": "Accelerated first-order optimization under nonlinear constraints", "authors": [ "Michael Muehlebach", "Michael I. Jordan" ], "published_online": "2025-04-21", "volume": "215", "issue": "1-2", "pages": "407-452", "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02224-1/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Introduction", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "where I either has the form \\(I=[n_\\text {g} ]:=\\{1,2,\\dots ,n_\\text {g}\\}\\) or the form \\(I_x:=\\{~i\\in [n_\\text {g}]~|~g_i(x)\\le 0\\}\\). The former version includes every constraint at every iteration, while the latter version includes only constraints that are violated at the current iterate x. The analysis of algorithms is significantly more challenging when \\(I=I_x\\) compared to \\(I=[n_\\text {g}]\\). In the following we will consider both variants, however, non-asymptotic linear rates in discrete time will only be derived for \\(I=[n_\\text {g}]\\). We conjecture that the same rates can be achieved asymptotically for \\(I=I_x\\), which has been shown in the non-accelerated situation in earlier work, see [1]. Our convergence results span both continuous-time and discrete-time models and are summarized in Table 1 (see the corresponding theorems for the precise statements)." }, { "section": "Convergence analysis", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "for some \\({\\xi _k^i}\\) between \\(x_k\\) and \\(y_k\\), where Taylor’s theorem has been used in the second step. These modifications simplify the analysis and will enable us to derive accelerated non-asymptotic convergence rates for the discrete algorithm (22). We conjecture (based on numerical experiments, see Sect. 5) that the modification of the right-hand side of the inequality constraints in (22) is an artifact of our analysis." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02209-0", "title": "A primal-dual extension of the Goemans–Williamson algorithm for the weighted fractional cut-covering problem", "authors": [ "Nathan Benedetto Proença", "Marcel K. de Carli Silva", "Cristiane M. Sato", "Levent Tunçel" ], "published_online": "2025-04-23", "volume": "215", "issue": "1-2", "pages": "1-56", "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02209-0/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02228-x", "title": "Correction: An accelerated distributed stochastic gradient method with momentum", "authors": [ "Kun Huang", "Shi Pu", "Angelia Nedić" ], "published_online": "2025-04-24", "volume": "215", "issue": "1-2", "pages": "859-861", "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02228-x/fulltext.html", "article_kind": "non_research_item", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02226-z", "title": "Optimizing over path-length matrices of unrooted binary trees", "authors": [ "Daniele Catanzaro", "Raffaele Pesenti", "Allan Sapucaia", "Laurence Wolsey" ], "published_online": "2025-04-25", "volume": "215", "issue": "1-2", "pages": "453-505", "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02226-z/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02229-w", "title": "Geometric and computational hardness of bilevel programming", "authors": [ "Jérôme Bolte", "Quốc-Tùng Lê", "Edouard Pauwels", "Samuel Vaiter" ], "published_online": "2025-04-29", "volume": "215", "issue": "1-2", "pages": "539-574", "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02229-w/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02222-3", "title": "A parametric approach for solving convex quadratic optimization with indicators over trees", "authors": [ "Aaresh Bhathena", "Salar Fattahi", "Andrés Gómez", "Simge Küçükyavuz" ], "published_online": "2025-05-02", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02222-3/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02223-2", "title": "A characterization for tightness of the sparse Moment-SOS hierarchy", "authors": [ "Jiawang Nie", "Zheng Qu", "Xindong Tang", "Linghao Zhang" ], "published_online": "2025-05-02", "volume": "215", "issue": "1-2", "pages": "369-405", "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02223-2/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Conclusions and discussions", "element": "p", "matched_phrases": [ "future_work" ], "passage": "Here are some interesting questions for future work." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02221-4", "title": "A better-than-1.6-approximation for prize-collecting TSP", "authors": [ "Jannis Blauth", "Nathan Klein", "Martin Nägele" ], "published_online": "2025-05-12", "volume": "216", "issue": "1-2", "pages": "87-109", "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02221-4/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [ { "section": "Introduction", "element": "p", "matched_phrases": [ "remains_open" ], "passage": "As mentioned, our improved approximation ratio of 1.599 for PCTSP improves upon the previous 1.774-approximation by Blauth and Nägele [13]. It remains open whether there is an efficient algorithm for PCTSP that matches—in terms of the approximation factor—the \\(3\\big /2\\)-approximation for TSP by Christofides [19] and Serdyukov [39] (also see [18, 42]), or the current best known approximation guarantee for TSP, which is just slightly below \\(3\\big /2\\) [27, 28]. The ideal result for PCTSP would be an algorithm that, given an \\(\\alpha \\)-approximation for TSP, produces an \\(\\alpha \\)-approximation for PCTSP (or possibly an \\((\\alpha +\\varepsilon )\\)-approximation for every \\(\\varepsilon >0\\)). Such a result was recently shown for Path TSP [41], and as approximation algorithms for PCTSP begin to approach the threshold \\(3\\big /2\\), this possibility feels less out of reach." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02227-y", "title": "$$\\mathcal {V}$$-polyhedral disjunctive cuts", "authors": [ "Egon Balas", "Aleksandr M. Kazachkov" ], "published_online": "2025-05-14", "volume": "215", "issue": "1-2", "pages": "507-537", "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02227-y/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Appendices", "element": "p", "matched_phrases": [ "future_work" ], "passage": "Next, via Table 6, we discuss our objective function choices, including statistics on the frequency of failures as a function of disjunction size. A caveat is that our analysis of objectives functions is somewhat limited by our relatively conservative strategy in selecting objectives to use, in order to limit time spent on cut generation and the types of failures discussed in Sect. 4. The objectives we choose utilize the results of Kazachkov et al. [49], suggesting that a successful class of objectives is the set of points and rays of the point-ray collection, but, motivated by Proposition 10, we only do this for cuts tight at \\(p^{*}\\) (the loop starting at step 14 of Algorithm 1). It is natural to consider cuts that only lie on deeper points, but this, as well as cuts that are satisfied by \\(\\bar{x}\\), remains a topic for future research." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02230-3", "title": "Homogeneous second-order descent framework: a fast alternative to Newton-type methods", "authors": [ "Chang He", "Yuntian Jiang", "Chuwen Zhang", "Dongdong Ge", "Bo Jiang", "Yinyu Ye" ], "published_online": "2025-05-15", "volume": "215", "issue": "1-2", "pages": "575-636", "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02230-3/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Appendices", "element": "p", "matched_phrases": [ "future_work" ], "passage": "We are aware of other techniques in [14, 15, 30] beyond simply choosing an appropriate \\(\\gamma _k\\). As a result, designing an efficient way to realize the gradient regularized Newton method remains an interesting future work." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02219-y", "title": "Polyhedral Newton-min algorithms for complementarity problems", "authors": [ "Jean-Pierre Dussault", "Mathieu Frappier", "Jean Charles Gilbert" ], "published_online": "2025-05-19", "volume": "215", "issue": "1-2", "pages": "269-324", "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02219-y/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02234-z", "title": "Lower bounds on the complexity of mixed-integer programs for stable set and knapsack", "authors": [ "Jamico Schade", "Makrand Sinha", "Stefan Weltge" ], "published_online": "2025-05-19", "volume": "216", "issue": "1-2", "pages": "135-176", "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02234-z/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [ { "section": "Introduction", "element": "p", "matched_phrases": [ "open_question", "we_leave" ], "passage": "We remark that since information complexity arguments are typically robust to approximations, we believe that with some amount of work, the approach in [15] can be made to yield Theorem 1.2 above. However, our proof is very much in line with the previous lower bounds for the cut and matching polytope which ultimately reduce the problem to understanding the nonnegative rank of a certain matrix called the unique disjointness matrix, possibly through a randomized reduction. Our proof in fact suggests that one may be able to prove all of these lower bounds — for matching, stable set, and knapsack polytopes — in a unified way, via a randomized reduction to unique disjointness. We leave this as an interesting open question for follow-up work." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02233-0", "title": "A $$\\nicefrac {4}{3}$$-approximation for the maximum leaf spanning arborescence problem in DAGs", "authors": [ "Meike Neuwohner" ], "published_online": "2025-05-24", "volume": "216", "issue": "1-2", "pages": "111-133", "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02233-0/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [ { "section": "Conclusion", "element": "p", "matched_phrases": [ "future_work" ], "passage": "Whether a better guarantee than \\(\\frac{4}{3}\\) can be, for example, obtained via a reduction to the hereditary k-set packing problem with \\(k\\ge 4\\) and an algorithm that considers local improvements of super-constant size remains a question for future research. Note that the state-of-the-art approximation algorithms for the unweighted k-set packing problem crucially rely on also considering well-structured local improvements of logarithmic size [4, 11]." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02236-x", "title": "Low solution rank of the matrix LASSO under RIP with consequences for rank-constrained algorithms", "authors": [ "Andrew D. McRae" ], "published_online": "2025-05-24", "volume": "215", "issue": "1-2", "pages": "717-741", "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02236-x/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Introduction and main result", "element": "p", "matched_phrases": [ "future_work" ], "passage": "A caveat to our result is that the RIP assumption is quite strong. It is well-known that it can hold (with high probability) when \\(\\mathcal {A}\\) is constructed from \\(O(r^*(d_1 + d_2))\\) random dense sensing matrices (e.g., with i.i.d. Gaussian entries) [4], but such an \\(\\mathcal {A}\\) is computationally cumbersome in practice. For accurate recovery of \\(M_*\\) in Frobenius norm, weaker conditions and more practical measurements suffice. For example, the paper [5] gives similar recovery guarantees under an \\(\\ell _1\\) RIP (i.e., \\(\\Vert \\mathcal {A}(M)\\Vert _1 \\approx \\Vert M\\Vert _{\\textrm{F}}\\) for low-rank M). One can construct \\(\\mathcal {A}\\) from \\(O(r^*(d_1 + d_2))\\) random rank-1 measurements and have, with high probability, this \\(\\ell _1\\) condition but not the classical \\(\\ell _2\\) RIP that we require (again, see [5]). It is not clear how to modify our proof of Theorem 1 to handle such a weaker assumption. Furthermore, the results on low-rank algorithms in Section 2 depend critically on the classical RIP (as does the considerable literature on which we build—see Section 3.3). Finding a result similar to Theorem 1 with weaker assumptions on the measurements \\(\\mathcal {A}\\) is a very interesting problem for future work." }, { "section": "Guarantees for nonconvex rank-constrained optimization", "element": "p", "matched_phrases": [ "open_question" ], "passage": "One limitation of Theorems 2 and 3 is that the maximum rank r must not be so big that \\(\\mathcal {A}\\) fails to have \\((2r, \\delta )\\)-RIP. One might expect the landscape of (7) to “improve” monotonically as r increases, but it is not clear that this is the case. Certainly, for sufficiently large r (e.g., \\(r \\ge \\min \\{d_1, d_2\\}\\)) we can make similar guarantees, but then the restriction is trivial. What the situation is for intermediate values of r is still an open question. Beyond our specific problem, such “overparametrization” is a significant theoretical issue in the literature (see Section 3.3 for further discussion). In practice, however, this should not be too much of a problem: because \\(\\mathcal {A}\\) is known, we ought to know (perhaps approximately) up to what rank \\(\\mathcal {A}\\) has RIP, and there would be no benefit to setting r larger than this." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02237-w", "title": "A minimal face constant rank constraint qualification for reducible conic programming", "authors": [ "Roberto Andreani", "Gabriel Haeser", "Leonardo M. Mito", "Héctor Ramírez" ], "published_online": "2025-05-24", "volume": "215", "issue": "1-2", "pages": "743-769", "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02237-w/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02231-2", "title": "Characterizations of the Aubin property of the solution mapping for nonlinear semidefinite programming", "authors": [ "Liang Chen", "Ruoning Chen", "Defeng Sun", "Liping Zhang" ], "published_online": "2025-06-02", "volume": "215", "issue": "1-2", "pages": "637-668", "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02231-2/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Introduction", "element": "p", "matched_phrases": [ "open_question" ], "passage": "According to [6, Theorem 5.24], the strong regularity of the KKT system (1.3) at a solution \\((\\bar{x};\\bar{y})\\) implies that the constraint nondegeneracy condition holds at \\(\\bar{x}\\) (or \\(\\bar{x}\\) is nondegenerate, cf. (2.13)). With the help of this result, Bonnans and Ramírez [5] established a counterpart of [6, Proposition 5.38] for the NLSOCP problem. For the NLSDP problem, Sun [45] defined the SSOSC by introducing an approximation set, and finally obtained a collection of equivalent characterizations of the strong regularity condition, including the SSOSC accompanied by the constraint nondegeneracy. These results properly extend the characterizations of the strong regularity from the conventional nonlinear programming to problem (1.1) with \\(\\mathcal {K}\\) being a non-polyhedral set. Nevertheless, for both the NLSOCP and the NLSDP problems, obtaining such an extension for characterizing the Aubin property has been an open question for a long time." }, { "section": "Characterizations of the Aubin property for NLSDP", "element": "p", "matched_phrases": [ "unknown_whether" ], "passage": "It should be emphasized that, for (1.1) with \\(\\mathcal {K}\\) being an arbitrary \\(C^2\\)-cone reducible set, it is still unknown if the strong regularity of the KKT system (1.7) is equivalent to the constraint nondegeneracy (2.13) combined with a certain second-order optimality condition similar to (2.18). The currently known cases include the nonlinear programming [13], the NLSOCP [5], the NLSDP problem [45] and a composite matrix programming regarding matrix eigenvectors [9]." }, { "section": "Conclusions", "element": "p", "matched_phrases": [ "open_question", "we_leave", "future_work" ], "passage": "In this paper, we prove that at a locally optimal solution to the nonlinear semidefinite programming problem (1.8), the Aubin property of \\({\\mathbb S}_\\textrm{KKT}\\) (1.6) is equivalent to the strong second-order sufficient condition plus the constraint nondegeneracy. This enables us to derive a series of equivalent characterizations of the Aubin property, which includes the strong regularity of the Karush-Kuhn-Tucker system (1.3). As a byproduct, for nonlinear semidefinite programming, this paper answers the open question posed in [11, Section 5] if the Aubin property can be characterized by an exact form of a certain second-order optimality condition together with the constraint nondegeneracy. It should be noted that our analysis for nonlinear semidefinite programming (and also for nonlinear second-order cone programming in [8]) relies on the explicit formulas of the coderivative for the underlying normal cone mapping. Currently, it is not clear to us how to extend these results to generic non-polyhedral \\(C^2\\)-cone reducible constrained optimization problems. We leave this as our future research topic." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02238-9", "title": "Assortment optimization with visibility constraints", "authors": [ "Théo Barré", "Omar El Housni", "Marouane Ibn Brahim", "Andrea Lodi", "Danny Segev" ], "published_online": "2025-06-07", "volume": "216", "issue": "1-2", "pages": "177-220", "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02238-9/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02232-1", "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" ], "published_online": "2025-06-09", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02232-1/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [ { "section": "Introduction", "element": "p", "matched_phrases": [ "open_question" ], "passage": "Many optimization problems have a clear disjunctive structure, where one decision leads to some constraints and another decision results in another set of constraints. In fact, any convex mixed-integer program (MIP), whether linear or convex nonlinear, can be viewed as a disjunctive program with the single disjunctive constraint that the solution must be within at least one convex set out of a collection of convex sets, i.e., within the union of the convex sets [9]. Disjunctive constraints are in general nonconvex, as the union of convex sets is generally not a convex set. Handling these nonconvex disjunctive constraints is a fundamental question in MIP. Disjunctive programming was pioneered by Balas [4,5,6, 8] and generalized by Grossmann and coworkers [29, 44, 56, 58, 65]. In general, disjunctive programming offers an alternative view of MIP that may be valuable for analyzing problems, creating strong extended formulations [8], and deriving cutting planes [10, 11, 23, 39, 68]. However, the best encoding of a disjunctive program, or disjunctive constraint(s), as an MIP is an open question." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02240-1", "title": "Extending the primal-dual 2-approximation algorithm beyond uncrossable set families", "authors": [ "Zeev Nutov" ], "published_online": "2025-06-09", "volume": "216", "issue": "1-2", "pages": "255-274", "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02240-1/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02245-w", "title": "Inexact subgradient methods for semialgebraic functions", "authors": [ "Jérôme Bolte", "Tam Le", "Eric Moulines", "Edouard Pauwels" ], "published_online": "2025-06-20", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02245-w/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02244-x", "title": "A zonogon approach for computing small convex polygons of maximum perimeter", "authors": [ "Bernd Mulansky", "Andreas Potschka" ], "published_online": "2025-06-21", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02244-x/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [ { "section": "Abstract", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "We derive a mixed integer nonlinear programming formulation for the problem of finding a convex polygon with n vertices that is small (diameter at most one) and has maximum perimeter provided a stated conjecture is true. The formulation is based on a geometric construction using centrally symmetric polygons (zonogons). The resulting zonogons can be characterized by equivalence classes (under the action of the dihedral group) of 2n-vectors with entries either plus or minus one and a self-duality property and we study the number of these codes. We propose a two-phase computational approach. Phase I comprises the solution of a purely combinatorial problem. Under assumption of Mossinghoff’s conjecture of axial symmetry, the Phase I problem can be reduced to a Subset-Sum Problem. Without Mossinghoff’s conjecture, a generalized Subset-Sum Problem needs to be solved, which consists of picking n non-opposing 2n-th roots of unity such that their sum is as small as possible. Phase II consists of a non-combinatorial Nonlinear Programming Problem. We develop and implement a Lagrange-Newton-type method with multi-precision floating point arithmetic to solve these problems to high accuracy. We provide extensive numerical results including solutions for polygons with 64 and 128 vertices that are accurate to 90 decimal digits." }, { "section": "Introduction", "element": "p", "matched_phrases": [ "remains_open" ], "passage": "Let \\(\\textrm{SP}_{n}\\) denote the set of convex polygons in the Euclidean plane with \\(n \\ge 3\\) vertices that have diameter at most one. As usual, the diameter of a set is defined as the supremum of the distance of any two points contained in the set. The elements of \\(\\textrm{SP}_{n}\\) are called small n-gons. More than one hundred years ago, Reinhardt [1] asked which (convex) polygons in \\(\\textrm{SP}_{n}\\) have maximum area or maximum perimeter. Reinhardt gave an answer to the area problem for odd n (the regular n-gon) and for the perimeter problem for all n with an odd divisor (via a construction with Reuleaux polygons). But even after a century of efforts, the perimeter problem is still open for the case when \\(n > 8\\) is a power of two." }, { "section": "Introduction", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "The case \\(n = 4\\) can be found in [1]. The computational results in [4] for hexagons and in [5] for octagons, for instance, justified the conjecture that the small polygons of maximum area with an even number n of vertices have a cycle of \\(n-1\\) diameter chords with one pending diameter chord attached. This conjecture was finally proved in [6]. Such a result is missing for the perimeter problem, which motivates us to study novel computational methods to solve the perimeter problem for moderate sizes of n." }, { "section": "Introduction", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "Reinhardt then cuts the Reuleaux polygon at its vertices and rotates and moves its arc segments to align with a semi-circle of radius 1 preserving their original order. The vertices stay pinned to the arcs. Our workings in this paper hinge on a conjecture, which we have neither been able to prove nor disprove so far:" }, { "section": "Introduction", "element": "h3", "matched_phrases": [ "conjecture" ], "passage": "Conjecture 1" }, { "section": "Introduction", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "Reinhardt [1, pages 257f] then proves this conjecture to be true for the case \\(n \\ne 2^s \\ge 3\\), where the optimal polygons must be equilateral. He also proves that they cannot be equilateral in the case \\(n=2^s\\) [1, page 262f]. It is well-known that Conjecture 1 holds for \\(n = 4\\) and \\(n=8\\) (see below) and our exhaustive numerical experiments yielded no counterexamples for \\(n = 16\\) and \\(n = 32\\)." }, { "section": "Introduction", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "Reinhardt’s semi-circle construction in comparison to the proposed zonogon construction for an example with \\(n=8\\), \\(\\nu =5\\), and satisfying the necessary optimality condition of Conjecture 1. Left: Reuleaux pentagon (multi-colored) with its defining star polygon (thick white lines). Dangling diameters are depicted in thinner white lines. The corresponding polygon \\(P_8\\) is depicted in black. The intersection of the five colored sectors is depicted in gray in the center. Middle: Reinhardts semi-circle, constructed by bending the circumference of the Reuleaux polygon in the vertices. The uni-colored sectors of the Reuleaux polygon are rotated. The black polygonal chain has the same length as the perimeter of \\(P_8\\). Right: The uni-colored sectors of the Reuleaux polygon are translated to the origin without rotation. The circle with radius 1 can be completely filled by adding the flipped sectors (with checkerboard pattern). Depicted in black is the corresponding zonogon, whose perimeter is exactly twice the perimeter of the polygon \\(P_8\\)" }, { "section": "Introduction", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "We illustrate Reinhardt’s construction in Fig. 1 (middle) for an example that satisfies the necessary optimality property of Lemma 1 and Conjecture 1. We shall propose an alternative construction, in which the segments are only flipped and translated but not rotated, see Fig. 1 (right). Lemma 1 together with Conjecture 1 imply that the diameter graph of a convex small polygon with maximum perimeter consists of a cycle of odd length \\(\\nu \\) and \\(n-\\nu \\) nodes of degree 1 that are connected to nodes on the cycle." }, { "section": "Introduction", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "Let us return to our alternative construction: We propose to map the arcs of the Reuleaux polygon \\(R_\\nu \\) and thus, by Lemma 1, also the vertices of polygon \\(P_n\\) (which include the vertices of the star polygon if Conjecture 1 holds) as depicted in Fig. 1 (right): We translate each arc segment of \\(R_\\nu \\) (and all vertices of \\(P_n\\) on this arc) such that its center (the “opposite” vertex \\(v_j\\) in the star polygon) lies in the origin. This divides the unit circle into \\(2\\nu \\) sectors, which are congruent either to a sector of \\(R_\\nu \\) or of its flipped image \\(-R_\\nu \\). At the same time, if \\(v_i\\) is a vertex of \\(P_n\\) on the considered arc segment, then \\(v_i - v_j\\) and \\(v_j - v_i\\) are two opposite points on the unit circle. The set of all these points yields the vertices of a zonogon (i.e., a centrally symmetric polygon) \\(Z_{n}\\) with 2n vertices inscribed in the unit circle." }, { "section": "Introduction", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "Alternatively, \\(Z_n\\) could be constructed as the convex hull of the discrete set \\(\\{ v_j - v_i \\mid i, j = 1, \\dotsc , n \\}\\), which can be written as the Minkowski sum of \\(V_n = \\{ v_i \\mid i = 1, \\dotsc , n \\}\\) with \\(-V_n\\). Because the Minkowski sum is compatible with taking convex hulls [11, Thm. 1.1.2], we obtain that \\(Z_n = P_n - P_n\\), also known as the difference body of \\(P_n\\) [11, Chap. 3]. It is also easy to see that the convex hull of \\(V_n - V_n\\) is equal to the convex hull of the union of the translated polygons \\(P_n - v_i\\), \\(i = 1, \\dotsc , n\\). We would like to point out that the zonogon construction via \\(Z_n = P_n - P_n\\) also works without assuming Conjecture 1 to hold. In this case, however, pairs of vertices of \\(Z_n\\) would come to lie inside of the unit circle without touching it, i.e., \\(Z_n\\) would not be inscribed in the unit circle." }, { "section": "Introduction", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "In comparison to the characterization of optimal polygons via diameter graphs or dihedral compositions, our codes have the following properties. First, they have the necessary optimality condition of Lemma 1 and Conjecture 1 built in, which reduces the number of possible codes with respect to other approaches. In [5] for instance, 31 possible diameter graphs (thrackleations) were considered for the case \\(n=8\\), while there are only eleven codes. These codes correspond to the diameter graphs 21–31 in [5, Fig. 1] (while the diameter graphs 1–20 lack an odd cycle). Second, optimization problem formulations based on dihedral compositions have a dimension that depends on the number \\(\\nu \\) of integers that add up to n and one has to set up several problems for varying \\(\\nu = 3, \\dotsc , n\\), while we only need to formulate one master problem, see MINLP (4) below." }, { "section": "Introduction", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "We present a Mixed-Integer Nonlinear Programming Problem (MINLP) that is equivalent to the perimeter problem under Conjecture 1. We count the number of possible codes (the integer part of the MINLP solutions) for given \\(n\\ge 3\\) (see Tab. 1). For a fixed code, the MINLP reduces to a continuous Nonlinear Programming Problem (NLP) with \\(n-1\\) variables, two equality constraints and n inequality constraints. We did not discover multiple local extrema for an NLP with fixed code numerically, but we are not able to provide a proof for uniqueness of the NLP solution. For \\(n=8\\), we display the local maxima for each of the eleven distinct codes in Fig. 2 along with their perimeters in Tab. 2. In particular, we covered all the known global solutions. We also enumerated all codes up to \\(n=32\\) and computed local maxima for each code. Fig. 3 shows the three local solutions for \\(n=32\\) with the largest perimeters, which coincide in the first thirteen decimal places. Off-the-shelf NLP solvers can just barely handle such high accuracy requirements and they fail to provide reliable solutions for larger n due to double precision floating point round-off errors. We wrote a customized solver for the fixed-code NLPs that uses multiple-precision arithmetic, which means that the number of mantissa bits can be set to an arbitrary number (such as 300) instead of the 53 bits available in IEEE double precision. A challenge lies in good numerical structure exploitation, because computations with multi-precision floating point numbers are much slower in comparison to computations with double precision, for which dedicated floating point units (FPUs) on modern central processing units (CPUs) can be exploited." }, { "section": "Introduction", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "We further propose a two-phase algorithm to compute excellent suboptimal solutions of the MINLP by solving first a linear combinatorial Subset-Sum Problem (SSP) to generate codes and, second, solving a continuous nonlinear problem with fixed code using Newton-type methods with multi-precision arithmetic. The derivation of the SSP exploits Mossinghoff’s conjecture [22] that optimal small n-gons are axially symmetric. The resulting solutions for \\(n \\le 32\\) coincide with the best local maxima over all possible fixed codes/diameter graphs. We confirm the current record of [24], but are able to compute the perimeter to arbitrary precision for the first time. We also provide a candidate polygon with large perimeter, which is as close as \\(5.1 \\cdot 10^{-42}\\) to the upper bound \\(\\bar{u}_{128}\\). Such computations were not possible with the computational methods available before. The resulting polygons are depicted in Fig. 4 along with the distance of their respective perimeters to the upper bound \\(\\bar{u}_n\\) in Tab. 3." }, { "section": "MINLP derivation using zonogons", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "Let \\(P \\in \\textrm{SP}_{n}\\) satisfy the necessary optimality condition of Lemma 1 and Conjecture 1 and consider its corresponding zonogon Z. We denote the vertices of Z as \\(z_j\\), \\(j = 1, \\dotsc , 2n\\) and take the representative code \\(c \\in \\{\\pm 1\\}^{2n}\\) starting with the directed zonogon facet \\(z_2 - z_1\\), all in counterclockwise direction and order. Then, the zonogon construction informs us that \\(z_{j+1} - z_j\\) is a counterclockwise directed facet of P if \\(c_j = 1\\) and a counterclockwise directed facet of \\(-P\\) if \\(c_j = -1\\). Hence, the sequence \\(f_j = c_j (z_{j+1} - z_j)\\), \\(j = 1, \\dotsc , 2n\\), contains each facet of P exactly twice.Footnote 1 Due to central symmetry of the zonogon Z and self-duality of c, we immediately see that" }, { "section": "MINLP derivation using zonogons", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "Taking the real and imaginary part of (3) as constraints, we can formulate the perimeter problem (equivalently if Conjecture 1 holds) as the MINLP" }, { "section": "Two-phase algorithm using multi-precision arithmetic", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "The computations become easier if we assume Mossinghoff’s conjecture [22] that a small convex polygon with maximal perimeter admits an axial symmetry: We can then resort to codes with the additional symmetry" }, { "section": "Two-phase algorithm using multi-precision arithmetic", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "The NLP (7) can be reduced further by exploiting structure in the code \\(c^*\\), e.g., code symmetry as in Mossinghoff’s conjecture or by exploiting that pending diameters dissect the diameter graph cycle’s angles equally. We shall see in Sec. 5 that the real bottleneck of the two-phase approach is Phase I, so that further improvements for Phase II will not yield considerable computational improvements." }, { "section": "Computational experience", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "Up to \\(n=32\\), we enumerated all possible codes (without the additional symmetry of Mossinghoff’s conjecture) using the C program accompanying the paper [28]." }, { "section": "Computational experience", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "For \\(n \\ge 64\\), the default settings of Gurobi lead to acceptance of non-optimal iterates, which is due to double precision arithmetic incurring too large numerical errors. Experiments with tweaking the options NumericFocus to its maximum of 3 and enabling Quad for quadruple precision in the Simplex method were not enough to obtain reliable results. This applies also for the SSP problem (6) resulting from assuming Mossinghoff’s conjecture to be true." }, { "section": "Conclusions", "element": "p", "matched_phrases": [ "open_problem", "conjecture" ], "passage": "We derived a new MINLP formulation for computing lower bounds for Reinhardt’s perimeter problem, which is still an open problem after more than 100 years. We showed that the derived lower bounds are strict if Conjecture 1 is true. We propose a two-phase approach that consists of a (generalized) SSP to compute the best code, for which the regular n-gon, which realizes Reinhardt’s upper bound, is almost feasible. Then we refine the regular n-gon in the second phase to a feasible polygon with large perimeter using a local multi-precision Lagrange-Newton-type method. We share ample computational experience and provide high-accuracy solution candidates up to \\(n=64\\) and a probably suboptimal candidate for \\(n=128\\) with large perimeter. We further show that the perimeter of the small convex octagon with largest perimeter is an algebraic number and provide the corresponding integer polynomials of degree 48." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02246-9", "title": "The L-shaped method for stochastic programs with decision-dependent uncertainty", "authors": [ "Giovanni Pantuso", "Mike Hewitt" ], "published_online": "2025-06-27", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02246-9/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [ { "section": "Conclusions and future work", "element": "p", "matched_phrases": [ "future_work" ], "passage": "We see multiple avenues for future work. First, we recall that the validity of the cuts presented for the case of second stage subproblems that are mixed integer linear programs requires an additional assumption regarding the domains of first stage decision variables. Thus, deriving cuts that do not require this assumption would leave a method that is applicable to a broad class of problems. The mechanism we provide for creating distribution-specific cuts can be easily adapted to cuts applicable to different types of first stage variables. Second, the method involves generating distribution-specific inequalities. While we also propose inequalities that are distribution-independent, one can envision enhancing the method with techniques that adapt the inequalities generated for one distribution to be valid for another. Third, there has been a tremendous amount of research recently on techniques for speeding up the L-Shaped method when solving stochastic programs in which uncertainty is exogenous. Adapting such techniques to the proposed method in this paper is a promising line of research. One example of such a technique is a multi-cut version of the proposed method in which cuts are generated for each distribution and/or each of its scenarios. Fourth, there are many practical applications in which uncertainty is endogenous. Thus, another line of future work is to adapt the proposed method to problems other than the production planning problem considered in this paper. Tailoring the method to specific problems may enable further enhancements as problem structure may be exploited. Fifth, preprocessing techniques for identifying when a partition of the feasible region can not contain an optimal solution will likely enable the method to converge much more quickly. The avenues just outlined are primarily methodological. We also believe there is value in extending the well-known concepts of Value of the Stochastic Solution and Expected Value of Perfect Information to problems with decision-dependent uncertainty." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02242-z", "title": "Sharper exponential convergence rates for Sinkhorn’s algorithm in continuous settings", "authors": [ "Lénaïc Chizat", "Alex Delalande", "Tomas Vaškevičius" ], "published_online": "2025-07-01", "volume": "215", "issue": "1-2", "pages": "809-858", "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02242-z/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Introduction", "element": "p", "matched_phrases": [ "open_question" ], "passage": "It is well-known that \\(\\delta _{t}\\) converges to zero under general conditions. However, despite the extensive history of Sinkhorn’s algorithm and its widespread adoption in modern applications, several open questions remain regarding its convergence speed." }, { "section": "Main result: exponential convergence with robust contraction constants", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "The convexity assumption made on the support \\(\\mathcal {X}\\) of \\(\\mu \\) is restrictive. This assumption is needed in only one of the intermediate results, Proposition 5.1 below, that proves a strong-concavity estimate for the entropic Kantorovich functional. It was shown however in [8] that such strong-concavity estimates could be obtained under milder assumptions on \\(\\mathcal {X}\\), e.g. by assuming only that \\(\\mathcal {X}\\) is a finite connected union of convex sets. Overall, we conjecture that assuming that \\(\\mu \\) satisfies a Poincaré inequality should be a sufficient condition for our results to hold." }, { "section": "Numerical experiments", "element": "p", "matched_phrases": [ "we_leave", "future_work" ], "passage": "The numerical results presented in Figure 1 empirically validate Theorems 1.1, 1.3 and the results proved for approximate \\((c,\\lambda )\\)-transforms in Appendix C. On the other hand, when neither of the marginal measures are log-concave nor approximations of log-concave measures, we were only able to prove exponential convergence with contraction rate \\(1-\\Theta (\\lambda ^2/c_{\\infty }^2)\\) (see Theorem 1.2). The numerical findings in Figure 2 suggest that log-concavity may not be a strict requirement for achieving a contraction rate of order \\(1 -\\Theta (\\lambda /c_{\\infty })\\). This indicates that the main results of this paper might extend to more general settings, which we leave for future work." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02248-7", "title": "Branch-and-Bound versus Lift-and-Project Relaxations in Combinatorial Optimization", "authors": [ "Gérard Cornuéjols", "Yatharth Dubey" ], "published_online": "2025-07-01", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02248-7/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02249-6", "title": "On solving integer robust optimization problems with integer decision-dependent uncertainty sets", "authors": [ "Leonardo Lozano", "Juan S. Borrero" ], "published_online": "2025-07-01", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02249-6/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02241-0", "title": "A Practical and Optimal First-Order Method for Large-Scale Convex Quadratic Programming", "authors": [ "Haihao Lu", "Jinwen Yang" ], "published_online": "2025-07-02", "volume": "215", "issue": "1-2", "pages": "771-808", "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02241-0/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "PDQP.jl: A Julia implementation of rAPDHG for QP", "element": "li", "matched_phrases": [ "conjecture" ], "passage": "Adaptive restart does not readily preserve convergence guarantee. Still, we conjecture that a proof of convergence could be established based on the theoretical guarantees derived for restarting with a fixed frequency." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02252-x", "title": "On covering Euclidean space with Q-arrangements of cones", "authors": [ "Khalil Ghorbal", "Christelle Kozaily" ], "published_online": "2025-07-10", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02252-x/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02253-w", "title": "Pairwise-independent contention resolution", "authors": [ "Anupam Gupta", "Jinqiao Hu", "Gregory Kehne", "Roie Levin" ], "published_online": "2025-07-10", "volume": "216", "issue": "1-2", "pages": "295-338", "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02253-w/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02254-9", "title": "Logarithmic integral optimization via adaptive importance sampling based surrogation methods", "authors": [ "Ziyu He", "Junyi Liu", "Jong-Shi Pang" ], "published_online": "2025-07-11", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02254-9/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Analysis of the AIS-based surrogation algorithm", "element": "p", "matched_phrases": [ "remains_to_determine" ], "passage": "To make our discussion self-contained, we present the following convergence result for the deterministic surrogation method as a reference for the intuition behind the formal analysis of AIS-based surrogation. The main idea is that our method can be treated as a combination of deterministic surrogation and some stochastic error subjected to our sampling scheme. While the convergence of deterministic surrogation method can be straightforwardly established as below, what remains to be shown is that the accumulated stochastic error will diminish asymptotically." }, { "section": "Analysis of the AIS-based surrogation algorithm", "element": "li", "matched_phrases": [ "we_leave", "future_work", "conjecture" ], "passage": "4. As suggested by one of the referees, our AIS-based method can alternatively be implemented by only sampling once per step \\(\\zeta ^t|x^t\\sim \\pi _{\\text {IS}}^{H(x^t,\\bullet )}\\), and solving subproblems that are similar to (20) but with the “log-summation” terms constructed by all previous samples: $$\\begin{aligned} \\log \\left( \\frac{1}{t}\\sum _{\\tau =1}^t\\frac{\\exp \\widehat{H}(x,\\zeta ^\\tau ;x^t)}{\\pi _{\\text {IS}}^{H(x^\\tau ,\\bullet )}(\\zeta ^\\tau )}\\right) . \\end{aligned}$$ (37) We will refer to this method as “historic-sample-reuse”, and a related method under a different context can be found in [29]. While this alternative scheme offers savings in sampling time, which is not negligible based on our experiments, we choose to focus on our current AIS scheme for the following reasons. First, unlike (20), we cannot use logarithmic manipulation to eliminate the intractable terms \\(Z(x^\\tau )\\) from each \\(\\displaystyle \\pi _{\\text {IS}}^{H(x^\\tau ,\\bullet )}(\\zeta ^\\tau ) = \\frac{\\exp H(x^\\tau , \\zeta ^\\tau )}{Z(x^\\tau )}\\) displayed in (37), thus the historic-sample-reuse approach poses additional challenges in computation. Second, as discussed in Appendix D, our AIS scheme can be improved so that \\( N_t \\) grows linearly in \\( t \\). As a result, the sample sizes used in the subproblmes for both our AIS scheme and the historical-sample-reuse approach are comparable. Finally, while both schemes require a similar number of samples per step to construct their subproblems, our AIS method achieves zero variance at \\( x^t \\), whereas the historical-sample-reuse scheme does not, as it incorporates samples from previous iterations. Recall that at the end of Section 3.2, we demonstrated that, compared to any other \\( \\pi \\), our AIS scheme achieves lower variance in the sampled surrogate function \\( \\widehat{Z}^N_{\\pi ^*(x^t)}(x;x^t) \\), not only at \\( x = x^t \\) but also for all \\( x \\) within a neighborhood of \\( x^t \\). Given this result, it is natural to conjecture that a similar local variance reduction property may hold when comparing AIS with the historical-sample-reuse scheme. However, a rigorous analysis of this conjecture is beyond the scope of this paper, and we leave it for future work. \\(\\square \\)" }, { "section": "Analysis of the AIS-based surrogation algorithm", "element": "li", "matched_phrases": [ "we_leave", "future_work", "conjecture" ], "passage": "Finally, while both schemes require a similar number of samples per step to construct their subproblems, our AIS method achieves zero variance at \\( x^t \\), whereas the historical-sample-reuse scheme does not, as it incorporates samples from previous iterations. Recall that at the end of Section 3.2, we demonstrated that, compared to any other \\( \\pi \\), our AIS scheme achieves lower variance in the sampled surrogate function \\( \\widehat{Z}^N_{\\pi ^*(x^t)}(x;x^t) \\), not only at \\( x = x^t \\) but also for all \\( x \\) within a neighborhood of \\( x^t \\). Given this result, it is natural to conjecture that a similar local variance reduction property may hold when comparing AIS with the historical-sample-reuse scheme. However, a rigorous analysis of this conjecture is beyond the scope of this paper, and we leave it for future work. \\(\\square \\)" } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02239-8", "title": "Fast Combinatorial Algorithms for Efficient Sortation", "authors": [ "Madison Van Dyk", "Kim Klause", "Jochen Koenemann", "Nicole Megow" ], "published_online": "2025-07-17", "volume": "216", "issue": "1-2", "pages": "221-254", "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02239-8/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02243-y", "title": "Capacitated facility location with outliers and uniform facility costs", "authors": [ "Neelima Gupta", "Rajni Dabas", "Naveen Garg" ], "published_online": "2025-07-17", "volume": "216", "issue": "1-2", "pages": "275-294", "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02243-y/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02250-z", "title": "A simple uniformly optimal method without line search for convex optimization", "authors": [ "Tianjiao Li", "Guanghui Lan" ], "published_online": "2025-07-21", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02250-z/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Introduction", "element": "p", "matched_phrases": [ "unresolved" ], "passage": "In this paper, we attempt to address the aforementioned unresolved issue in the development of optimal methods for convex optimization. Our major contributions are briefly summarized as follows. Firstly, we introduce a novel first-order algorithm, called Auto-Conditioned Fast Gradient Method (AC-FGM), which can achieve the optimal \\(\\mathcal {O}(1/k^2)\\) convergence rate for smooth convex optimization without knowing any problem parameters or resorting to any line search procedures. The algorithmic framework, being as simple as the AGD method, can completely adapt to the problem structure. The major novelty of AC-FGM lies in the management of two extrapolated sequences of search points, with one serving as the “prox-centers” and the other used for gradient computation, respectively. This design plays an essential role in getting rid of line search required by [40] and removing the assumption of a bounded feasible region used in [19]. Moreover, we propose a novel stepsize policy that is highly adaptive to the local smoothness level, which can lead to more efficient empirical performance compared with other existing adaptive methods. Secondly, we show that AC-FGM can be extended to solve convex problems with Hölder continuous gradients, requiring no prior knowledge of problem parameters except for the desired accuracy of the solution. Our analysis demonstrates that AC-FGM is uniformly optimal for all smooth, weakly smooth, and nonsmooth objectives with \\(\\nu \\in [0,1]\\). Thirdly, we confirm the theoretical advantages of AC-FGM through a series of numerical experiments. Our results show that AC-FGM outperforms the previously developed parameter-free methods, such as adaptive gradient descent [33], FISTA [3], and Nesterov’s FGM [40], across a wide range of testing problems." }, { "section": "Concluding remarks", "element": "p", "matched_phrases": [ "future_work" ], "passage": "Our work also opens up several interesting directions for future research. First, in our experiments, the best-performing instantiations of AC-FGM consistently correspond to choosing \\(\\alpha \\in \\{0, 0.1\\}\\), whereas the worst-case convergence guarantees in Corollary 2 suggest using a larger \\(\\alpha \\). Bridging this gap between theoretical guidance and empirical performance remains an important question. Second, it would be interesting to investigate the theoretical guarantees and practical performance of AC-FGM (and other algorithms) when combined with more sophisticated adaptive restart strategies, such as those proposed in [11, 21, 46]." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02251-y", "title": "A hierarchy of eigencomputations for polynomial optimization on the sphere", "authors": [ "Benjamin Lovitz", "Nathaniel Johnston" ], "published_online": "2025-07-21", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02251-y/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Introduction", "element": "p", "matched_phrases": [ "unresolved" ], "passage": "Notably, an optimization-free harmonic hierarchy is developed in [14]. This is closely related to harmonic analysis on spheres, and builds on previous work of [8, 30, 63, 64]. This hierarchy depends on a choice of cubature rule and kernel, and the authors give an explicit choice for which the harmonic hierarchy converges (multiplicatively) as \\(O(1/k^2)\\) in the level k, with the k-th level minimizing a certain degree-2k polynomial (obtained from p and a choice of kernel) over a cubature rule of size \\(2(k+1)^{n-1}\\). While their convergence in the level is faster than ours, the discrete optimization required at each level is prohibitively expensive when n is large. It is possible that a smaller cubature rule could lead to computational savings, although the problem of explicitly constructing small cubature rules is largely unresolved (see [14] and the references therein). Numerical experiments in Section 5 show that even for small examples such as the Motzkin polynomial, the harmonic hierarchy is significantly slower than ours in terms of time budget needed to achieve a given bound. We note that our hierarchy has the additional advantages of being completely canonical, and generalizing easily to optimization over the Segre-Veronese variety." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02255-8", "title": "Globally convergent derivative-free methods in nonconvex optimization with and without noise", "authors": [ "Pham Duy Khanh", "Boris S. Mordukhovich", "Dat Ba Tran" ], "published_online": "2025-07-21", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02255-8/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02261-w", "title": "Verification of first-order methods for parametric quadratic optimization", "authors": [ "Vinit Ranjan", "Bartolomeo Stellato" ], "published_online": "2025-07-28", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02261-w/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02262-9", "title": "Lipschitz minimization and the Goldstein modulus", "authors": [ "Siyu Kong", "A. S. Lewis" ], "published_online": "2025-07-29", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02262-9/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02235-y", "title": "Second-order methods for quartically-regularised cubic polynomials, with applications to high-order tensor methods", "authors": [ "Coralia Cartis", "Wenqi Zhu" ], "published_online": "2025-07-31", "volume": "215", "issue": "1-2", "pages": "669-715", "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02235-y/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Abstract", "element": "p", "matched_phrases": [ "open_question" ], "passage": "There has been growing interest in high-order tensor methods for nonconvex optimization, with adaptive regularization, as they possess better/optimal worst-case evaluation complexity globally and faster convergence asymptotically. These algorithms crucially rely on repeatedly minimizing nonconvex multivariate Taylor-based polynomial sub-problems, at least locally. Finding efficient techniques for the solution of these sub-problems, beyond the second-order case, has been an open question. This paper proposes a second-order method, Quadratic Quartic Regularisation (QQR), for efficiently minimizing nonconvex quartically-regularized cubic polynomials, such as the ARp sub-problem (Birgin et al. Math Program 163:359–368, 2017) with \\(p=3\\). Inspired by Nesterov (Quartic regularity, 2022), QQR approximates the third-order tensor term by a linear combination of quadratic and quartic terms, yielding (possibly nonconvex) local models that are solvable to global optimality. In order to achieve accuracy \\(\\epsilon \\) in the first-order criticality of the sub-problem in finitely many iterations, we show that the error in the QQR method decreases either linearly or by at least \\(\\mathcal {O}(\\epsilon ^{4/3})\\) for locally convex iterations, while in the nonconvex case, by at least \\(\\mathcal {O}(\\epsilon )\\); thus improving, on these types of iterations, the general cubic-regularization bound. Preliminary numerical experiments indicate that two QQR variants perform competitively with state-of-the-art approaches such as ARC (also known as ARp with \\(p=2\\)), achieving either lower objective value or iteration counts." }, { "section": "The QQR method", "element": "p", "matched_phrases": [ "future_work" ], "passage": "Our complexity proofs for Theorem 2.11 and Theorem 2.12 follow a similar structure as the complexity proof for ARC. These results were first obtained by Nesterov and Polyak [41, Thm 3.3.5]. Other works can be found in [5, 13]. To conduct the complexity analysis, we employ a typical construction that involves demonstrating lower bounds on the model decrease and the length of the step at each iteration. The connection between the gradient of model decrease and the gradient of function value decrease is established using (52). While our proof assumes exact minimization of \\(M_{\\alpha ^+}\\) such that \\(\\nabla M_{\\alpha ^+}(s^{(i)}, s^{(i)}_+) = 0\\), the same order of complexity bounds as in Theorem 2.11 and in Theorem 2.12 can be derived when considering approximate minimization with a step-termination condition, such as \\(\\Vert \\nabla M_{\\alpha ^+}(s^{(i)}_+)\\Vert \\le \\theta \\Vert s^{(i)}_+\\Vert ^3\\) for some \\(\\theta \\in (0, 1)\\). This step-termination condition is also utilized in [13], and alternative variants exist [10, 22]. The proof incorporates standard techniques found in [5, Lemma 2.3] and in [13, Lemma 3.3], which demonstrate that the step cannot be arbitrarily small compared to the criticality conditions. We delegate the use of approximate termination conditions in the subproblem solution to future work and to current numerical results." }, { "section": "Numerical results: comparison across methods", "element": "p", "matched_phrases": [ "future_work" ], "passage": "Lastly, we note that in terms of CPU time, the performance of AR3 + QQR and AR3 + ARC is comparable. The primary reason is that AR3 + QQR requires fewer iterations and evaluations per subproblem compared to AR3 + ARC, with AR3 + QQR averaging 1.92 iterations and 1.80 evaluations, while AR3 + ARC averages 6.47 iterations and 4.08 evaluations. When compared to second-order methods, the AR3 method generally requires more CPU time than the ARC method. Currently, both the quadratic-quartic model and the ARC model are solved using Newton’s method applied to a secular equation, which then uses Cholesky factorization of the ensuing linear system. Other alternatives to the latter are possible, such as computing, for small-scale problems, the complete eigenvalue-eigenvector decomposition. Similarly to existing approaches for trust-region and regularization subproblems ([10, Ch. 6] and [15, Ch. 8, 10]), scalable iterative algorithms could be designed here, based on Krylov methods, eigenvalue formulations, or subspace optimization. A detailed analysis of these alternative approaches is deferred to future work." }, { "section": "Notes", "element": "li", "matched_phrases": [ "remains_to_determine" ], "passage": "It remains to be seen if such a large pre-defined regularization parameter as in (19) would be detrimental to the theoretical complexity of AR3." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02263-8", "title": "Extended formulations for control languages defined by finite-state automata", "authors": [ "Christoph Buchheim", "Maximilian Merkert" ], "published_online": "2025-08-04", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02263-8/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [ { "section": "Conclusion", "element": "p", "matched_phrases": [ "open_question", "future_work" ], "passage": "Future research may go further in this direction, possibly discovering a more restrictive pumping lemma that yields an equivalent characterization for regular control languages. Moreover, from a theoretical standpoint, it is an interesting open question whether assuming determinism reduces the representable class of languages. Going in a different direction, one may aim to further generalize the construction from Theorem 4.7. For instance, allowing instantaneous switches (i.e., state switches without processing any input) under certain restrictions would make the concept even more versatile for modeling. However, we do not see a direct way to incorporate such transitions in our proof. Finally, we hope that computational studies will confirm the usefulness of our results in applications." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02266-5", "title": "Composite optimization with indicator functions: stationary duality and a semismooth newton method", "authors": [ "Penghe Zhang", "Naihua Xiu", "Houduo Qi" ], "published_online": "2025-08-06", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02266-5/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Introduction", "element": "p", "matched_phrases": [ "future_work" ], "passage": "The paper is organized as follows. Some frequently used symbols and notations are introduced in Section 2. The one-to-one correspondence between solutions of primal and dual problems is established in Section 3. To find a dual solution, we describe the framework of SGSN and prove its global convergence with a local superlinear (or quadratic) rate in Section 4. The numerical experiments on AUC maximization and multi-label classification are conducted in Section 5. We conclude the paper in Section 6 with some discussion on possible future research." }, { "section": "Conclusions", "element": "p", "matched_phrases": [ "we_leave", "future_work" ], "passage": "Although the developed algorithm SGSN explicitly requires the gradient information of the conjugate function \\(f^*(\\cdot )\\), the stationary duality result does not rely on the availability of this piece of information. In theory, the stationary duality applies to general convex function f as long as the dual problem admits an optimal solution. Numerically, this calls for further investigation when the gradient function cannot be cheaply computed. For instance, it would be interesting to see if derivative-free algorithms [17, 37] is a viable choice for the dual problem in this situation. If successful, it would significantly enlarge the applications of the stationary duality approach. We also note that the dual problem provides valuable information for the Lagrangian multiplier of the primal problem. It would be interesting to investigate whether a primal-dual method is possible. Those questions are not easy to answer and we leave them to our future research." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02258-5", "title": "Computer-assisted design of accelerated composite optimization methods: OptISTA", "authors": [ "Uijeong Jang", "Shuvomoy Das Gupta", "Ernest K. Ryu" ], "published_online": "2025-08-12", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02258-5/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02260-x", "title": "Parabolic target-space interior-point algorithm for monotone weighted linear complementarity problem", "authors": [ "Marianna E.-Nagy", "Tibor Illés", "Yurii Nesterov", "Petra Renáta Rigó" ], "published_online": "2025-08-12", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02260-x/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Numerical results", "element": "p", "matched_phrases": [ "open_question" ], "passage": "In the end of the process, the algorithm often demonstrates a good linear rate. However, the possibility to achieve a local super-linear rate of convergence for methods of this type remains an interesting open question." }, { "section": "Conclusion", "element": "p", "matched_phrases": [ "open_question", "future_work" ], "passage": "In this paper, we introduced an IPA for monotone WLCPs (see Algorithm 1) by extending the PTS framework proposed by Nesterov (2008) in [25] for primal-dual LP problems. The method performs two types of steps. In the predictor stage, the goal is to use a search direction in the PTS which ensures a large enough decrease in the stopping criteria. The goal of the corrector stage is to ensure that the computed solution belongs to the neighbourhood \\(\\mathcal {N}_{\\lambda }(\\beta )\\) of the current point \\(\\textbf{u}(\\textbf{t})\\). The algorithm works in a wide neighborhood, allowing very large steps. This results in a significant acceleration in the end of the process, which was also illustrated in our preliminary numerical experiments. Furthermore, we showed that the proposed PTS IPA has the best known worst-case complexity bound. One direction for future research is to generalize the algorithm to more general classes of problems, such as \\(P_*(\\kappa )\\)-WLCPs (sufficient-WLCPs), or WLCPs over symmetric cones. Considering the local super-linear rate of convergence for the method is another open question. Furthermore, analysing other strategies instead of the greedy step in the PTS is also an interesting future research topic." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02270-9", "title": "Minimizing Quasi-Self-Concordant Functions by Gradient Regularization of Newton Method", "authors": [ "Nikita Doikov" ], "published_online": "2025-08-12", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02270-9/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Discussion", "element": "p", "matched_phrases": [ "open_question" ], "passage": "There are several open questions that are closely related to our results. First, it would be interesting to investigate the lower complexity bounds for our problem class. As it was shown in [6], the power 2/3 is optimal for the methods based on the ball-minimization oracle. Consequently, these optimality results could also hold significant implications for the minimization of quasi-self-concordant functions." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02271-8", "title": "Frank-Wolfe meets Shapley-Folkman: a systematic approach for solving nonconvex separable problems with linear constraints", "authors": [ "Benjamin Dubois-Taine", "Alexandre d’Aspremont" ], "published_online": "2025-08-12", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02271-8/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02256-7", "title": "Non-asymptotic global convergence rates of BFGS with exact line search", "authors": [ "Qiujiang Jin", "Ruichen Jiang", "Aryan Mokhtari" ], "published_online": "2025-08-19", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02256-7/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02264-7", "title": "A first order method for linear programming parameterized by circuit imbalance", "authors": [ "Richard Cole", "Christoph Hertrich", "Yixin Tao", "László A. Végh" ], "published_online": "2025-08-19", "volume": "216", "issue": "1-2", "pages": "339-377", "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02264-7/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02273-6", "title": "Interdiction of minimum spanning trees and other matroid bases", "authors": [ "Noah Weninger", "Ricardo Fukasawa" ], "published_online": "2025-08-19", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02273-6/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02259-4", "title": "Learning and decision-making with data : optimal formulations and phase transitions", "authors": [ "Amine Bennouna", "Bart P. G. Van Parys" ], "published_online": "2025-08-25", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02259-4/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02265-6", "title": "Sensitivity analysis for mixed binary quadratic programming", "authors": [ "Diego Cifuentes", "Santanu S. Dey", "Jingye Xu" ], "published_online": "2025-08-25", "volume": "216", "issue": "1-2", "pages": "379-424", "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02265-6/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02274-5", "title": "Factorized binary polynomial optimization", "authors": [ "Alberto Del Pia" ], "published_online": "2025-08-25", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02274-5/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02275-4", "title": "Approximation algorithm for unrooted prize-collecting forest with multiple components and its application on prize-collecting sweep coverage", "authors": [ "Wei Liang", "Shaojie Tang", "Zhao Zhang" ], "published_online": "2025-08-26", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02275-4/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02272-7", "title": "Polyhedral analysis of quadratic optimization problems with Stieltjes matrices and indicators", "authors": [ "Peijing Liu", "Alper Atamtürk", "Andrés Gómez", "Simge Küçükyavuz" ], "published_online": "2025-08-27", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02272-7/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02268-3", "title": "The Price of Adaptivity in Stochastic Convex Optimization", "authors": [ "Yair Carmon", "Oliver Hinder" ], "published_online": "2025-09-03", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02268-3/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Notes", "element": "li", "matched_phrases": [ "remains_open" ], "passage": "The in-expectation and high-probability case both remain open." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02267-4", "title": "Low-rank mirror-prox methods for nonsmooth and low-rank matrix optimization problems", "authors": [ "Dan Garber", "Atara Kaplan" ], "published_online": "2025-09-04", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02267-4/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Appendices", "element": "p", "matched_phrases": [ "remains_to_determine" ], "passage": "The generalized first order optimality condition for unconstrained minimization implies that there exists some \\({\\textbf{G}}^*\\in \\partial g({\\textbf{X}}^*)\\) for which \\(\\textbf{0}= {\\textbf{G}}^*-{\\textbf{Z}}^*-s^*{\\textbf{I}}\\). It remains to be shown that \\(\\langle {\\textbf{X}}-{\\textbf{X}}^*,{\\textbf{G}}^*\\rangle \\ge 0\\) for all \\({\\textbf{X}}\\in \\mathcal {S}_n\\)." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02277-2", "title": "A Monte Carlo Policy Gradient Method with Local Search for Binary Optimization", "authors": [ "Cheng Chen", "Ruitao Chen", "Tianyou Li", "Ruicheng Ao", "Zaiwen Wen" ], "published_online": "2025-09-08", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02277-2/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02278-1", "title": "On the strength of Burer’s lifted convex relaxation to quadratic programming with ball constraints", "authors": [ "Fatma Kılınç-Karzan", "Shengding Sun" ], "published_online": "2025-09-09", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02278-1/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Abstract", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "We study quadratic programs with m ball constraints, and the strength of a lifted convex relaxation for it recently proposed by Burer (2024). Burer shows this relaxation is exact when \\(m=2\\). For general m, Burer (2024) provides numerical evidence that this lifted relaxation is tighter than the Kronecker product based Reformulation Linearization Technique (RLT) inequalities introduced by Anstreicher (2017), and conjectures that this must be theoretically true as well. In this note, we provide an affirmative answer to this question and formally prove that this lifted relaxation indeed implies the Kronecker inequalities in the original space. Our proof is based on a decomposition of non-rank-one extreme rays of the lifted relaxation for each pair of ball constraints. Burer (2024) also numerically observes that for this lifted relaxation, an RLT-based inequality proposed by Zhen et al. (2021) is redundant, and conjectures this to be theoretically true as well. We also provide a formal proof that Zhen et al. (2021)’s as well as Jiang and Li (2019)’s SST inequalities are redundant for this lifted relaxation. In addition, we establish that Burer’s lifted relaxation is a particular case of the moment-sum-of-squares hierarchy." }, { "section": "Introduction", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "There has been much work [8, 19, 20] on studying exact or tight description of \\(\\mathcal{D}\\) for certain classes of S. Recently, when S is given by the intersection of two ball constraints, Kelly et al. in [14] gave an exact disjunctive SDP reformulation. Inspired by this result, for arbitrary number of ball constraints, Burer in [6] propose a lifted SOC-RLT-strengthened convex relaxation which involves only one more additional variable for lifting, and showed that although it is not an exact relaxation for general \\(m>2\\), its admits a very good numerical performance in terms of both relaxation quality and efficiency of computation. In [6], it was conjectured that this lifted relaxation is provably tighter than the Kronecker RLT inequalities and Zhen et al.’s RLT inequalities are redundant for this lifted relaxation. We answer these two conjectures affirmatively. In particular, in Theorem 1, by proving a decomposition theorem (see Theorem 4) for the non-rank-one extreme rays (extreme rays generated by matrices of rank at least two) of the lifted relaxation proposed in [6] and studying the properties of both rank-one and non-rank-one extreme rays, we show that Kronecker RLT inequalities are redundant for the projection of this lifted relaxation. In Proposition 3 we formally show that the RLT inequalities by Jiang and Li’s as well as Zhen et al. are implied by Kronecker RLT inequalities, and hence are also redundant for Burer’s lifted SOC-RLT-strengthened relaxation. Finally, we close by giving a new interpretation of Burer’s lifted relaxation using the techniques from moment-sum-of-squares (moment-SOS) hierarchy." }, { "section": "Domination of Kronecker RLT inequalities", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "The numerical study in [6] indicates that the lifted relaxation (3) is stronger than the Kronecker inequalities (5), and as a result [6] conjectures that this must be true theoretically as well. In this note we resolve this conjecture and show that this is indeed the case." }, { "section": "Domination of Kronecker RLT inequalities", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "In this section we examine two inequalities, respectively proposed by Jiang and Li in [13] and Zhen et al. in [21], to strengthen the Shor SDP relaxation of the sets of form S. In the numerical study of [6], it was identified that the inequality proposed by Zhen et al. is redundant for \\(\\mathcal{C}_m\\), and it was conjectured in [6] that this must be theoretically true as well." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02279-0", "title": "Hyperparameter tuning via trajectory predictions: stochastic prox-linear methods in matrix sensing", "authors": [ "Mengqi Lou", "Kabir Aladin Verchand", "Ashwin Pananjady" ], "published_online": "2025-09-17", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02279-0/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [ { "section": "Discussion", "element": "p", "matched_phrases": [ "open_question" ], "passage": "In this paper, we derived deterministic trajectory predictions for the stochastic prox-linear method when applied to rank one matrix sensing with Gaussian data. We then used these predictions to derive concrete, two-sided convergence guarantees elucidating the trade-off between batch-size and step-size as well as the sensitivity of the convergence guarantees to the inherent noise in the problem. Several interesting open questions remain and we detail a few here." }, { "section": "Discussion", "element": "p", "matched_phrases": [ "open_question" ], "passage": "The first family of questions concerns the minibatching scheme. While our one-step guarantees in Theorem 1 hold for all batch-sizes \\(1 \\le m \\le d\\), our convergence guarantees in Theorem 2 require batch-sizes which scale poly-logarithmically in the dimension d. This deficiency stems from bounds on the deviation around the deterministic predictions which incur a \\(\\text {polylog}(d)\\) factor. Removing these logarithmic factors in Theorem 1 would immediately extend our convergence guarantees to constant batch-size settings. In a similar spirit, while our deterministic trajectory predictions—which we illustrate in Section 2.3—demonstrate excellent adherence to the empirical trajectory, our results only show that the empirical trajectory and deterministic trajectory enjoy the same convergence rate. It would be interesting to show exact convergence of the empirical trajectory to the deterministic trajectory. On an unrelated note, our theoretical guarantees require the mini-batches to be generated in an online fashion. It does not permit re-using samples across iterations. An interesting open question in this direction is to understand the impact of sample re-use and to develop trajectory-based predictions under such batching schemes." }, { "section": "Discussion", "element": "p", "matched_phrases": [ "future_work" ], "passage": "A second promising direction is to investigate hyperparameter tuning when the measurement covariance matrix is not (a multiple of) the identity. In this more general setting, increasing the batch-size may improve the dependence of the convergence rate on the condition number of the covariance matrix [39], rendering the hyperparameter tuning problem even more delicate. On a related note, we observe that additional phenomena are at play in this case, and we illustrate for the case when the measurement covariance follows a power-law spectrum. Specifically, consider the case where the covariance matrix is diagonal with eigenvalues \\(\\sigma _i = i^{-\\kappa }\\) for \\(i \\in [d]\\) and \\(\\kappa > 0\\). As illustrated in Figure 8, the convergence rate appears to become increasingly sensitive to the step-size as the decay exponent \\(\\kappa \\) grows. Understanding how to optimally choose the step-size under general covariance structures is a compelling direction for future work. We emphasize that addressing these challenges requires significant new theoretical developments, as our current analysis does not directly apply to these more general settings. We view these questions as important and exciting avenues for future research." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02247-8", "title": "On the complexity of a simple primal-dual coordinate method", "authors": [ "Ahmet Alacaoglu", "Volkan Cevher", "Stephen J. Wright" ], "published_online": "2025-09-22", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02247-8/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02276-3", "title": "Truly asymptotic lower bounds for online vector bin packing", "authors": [ "János Balogh", "Ilan Reuven Cohen", "Leah Epstein", "Asaf Levin" ], "published_online": "2025-09-22", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02276-3/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02282-5", "title": "Quadratic-form optimal transport", "authors": [ "Ruodu Wang", "Zhenyuan Zhang" ], "published_online": "2025-09-22", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02282-5/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Appendices", "element": "p", "matched_phrases": [ "we_do_not_know" ], "passage": "For the minimizers, we only obtain marginals in the same location-scale class as in the first item. For the maximizers, we do not need to assume identical marginals, but we only have asymptotic results as in the second and the third items. For arbitrary \\(\\mu ,\\nu \\) and fixed \\(\\gamma >0\\), we do not know either the minimizer or the maximizer for the QOT problem in general." }, { "section": "Appendices", "element": "p", "matched_phrases": [ "we_do_not_know" ], "passage": "For the maximizers of \\(\\iint c_\\gamma \\,\\textrm{d}\\pi \\otimes \\textrm{d}\\pi \\), we do not know explicit forms even for the case \\(\\mu =\\nu =\\textrm{U}(0,1)\\), so we consider these marginals in the numerical scheme. In Figure 4, we report the approximate maximizers with parameters \\(\\gamma \\in \\{0.3,\\,2,\\,6\\}\\). Maximizers for smaller \\(\\gamma \\) appear closer to \\(\\pi _{\\textrm{dia}}\\) (thus reassuring Proposition 17), and for larger \\(\\gamma \\) appear closer to \\(\\pi _{\\textrm{ind}}\\) (thus reassuring Proposition 19). The support of the optimizer seems to be contained in a certain symmetric convex shape \\(E_\\gamma \\) in \\([0,1]^2\\). As \\(\\gamma \\) increases, the support expands, and less mass is concentrated near the boundary of \\(E_\\gamma \\) but more mass in the interior of \\(E_\\gamma \\)." }, { "section": "Appendices", "element": "li", "matched_phrases": [ "we_do_not_know", "conjecture" ], "passage": "(iv) There are many simple cost functions for which we do not have an explicit solution to the corresponding QOT problem. We list a few examples below. (a) We wonder whether Theorem 12 extends to marginal distributions that are not symmetric. We conjecture that some “diamond-type” coupling is the minimizer of the corresponding QOT problem. Such a coupling is a combination of four comonotone and antitone pieces, and it is numerically supported by Figure 5. (b) In Theorem 14, we explicitly solved a class of QOT problems with cost function \\(|(x-x')(y-y')|^q,\\,12\\) also deserves future study, as it is equivalent to maximization of the (2, q)-GW transport cost in (3.3). (c) In addition to the results we obtained and the conjectures above, many other cost functions may yield explicit optimizers of the QOT, which need to be further explored. For instance, we do not know the QOT minimizers for the type-XX cost function \\(\\min \\{|x-x'|,|y-y'|\\}\\), although the QOT minimizers for similar cost functions \\(\\max \\{|x-x'|,|y-y'|\\}\\) and \\(\\min \\{x-x',y-y'\\}\\) are solved in Example 12. As another example, we do not know the QOT minimizers for the cost function \\(|(x-x')(y-y')|^q,\\,12\\) also deserves future study, as it is equivalent to maximization of the (2, q)-GW transport cost in (3.3)." }, { "section": "Appendices", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "In Theorem 14, we explicitly solved a class of QOT problems with cost function \\(|(x-x')(y-y')|^q,\\,12\\) also deserves future study, as it is equivalent to maximization of the (2, q)-GW transport cost in (3.3)." }, { "section": "Appendices", "element": "li", "matched_phrases": [ "we_do_not_know", "conjecture" ], "passage": "(c) In addition to the results we obtained and the conjectures above, many other cost functions may yield explicit optimizers of the QOT, which need to be further explored. For instance, we do not know the QOT minimizers for the type-XX cost function \\(\\min \\{|x-x'|,|y-y'|\\}\\), although the QOT minimizers for similar cost functions \\(\\max \\{|x-x'|,|y-y'|\\}\\) and \\(\\min \\{x-x',y-y'\\}\\) are solved in Example 12. As another example, we do not know the QOT minimizers for the cost function \\(|(x-x')(y-y')|^q,\\,10\\), it is NP-hard to even approximate (the leader’s strategy in) a \\(\\delta \\)-RSE; this inapproximability result rules out the possibility of a fully polynomial time approximation scheme (FPTAS), assuming P \\(\\ne \\) NP. Our proof employs a highly nontrivial reduction from the eXact 3-set Cover (X3C) problem. The reduction is combinatorial in nature despite computing (continuous) mixed strategies, and a key technical challenge in the proof is to relate the continuous game strategy space to the combinatorial solution space of X3C. On the positive side, we present a quasi-polynomial approximation scheme (QPTAS) to compute an approximate RSE. Our proof employs the probabilistic method [2] to prove the existence of an approximate \\(\\delta \\)-RSE, which is similar to [34] for proving the existence of an approximate Nash equilibrium with a simple format termed uniform strategy (which can be enumerated in quasi-polynomial time). However, our algorithm for identifying the approximate \\(\\delta \\)-RSE is significantly different — in fact, the approximate \\(\\delta \\)-RSE is not even a uniform strategy, but instead is some strategy nearby. This is due to the nature of bi-level optimization in Stackelberg games. While a uniform leader strategy \\(\\overline{ \\varvec{ x } }\\) and any nearby strategy \\(\\varvec{ x }\\) will lead to similar leader utilities, they will lead to different follower responses, which in turn affects what leader utility is induced in the equilibrium. This challenge forces us to efficiently search the nearby region of a uniform strategy \\(\\overline{ \\varvec{ x } }\\) for a leader strategy \\(\\varvec{ x }\\) that induces the most favorable follower response. This challenge brought by follower responses is not present in computing an approximate Nash equilibrium. We remark that it is an intriguing yet highly non-trivial open question to close the gap between the above hardness result and QPTAS.Footnote 2" }, { "section": "Analytic properties of RSE", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "We conclude this section with a discussion on the convexity of \\(u_{RSE }(\\delta )\\). Intuitively, one would conjecture that \\(u_{RSE }(\\delta )\\) is convex non-increasing, as the leader’s payoff should have diminishing margin as \\(\\delta \\) increases and follower chooses the worse responses. However, this intuition is only correct when \\(m=2\\). The function in general is neither convex nor concave, as illustrated in the plot in Figure 2." }, { "section": "Computational complexity of RSE", "element": "p", "matched_phrases": [ "open_problem" ], "passage": "We start with the result that it is NP-hard, in general, to obtain an \\(\\epsilon \\)-optimal \\(\\delta \\)-RSE leader strategy. We remark that, though \\(\\delta \\)-RSE appears to be a natural solution concept, we are not aware of any previous results on its computational complexity. The closest result we can find is the NP-hardness of computing an optimal leader strategy that is robust with respect to uncertainty about the follower’s utility matrix, studied by [32]. Despite some similarity in spirit, these two problems can not be seen as special cases of each other. Moreover, from a technical point of view, our hardness result also sheds light on the inapproximability of the problem whereas the proof technique of [32] only implies the hardness of exact computation and leaves inapproximability an open problem from their problem." }, { "section": "Computational complexity of RSE", "element": "p", "matched_phrases": [ "open_problem" ], "passage": "We now present a quasi-polynomial time approximation scheme (QPTAS) for computing an approximate \\(\\delta \\)-RSE, for any given \\(\\delta \\). The approximation ratio \\(\\delta \\) holds irrespective of any specific properties of the game, such as Lipschitz continuity or the inducibility gap, so the algorithm is applicable to a broad range of instances. We also leave an intriguing open problem here to close the gap between the efficiency of this algorithm and the above inapproximability result — specifically, to understand whether a PTAS exists for \\(\\delta \\)-RSE or whether Theorem 2 can be strengthened to the hardness of obtaining a constant additive approximation (possibly under some assumption like exponential time hypothesis as used by Rubinstein [45] to rule out PTAS for Nash equilibrium). Our preliminary investigation suggests that either direction seems to require significantly different ideas from our current techniques." }, { "section": "Conclusion", "element": "p", "matched_phrases": [ "future_work" ], "passage": "Our results open up possibilities for many other interesting questions. For example, our positive and negative computational results in Section 4 have a small gap due to the logarithmic exponent term in the computational complexity of the QPTAS. An immediate direction for future research is to close this gap by either showing a PTAS algorithm or strengthening the hardness result of inapproximability to a constant factor. In addition, our study concerns the most basic normal-form game setups. It is worth understanding the applicability of this robust solution concept to various applications of leader-follower or principal-agent game models, including pricing games [22, 38], security games [48], Bayesian persuasion [26] and contract design [24]. Future work can also study analogous robust solution concepts for more general Stackelberg game models such as these with Bayesian follower types [17] or with constrained follower strategy sets [23, 53]. Lastly, we would also like to study the learnability of RSEs in different feedback models, e.g., when the learner cannot observe the follower payoffs [6, 9, 32, 41]." }, { "section": "Notes", "element": "li", "matched_phrases": [ "open_problem" ], "passage": "Familiar readers may recall that there was a similar gap in the complexity landscape of computing a Nash equilibrium, which was an open problem for about 10 years. Specifically, around 2005, Lipton et al. [34] developed a QPTAS for finding an \\(\\epsilon \\)-Nash whereas Daskalakis et al. [20], Chen et al. [14] ruled out FPTAS for two-player Nash (assuming PPAD\\(\\not \\subseteq \\)P). The gap between these two results was open for about 10 years until Rubinstein [45] settled the lower bound that rules out PTAS for Nash and matches the QPTAS of [34], assuming the Exponential Time Hypothesis for PPAD." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02299-w", "title": "On tractability, complexity, and mixed-integer convex programming representability of distributionally favorable optimization", "authors": [ "Nan Jiang", "Weijun Xie" ], "published_online": "2025-11-24", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02299-w/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02301-5", "title": "Beyond submodular maximization via one-sided smoothness", "authors": [ "Mehrdad Ghadiri", "Richard Santiago", "Bruce Shepherd" ], "published_online": "2025-11-24", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02301-5/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Our results", "element": "p", "matched_phrases": [ "we_do_not_know" ], "passage": "Fractional approximation. Our main result in this part (Theorem 5) shows that a modified version of the continuous greedy process gives a \\((1-\\exp {(-(1-\\alpha )(\\frac{\\alpha }{\\alpha +1})^{2\\sigma })})\\)-approximation for maximizing a non-negative, monotone, \\(\\sigma \\)-OSS function subject to a downwards-closed polytopeFootnote 3, where \\(\\alpha \\) is an arbitrary number in [0, 1). We remark that for \\(\\sigma =0\\), our results recover the \\((1-1/e)\\)-approximation [1, 2] for maximizing the multilinear extension of a submodular function, by setting \\(\\alpha =0\\). For \\(\\alpha =0.5\\), our approximation is better than \\(\\frac{0.5}{3^{2\\sigma }+0.5}\\). For fixed \\(\\sigma \\) this gives a constant-factor approximation independent of n. At present, we do not know the correct dependence on \\(\\sigma \\). However, the dependence improves to linear with an additional assumption that third-order partials are non-positive. More precisely, we obtain a \\((1-\\exp {(-\\frac{1}{4\\sigma +2})})\\)-approximation; see Theorem 6. As mentioned in Section 1 this gives an \\(\\Omega (1/\\sigma )\\) approximation which is in fact tight within a constant factor (cf. Corollary 23 discussed below). One example of such functions are multilinear extensions of semi-metric diversity functions, i.e., whose Hessian is a \\(\\sigma \\)-semi-metric (discussed further in the rounding part)." }, { "section": "Our results", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "Hardness. In this part, we show that the hardness of approximation is also governed by the smoothness parameter of the function. More specifically, in Theorem 22 we show that assuming the exponential time hypothesis (ETH), for a constant \\(\\sigma \\), it is hard to approximate the maximum of a \\(\\sigma \\)-semi-metric diversity function subject to a cardinality constraint within a factor better than \\(2\\sigma \\) in polynomial time — the planted clique conjecture also implies the same hardness result. We also rule out the existence of a polynomial-time, slightly subpolynomial approximation algorithm under the ETH. More specifically, we show that for some absolute constant c, there is no polynomial-time algorithm achieving a \\(n^{1/\\log \\log ^c n} / 4\\) approximation." }, { "section": "Hardness", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "Given a graph G and integer k, the densest k-subgraph problem aims to find an induced subgraph of size k with the maximum number of edges. Let R be a subset of vertices of G and E(R) be the number of edges in the induced subgraph of R. The density of R is defined as \\(\\rho (R)=E(R)/{\\left( {\\begin{array}{c}|R|\\\\ 2\\end{array}}\\right) } \\le 1\\). A recent breakthrough [51] shows that assuming the exponential time hypothesis (ETH), there is no polytime algorithm with slightly subpolynomial approximation ratio for densest subgraph. More precisely, there is no polytime algorithm which can distinguish between two cases: (i) an instance G which contains a k-clique and (ii) an instance where the density of every k-subset S satisfies \\(\\rho (S) \\le n^{-1/(\\log \\log n)^c}\\), where c is a universal constant independent of n. In the following, we let \\(\\theta := n^{1/(\\log \\log n)^c}\\). Existence of constant-factor approximations had previously been ruled out under the unique games conjecture with small set expansion [52]." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02302-4", "title": "Breakdown points of Fermat–Weber problems under gauge distances", "authors": [ "Andrei Comăneci", "Frank Plastria" ], "published_online": "2025-11-24", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02302-4/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Introduction", "element": "p", "matched_phrases": [ "left_open" ], "passage": "In [16] the question was left open whether the same bounded set K may contain all Fermat–Weber points for samples corrupted by any \\(\\tau \\) below the breakdown point, a property we named uniform robustness. We show in section 5 that this does not hold in many cases, including the Euclidean distance, but that any polyhedral distance has uniform robustness for its Fermat–Weber points. Thus, we guarantee a consistent performance in robustness for polyhedral Fermat–Weber points across varying contamination levels, making them more predictable in worst-case or uncertain conditions. What is more, the displacement of the estimator of the corrupted sample is easier to analyze, especially when relating subsequently to majority rules in location problems." }, { "section": "Final remarks", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "We also conjecture that the computations of breakdown points of M-estimators reduces to the Fermat–Weber case. Recall that an M-estimator arises as the minimum of a function of the form" }, { "section": "Final remarks", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "Our analysis does not extend directly, as we lack the invariance of subdifferentials under scaling or linearity over segments. However, the breakdown occurs when some part of A is arbitrarily corrupted, so we have a movement to “infinity.” Such asymptotic behaviour is encoded by the horizon function \\(\\rho ^\\infty \\) as defined in [53, §3.C]. We remark that this is a gauge and we conjecture that the corresponding Fermat–Weber point has the same breakdown point with our M-estimator. We remark that this conjecture holds for the 1-dimensional case, according to the analysis in [54, §3.2.1]." }, { "section": "Final remarks", "element": "p", "matched_phrases": [ "future_work" ], "passage": "We note that the practical identification of corrupted versus uncorrupted subsets (such as the sets C and D considered in section 3) is closely related to the problem of outlier detection [55, 56]. Exploring such applications is a promising direction for future work and would likely require incorporating robust measures of dispersion alongside location. Nevertheless, we note that this connection is not straightforward: for example, certain cases of our Fermat–Weber estimators, such as M-quantiles, have been found less effective for outlier detection [57]." }, { "section": "Final remarks", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "We saw that the uniform robustness is related to the boundedness of the elementary hull. This is easy to check for polyhedral or locally strictly convex norms, but our approach does not extend to more general norms, e.g. cylindrical. In any case, we give the following conjecture which holds in dimension at most two." }, { "section": "Final remarks", "element": "h3", "matched_phrases": [ "conjecture" ], "passage": "Conjecture 9.1" }, { "section": "Final remarks", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "When we lose symmetry, analyzing the elementary hull is not sufficient. Theorem 3.4 and Examples 7 and 8 indicate that the weights and the skewness directions play an important role; the asymmetric case is highly more complex. Moreover, it is plausible that a similar statement to Conjecture 9.1 is false for asymmetric gauges. It might be that \\(\\gamma \\) is uniformly robust if it is polyhedral in a neighbourhood of \\(-{{\\,\\textrm{SD}\\,}}_\\gamma \\)." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02303-3", "title": "ODE-Based Learning to Optimize", "authors": [ "Zhonglin Xie", "Wotao Yin", "Zaiwen Wen" ], "published_online": "2025-11-24", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02303-3/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02298-x", "title": "Approximation algorithms for network design in non-uniform fault models", "authors": [ "Chandra Chekuri", "Rhea Jain" ], "published_online": "2025-12-01", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02298-x/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Abstract", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "The Survivable Network Design problem (SNDP) is a well-studied problem, (partly) motivated by the design of networks that are robust to faults under the assumption that any subset of edges up to a specific number can fail. We consider non-uniform fault models where the subset of edges that fail can be specified in different ways. We consider three models: the flexible graph connectivity model (Adjiashvili, D.: Fault-tolerant shortest paths - beyond the uniform failure model (unpublished).) (Adjiashvili, D. et al. in Math. Program. 192(1–2), 409–441 (2022)) (Boyd, S. et al. in Math. Program. pp. 1–24 (2023)) which was our initial motivation, the bulk-robust model (Adjiashvili, D. in Algorithms and Techniques (APPROX/RANDOM 2015), Leibniz International Proceedings in Informatics (LIPIcs), vol. 40, pp. 61–77. Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl, Germany (2015)) (Adjiashvili, D. et al,: in Math. Program. 149(1–2), 361–390 (2015)) and the relative survivable network design model (Dinitz, M. et al.: in Algorithms and Techniques (APPROX/RANDOM 2022), Leibniz International Proceedings in Informatics (LIPIcs), 245, pp. 41:1–41:19. Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl, Germany (2022)) (Dinitz, M. et al.: in Algorithms, pp. 190–204. Springer Nature Switzerland, Cham (2023)). While SNDP admits a 2-approximation (Jain, K.: in Combinatorica 21(1), 39–60 (2001)), the approximability of problems in these more complex models is much less understood even in special cases. We make two contributions. Our first set of results are in the flexible graph connectivity model. Motivated by a conjecture that a constant factor approximation is feasible when the robustness parameters are fixed constants, we consider two important special cases, namely the single pair case and the global connectivity case. For both these, we obtain constant factor approximations in several parameter ranges of interest. These are based on an augmentation framework and via decomposing the families of cuts that need to be covered into a small number of uncrossable families. Our second set of results are poly-logarithmic approximations for the bulk-robust model (Adjiashvili, D. in Algorithms and Techniques (APPROX/RANDOM 2015), Leibniz International Proceedings in Informatics (LIPIcs), vol. 40, pp. 61–77. Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl, Germany (2015)) when the “width” of the given instance (the maximum number of edges that can fail in any particular scenario) is fixed. Via this, we derive corresponding approximations for the flexible graph connectivity model and the relative survivable network design model. The results are obtained via two algorithmic approaches and they have different tradeoffs in terms of the approximation ratio and generality." }, { "section": "Introduction", "element": "p", "matched_phrases": [ "future_work" ], "passage": "Uncrossability of the violated sets does not hold for most of our settings we are interested in. To prove Theorem 1, we overcome this difficulty by decomposing the family of cuts to be covered in the augmentation problem into a sequence of cleverly chosen uncrossable subfamilies. Our structural results hold for certain ranges of values of p and q, and hint at additional structure that may be available to exploit in future work. Our proof of Theorem 2 follows a similar framework, though additional structure is needed. Boyd et al. [13] (also implicitly in [5]) show that (1, k)-Flex-SNDP and (k, 1)-Flex-SNDP can be reduced to corresponding capacitated network design problems with maximum capacity k. Capacitated network design generalizes standard edge connectivity by allowing each edge e to have an integer capacity \\(u_e \\ge 1\\); a feasible solution supports a flow of \\(k_i\\) between each \\(s_i\\)-\\(t_i\\) pair. One can reduce capacitated network design to standard edge connectivity by replacing each edge e with \\(u_e\\) parallel edges, blowing up the approximation factor by \\(\\max _e u_e\\). Although the connection to between flex-connectivity and capacitated network design does not hold for \\(p, q \\ge 2\\), it provides a useful starting point that we exploit." }, { "section": "Conclusion and open problems", "element": "p", "matched_phrases": [ "open_problem" ], "passage": "Non-uniform network design is a fairly new area of study with several open problems. We highlight one important one. Is there a constant factor approximation for fixed width/connectivity? Achieving this even for the special cases of global or single pair connectivity would be a significant step forward. We are also interested in hardness of approximation. Is it possible to rule out a constant factor approximation for fixed width Bulk-Robust SNDP? For large/unbounded width, is Bulk-Robust SNDP as hard as Label Cover, or can we hope for a polylogarithmic approximation? We note that although the Bulk-Robust model is fairly general and captures known hard problems such as Capacitated SNDP, the fact that the scenario set is given explicitly makes direct reductions from such problems tricky. For unbounded width, there are also some problems such as FGC for which a constant factor approximation is not ruled out. We are hopeful that progress on these questions will also lead to the development of new ideas and techniques. In fact, there have already been some interesting developments in extending algorithmic ideas such as primal-dual to larger class of problems — see Remark 1 and [9, 10, 51]. We also point to [43] and [42] for additional work on non-uniform models." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02307-z", "title": "Primal-dual proximal bundle and conditional gradient methods for convex problems", "authors": [ "Jiaming Liang" ], "published_online": "2025-12-01", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02307-z/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Appendices", "element": "p", "matched_phrases": [ "we_do_not_know" ], "passage": "It is worth noting that MPDS\\(({\\hat{x}}_0,{\\lambda })\\) is a conceptual method since we do not know \\(d_0\\) and hence \\({\\hat{h}}\\). We show equivalence between PDS\\(({\\hat{x}}_0,{\\lambda })\\) and MPDS\\((x_0,{\\lambda })\\), and only use MPDS\\(({\\hat{x}}_0,{\\lambda })\\) for analyzing the convegence." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02310-4", "title": "Fully-Dynamic Load Balancing", "authors": [ "Ayoub Foussoul", "Vineet Goyal", "Amit Kumar" ], "published_online": "2025-12-10", "volume": "216", "issue": "1-2", "pages": "573-603", "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02310-4/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02315-z", "title": "Online Algorithms for Spectral Hypergraph Sparsification", "authors": [ "Tasuku Soma", "Kam Chuen Tung", "Yuichi Yoshida" ], "published_online": "2025-12-10", "volume": "216", "issue": "1-2", "pages": "627-650", "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02315-z/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02294-1", "title": "A single-level reformulation of binary bilevel programs using decision diagrams", "authors": [ "Sebastián Vásquez", "Leonardo Lozano", "Willem-Jan van Hoeve" ], "published_online": "2025-12-16", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02294-1/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02305-1", "title": "Getting to the root of the problem: sums of squares for limits of trees", "authors": [ "Daniel Brosch", "Diane Puges" ], "published_online": "2025-12-16", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02305-1/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [ { "section": "Introduction", "element": "p", "matched_phrases": [ "remains_open", "conjecture" ], "passage": "Asymptotic extremal graph theory studies the behaviour of graphs whose number of vertices grows towards infinity. It is an old and well-studied area of graph theory in which many areas of mathematics meet, and where even simple-sounding conjectures can prove extremely challenging. A main topic in this area is the problem of finding the inducibility of a graph, i.e., its maximum density as an induced subgraph in an arbitrarily large graph [1]. It is a very hard problem that remains open even for many “easy” graphs despite being widely studied: for instance, the inducibility of the 5-cycle was determined in 2018 [2], more than 40 years after being conjectured by Pippenger and Golumbic [1]. An adjacent problem consists in determining the profile of a set of graphs: that is, the exact relations between the densities of these graphs in a common large graph. In 2017, Czabarka, Székely and Wagner [3] extended the notions of density and inducibility to leaf-labeled rooted binary trees, a type of trees stemming from phylogenetics. They obtained the first results and bounds for the inducibilities of these trees. In this paper, we follow up on their work and apply, for the first time, flag algebra theory to the setting of leaf-labeled rooted binary trees. Flag algebras are a powerful tool in extremal combinatorics, introduced in 2007 by Razborov [4]. They have been successfully used to tackle and solve several very challenging problems of graph theory [5,6,7,8,9,10,11], by allowing the application of tools from polynomial optimization, computationally in the form of semidefinite programming, to extremal combinatorics. Here, we use them to develop a computer-assisted way to obtain strong rigorous bounds on inducibilities of trees. As main results, we recover all known inducibilities of small trees (up to a small epsilon), obtain over 300 new bounds on inducibilities, and the first outer approximations of profiles of trees, for some of which we prove nonconvexity. To do so, we solve the first instance of a generalized problem of moments in flag algebras." }, { "section": "Introduction", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "Tree-profile of the caterpillar tree of size 4 and the even tree of size 6 with three known points and our conjecture for the upper boundary" }, { "section": "Density and inducibility of rooted leaf-labeled binary trees", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "Graph profiles were first introduced in 1979 by Erdős, Lovász, and Spencer [24], who investigated how the densities of two graphs behaved with respect to each other. In particular, they showed that the profile of any m connected graphs is full-dimensional in \\(\\mathbb {R}^m\\). However, graph profiles have proven to be very challenging to study; they are not necessarily convex or even semialgebraic sets, and still very little is known about them. A major breakthrough on the topic was obtained in 2008 by Razborov [5], who used his flag algebra theory to obtain the exact relation between the density of triangles in a graph and its edge density. In other words, he gave the first full description of a graph profile, the profile of the triangle and the edge graph. This problem was introduced by Turán in 1941 [25] and was until then still (mostly) unsolved. The case of edge versus the complete graph \\(K_4\\) was then solved by Nikiforov in 2010 [26]. The general case of \\(K_n\\) and the edge was solved in 2016 by Reiher [27], proving a conjecture of Lovász and Simonovits [28]." }, { "section": "Results", "element": "p", "matched_phrases": [ "future_work" ], "passage": "While we do not attempt a formal proof here, it appears reasonable to expect that these equations imply any optimal construction must be an even tree, up to a vanishing error as the number of leaves tends to infinity. A similar approach was recently employed to verify the uniqueness of the \\(D_4\\)-root system [50]. Moreover, one might aim to establish a stability result: any nearly optimal construction \\(\\mathcal {T}\\) must be close to the even tree, in a manner analogous to the stability phenomena first studied by Simonovits [51] and later extended to the flag algebra framework by Pikhurko [52]. A full formalization and detailed proof of these ideas are left for future work." }, { "section": "Results", "element": "p", "matched_phrases": [ "open_question", "remains_open" ], "passage": "In both [3] and [21], the authors raised the possibility that this tree may have an irrational inducibility - if rational, it would have to have a denominator of at least 89. Proving this could be possible using our flag algebra software, and would answer the still open question of the possible irrationality of the inducibility of a tree." }, { "section": "Results", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "The three outer approximations of the tree-profiles of \\(\\textrm{Cat}_{k}\\) and \\(E_6\\) for \\(k \\in \\{4,5,6\\}\\) are represented in the first row of Figure 13. Each level of the SDP hierarchy from 6 to 11 is drawn, the approximation becoming tighter as the level increases. We can thus clearly visualize the improvement brought the levels of the hierarchy, and note that they all seem to converge to tree-profiles with similar characteristics. The upper boundary appears to have two parts: we will give a conjecture for the expression of the right part. The right part of the upper bound turns non-convex and appears (numerically) sharp at levels 8, 9 and 10, respectively. The lower boundary, however, seems more complex: the approximations are composed of multiple different sections (some of which may not be visible yet), and each level of the hierarchy is able to substantially improve the approximation. This behavior is made obvious when plotting separately the improvement of the lower approximation with each level of the hierarchy. In each figure of the second row of Figure 13 are represented the respective gaps between the lower boundary obtained at levels 7, 8 and 8 respectively of the hierarchy and those obtained at each level from 6 to 11. The leftmost section of each lowerbound appears (close to) linear, and is followed by (at least) two \"scallops\". As soon as the level reaches 7,8 and 8 respectively, the bound appears to stabilize at the point where the two scallops meet, suggesting existence of a \"nice\" extremal configuration. In the scallops themselves, the bounds improve strictly at every level, but numerical issues become more visible, making it hard to claim anything specific. The apparent complexity of approximating these lower boundaries suggests that more advanced strategies (such as the variational methods Razborov used for the triangle-edge profile [5]) may be necessary to find exact bounds." }, { "section": "Results", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "We will now formulate a conjecture for the right part of the upper boundary of each of these tree-profiles and prove some points of interest on these boundaries, depicted in Figure 14 for \\(\\textrm{Cat}_{4}\\)." }, { "section": "Results", "element": "h3", "matched_phrases": [ "conjecture" ], "passage": "Conjecture 1" }, { "section": "Results", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "For every point of this curve, we are able to obtain a (numerically exact) sum of squares certificate that it indeed lies on the boundary of the profile. However, in the absence of information about the algebraic degree of the curve, this is not enough to prove Conjecture 1, for we would need a parametrized family of sum of squares certificates for each point." }, { "section": "Concluding remarks and open problems", "element": "p", "matched_phrases": [ "open_question", "conjecture" ], "passage": "Proving open conjectures on the inducibilities of trees. There are several conjectures about the maximizers of the density of a graph over all graphs of a fixed size. For example, in [3], Czabarka et al. made the following conjecture: for every \\(n\\ge k\\), \\(E_n\\) has the largest number of copies of \\(E_k\\) among all binary trees with n leaves. In a similar fashion to what has been done in [20], it may be possible to prove these conjectures using flag algebras. In [21], Dossou-Olory and Wagner make the hypothesis that some trees may have an irrational inducibility, following on a question stated in [3]. As we mentioned in Section 5.1, they consider the tree as a possible tree with irrational inducibility. If the inducibility of this tree is indeed irrational, proving it could be possible using our framework, and would answer this open question." }, { "section": "Concluding remarks and open problems", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "Further investigation on the tree-profiles. There is still a lot to be discovered about tree-profiles. Our next goal would be to compute the exact description of the tree-profiles in Figure 13. This means, on the one hand proving Conjecture 1, and on the other hand determining the lower boundary which seems, as shown before, much more challenging to approach. Studying other tree-profiles in detail would also be of great interest and could lead us to learn more about the behavior and characteristics and the densities of these trees. In particular, we notice in Figure 23 in Appendix C a sharp angle present on the x-axis at every level of the hierarchy above 1. This leads us to think that we could obtain an exact Turán number of excluding ." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02306-0", "title": "Efficient branching rules for optimizing range and order-based objective functions", "authors": [ "Bart van Rossum", "Rui Chen", "Andrea Lodi" ], "published_online": "2025-12-16", "volume": "216", "issue": "1-2", "pages": "539-572", "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02306-0/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02312-2", "title": "An FPTAS for Connectivity Interdiction", "authors": [ "Chien-Chung Huang", "Nidia Obscura Acosta", "Sorrachai Yingchareonthawornchai" ], "published_online": "2025-12-16", "volume": "216", "issue": "1-2", "pages": "605-626", "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02312-2/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [ { "section": "Introduction", "element": "p", "matched_phrases": [ "open_question" ], "passage": "In [1] Zenklusen has posed the open question whether it is possible to obtain an FPTAS with arbitrary edge costs." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02318-w", "title": "A fixed-point algorithm for matrix projections with applications in quantum information", "authors": [ "Shrigyan Brahmachari", "Roberto Rubboli", "Marco Tomamichel" ], "published_online": "2025-12-16", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02318-w/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02311-3", "title": "Exact approaches for convex adjustable robust optimization", "authors": [ "Henri Lefebvre", "Enrico Malaguti", "Michele Monaci" ], "published_online": "2025-12-17", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02311-3/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02316-y", "title": "On the partial convexification for low-rank spectral optimization: rank bounds and algorithms", "authors": [ "Yongchun Li", "Weijun Xie" ], "published_online": "2025-12-22", "volume": "216", "issue": "1-2", "pages": "651-708", "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02316-y/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02300-6", "title": "Approximation Algorithms for the Weighted Nash Social Welfare via Convex and Non-Convex Programs", "authors": [ "Adam Brown", "Aditi Laddha", "Madhusudhan Pittu", "Mohit Singh" ], "published_online": "2025-12-23", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02300-6/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Introduction", "element": "p", "matched_phrases": [ "open_problem" ], "passage": "The proof of Theorem 1 follows straightforwardly from applying convex duality and making a couple changes of variables. We defer the proof to Appendix 6.1 for those who are interested in the details. In Section 2 we show that the generalized mathematical programs are relaxations for the weighted Nash Social Welfare problem. A summary of the approximation algorithm and its analysis is given in Section 3. This should make it clear why we have arrived at the given approximation factor. Most of the technical work is contain in Section 4, in which we analyse the rounding algorithm on tree solutions. We end with conclusions and a few open problems in Section 5." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02308-y", "title": "Exponential convergence rates for momentum stochastic gradient descent in the overparametrized setting", "authors": [ "Benjamin Gess", "Sebastian Kassing" ], "published_online": "2026-01-07", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02308-y/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Introduction", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "Empirical observations [4,5,6] motivate the long-standing conjecture that including momentum improves the performance of stochastic optimization algorithms. In recent years, a large class of optimization algorithms has been proposed using combinations of various variants of momentum with other techniques such as adaptive step-sizes, preconditioning and batch-normalization [7,8,9,10]. However, theoretical understanding of these methods remains limited. Existing convergence guarantees typically apply only to deterministic or continuous-time dynamics [11,12,13,14], or to deterministic settings with strongly convex objectives [11, 15]. For stochastic momentum methods, most theoretical results are qualitative [5, 16], or focus on convex optimization with classical noise assumptions [5, 6]." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02314-0", "title": "Operator convexity along lines, self-concordance, and sandwiched Rényi entropies", "authors": [ "Kerry He", "James Saunderson", "Hamza Fawzi" ], "published_online": "2026-01-08", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02314-0/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Conclusion", "element": "p", "matched_phrases": [ "open_question" ], "passage": "Whether there exists a self-concordant barrier for the epigraph of \\(\\varPsi _\\alpha \\) for \\(\\alpha \\in (2, \\infty )\\) with optimal barrier parameter \\(1+2n\\) remains an open question. Since \\(x^\\alpha \\) for \\(\\alpha \\in (2,\\infty )\\) is not operator convex [23] it follows that \\(\\varPsi _\\alpha \\) is not operator convex along lines for \\(\\alpha \\in (2,\\infty )\\). Therefore we cannot appeal to Theorem 1 to construct a self-concordant barrier for the epigraph of \\(\\varPsi _\\alpha \\) for \\(\\alpha \\in (2,\\infty )\\). An alternative approach could be to directly try to establish compatibility properties of \\(\\varPsi _{\\alpha }\\) for \\(\\alpha \\in (2,\\infty )\\). For the scalar case, we can prove the following compatibility result." }, { "section": "Conclusion", "element": "h3", "matched_phrases": [ "conjecture" ], "passage": "Conjecture 1" }, { "section": "Conclusion", "element": "p", "matched_phrases": [ "open_question" ], "passage": "It remains an open question how to construct self-concordant barriers for the remaining scenarios, i.e., the subset of (ii) where \\(-1\\le q \\le p \\le 0\\) and \\(s>-\\frac{1}{p+q}\\), and scenario (iii). We note that scenario (iii) contains the \\(\\alpha \\in [1,\\infty )\\) range as a special case, and therefore a subset of this range may be amenable to a generalization of our approach in Section 4.2." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02320-2", "title": "A single-loop stochastic feasible interior-point algorithm for nonlinear inequality-constrained optimization", "authors": [ "Frank E. Curtis", "Xin Jiang", "Qi Wang" ], "published_online": "2026-01-08", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02320-2/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [ { "section": "Introduction", "element": "p", "matched_phrases": [ "future_work" ], "passage": "Most of the assumptions that are required for our convergence guarantees are standard in the literature on interior-point algorithms, specifically for that on so-called feasible interior-point methods that maintain feasibility (with respect to all of the constraints) at all algorithm iterates. However, we require one additional assumption that is not standard. This assumption essentially requires that, when an algorithm iterate is close to a constraint boundary, a direction of sufficient descent can be computed that moves sufficiently away from the boundary. It is, in essence, an assumption that combines two aspects: (a) a type of nondegeneracy assumption and (b) an assumption that the barrier parameter is initialized to be sufficiently large compared to a neighborhood parameter that is introduced in our algorithm. We demonstrate these aspects through some illustrative examples. Since, to the best of our knowledge, there are no other interior-point methods in the literature that possess convergence guarantees in a stochastic setting that is comparable to ours, we contend that our algorithm and analysis together constitute a notable contribution to the literature despite our method being a feasible interior-point method and our need to introduce a nonstandard assumption. Future research may reveal techniques that can offer convergence guarantees for a stochastic-gradient-based interior-point method in other settings." }, { "section": "Algorithm", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "Upon computation of the search direction, Algorithm 1 chooses a step size \\(\\alpha _k \\in \\mathbb {R}_{>0}\\). For simplicity, our analyses for the deterministic and stochastic settings consider conservative rules for selecting the step sizes that each depend on the barrier- and neighborhood-parameter sequences, \\(\\{\\mu _k\\}\\) and \\(\\{\\theta _k\\}\\). We conjecture that it would be possible to generalize our algorithm and analyses to allow more flexibility in the step-size selection rules, as is done in [8]. However, for our purposes here, we present conservative strategies that are sufficient for proving convergence." }, { "section": "Numerical experiments", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "Figure 2 shows a histogram of relative stationarity values over our test set of problems from the CUTEst collection. The results show that, generally speaking, the relative stationarity values were quite small, meaning that the final iterates in most runs appeared to be much closer to stationarity than the initial iterates, thus indicating that within the iteration budget the algorithm made substantial progress toward stationarity. We conjecture that the problems on which the relative stationarity value was higher may be ones that are very nonlinear and/or have constraints that are degenerate near the iterates generated by the algorithm." }, { "section": "Conclusion", "element": "p", "matched_phrases": [ "open_question" ], "passage": "A few significant open questions remain in the study of stochastic-gradient-based interior-point methods for solving nonlinearly constrained optimization problems. For example, our theoretical convergence guarantees requires that the ratio \\(\\mu _1/\\theta _0\\) be sufficiently large and that each computed search direction satisfies the conditions in Assumptions 2 and/ or 5, but, as we have explained throughout the paper, a computationally effective strategy for computing search directions that adhere to our theoretical guarantees is not straightforward to design. A related open question is whether it is possible to design a single-loop infeasible interior-point framework that possesses theoretical convergence guarantees that are at least on par with those that we offer for the algorithm proposed in this paper. The main challenge in the design of such an approach is that our theoretical guarantees, which employ a prescribed sequence \\(\\{\\mu _k\\} \\searrow 0\\), rely heavily on the fact that the iterates generated by our algorithm are feasible in every iteration. There exist stochastic-gradient-based infeasible Newton-type methods for solving equality-constrained optimization problems; see, e.g., [1,2,3,4, 9,10,11, 15, 16, 20]. Hence, one might expect it to be possible to employ such methods for solving the equality-constrained subproblems that arise in an infeasible interior-point method. However, even if one were to introduce slack variables for which the barrier functions are introduced, it remains unclear how to merge such an approach with our strategy for ensuring feasibility at all iterates. Concretely, suppose that inequality constraints \\(c(x) \\le 0\\) are reformulated with slack variables as \\(c(x) + s = 0\\) along with \\(s \\ge 0\\), where the latter inequalities are handled with a barrier function. Attempting to follow the strategy for bound-constrained optimization in [8], one is confronted with the challenge that the search direction in the slack variables, call it \\(d_k^s\\), depends on the search direction in the original variables, say \\(d_k^x\\), where it may be possible that \\(c(x_k) \\not \\le 0\\). It is difficult to follow a strategy such as ours to ensure, e.g., that there exists \\(\\gamma \\) uniformly bounded below such that \\(s_k + \\gamma \\alpha _k d_k^s\\) is sufficiently positive, say to stay within a prescribed neighborhood. The difficulties of designing such an approach led us to propose the feasible method that has been presented in this paper, although it is possible that such an approach, or one based on alternative strategies, could be designed with convergence guarantees." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02309-x", "title": "A novel catalyst scheme for stochastic minimax optimization", "authors": [ "Guanghui Lan", "Yan Li" ], "published_online": "2026-01-12", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02309-x/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02319-9", "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" ], "published_online": "2026-01-12", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02319-9/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [ { "section": "Regularity assumptions of proposition 1", "element": "p", "matched_phrases": [ "open_problem" ], "passage": "Metric subregularity is central to the analysis of implicit functions for solution mappings (see [9, 15]) and it is often invoked either implicitly or explicitly by assumption. In [19, Theorem 3.15] linear metric subregularity was shown to be necessary and sufficient for R-linear convergence in expectation of random function iterations for consistent stochastic feasibility. This result has been extended to nonlinear spaces with nonlinear gauges in [25, Theorem 16]. It is an open problem whether metric subregularity is necessary in the present setting, though we see no reason why it should not be." }, { "section": "Case Study: likelihood maximization and X-FEL imaging", "element": "p", "matched_phrases": [ "open_question", "future_work" ], "passage": "Characterization of the invariant distributions is a topic of future research, but intuitively it should be clear that the more observations \\(\\{y_1, \\dots , y_M\\}\\) and constraints that can be brought into the objective function (48), the better chance that the set of fixed points of Algorithm 1 will contain only meaningful critical points of (48). The reasoning is as follows: if there is no data, then any feasible parameters \\(x\\in C_0\\) will solve (48). As the number of observations grows, so too grow the number of functions \\(g_j\\) in the log-likelihood objective, and hence the set of critical points can only shrink. If the set of critical points of (48) consists only of globally optimal solutions, the invariant measures of the randomized algorithm (1) should be supported on these solutions. It is an open question to determine exactly when and in what way this holds." }, { "section": "Concluding remarks and open questions", "element": "p", "matched_phrases": [ "future_work" ], "passage": "Several issues and directions for future research were mentioned above. To conclude, we summarize these and add to this list. In the context of X-FEL imaging, the directions for further study include: improving the numerical model without increasing the computational complexity (too much); exploring more realistic parameterizations of the electron density; including constraints into the model (48); and testing this approach on experimental data. Regarding (54c), it was mentioned that a solution \\(x^*\\) can be used to assign a probability distribution to the rotations of each outcome \\(y_j\\), which can then be used for the weighted integral in (54c). This is appealing both from the perspective of the application as well as posing interesting mathematical challenges for the analysis of such an approach. What is not clear is how many scattering events in each image are required to produce weights on the orientations that are significantly different than a uniform weighting." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02281-6", "title": "Wasserstein distributionally robust estimation in high dimensions: performance analysis and optimal hyperparameter tuning", "authors": [ "Liviu Aolaritei", "Soroosh Shafiee", "Florian Dörfler" ], "published_online": "2026-01-15", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02281-6/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02287-0", "title": "Complexity of trust-region methods in the presence of unbounded Hessian approximations", "authors": [ "Youssef Diouane", "Mohamed Laghdaf Habiboullah", "Dominique Orban" ], "published_online": "2026-01-19", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02287-0/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02321-1", "title": "On the connected (sub)partition polytope", "authors": [ "Phablo F. S. Moura", "Hande Yaman", "Roel Leus" ], "published_online": "2026-01-19", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02321-1/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02304-2", "title": "Complexity of trust-region methods with potentially unbounded Hessian approximations for smooth and nonsmooth optimization", "authors": [ "Geoffroy Leconte", "Dominique Orban" ], "published_online": "2026-01-26", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02304-2/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Abstract", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "We develop a worst-case evaluation complexity bound for trust-region methods in the presence of unbounded Hessian approximations. We use the algorithm of Aravkin et al. (SIAM J Optim 32(2):900–929, 2022) as a model, which is designed for nonsmooth regularized problems, but applies to unconstrained smooth problems as a special case. Our analysis assumes that the growth of the Hessian approximation is controlled by the number of successful iterations. We show that the best known complexity bound of \\(\\epsilon ^{-2}\\) deteriorates to \\(\\epsilon ^{-2/(1-p)}\\), where \\(0 \\leqslant p < 1\\) is a parameter that controls the growth of the Hessian approximation. The faster the Hessian approximation grows, the more the bound deteriorates. We construct an objective that satisfies all of our assumptions and for which our complexity bound is attained, which establishes that our bound is sharp. To the best of our knowledge, our complexity result is the first to consider potentially unbounded Hessians and is a first step towards addressing a conjecture of Powell (IMA J Numer Anal 30(1):289–301, 2010) that trust-region methods may require an exponential number of iterations in such a case. Numerical experiments conducted in double precision arithmetic are consistent with the analysis." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-026-02327-3", "title": "Weakly polynomial-time algorithms to minimize 2/3-submodular functions", "authors": [ "Ryuhei Mizutani", "Yuki Yoshida" ], "published_online": "2026-01-26", "volume": "216", "issue": "1-2", "pages": "709-756", "publisher_url": "https://link.springer.com/article/10.1007/s10107-026-02327-3/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02322-0", "title": "A Lyapunov analysis of Korpelevich’s extragradient method with fast and flexible extensions", "authors": [ "Manu Upadhyaya", "Puya Latafat", "Pontus Giselsson" ], "published_online": "2026-01-28", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02322-0/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Introduction", "element": "p", "matched_phrases": [ "future_work" ], "passage": "In Sect. 2, we formally introduce the new Lyapunov analysis for Korpelevich’s extragradient method. Building on this framework, we present the three new algorithms in Sect. 3 and establish their global convergence under suitable assumptions. In Sect. 4, we provide detailed proofs of the results from the preceding section. Next, Sect. 5 focuses on superlinear convergence, including corresponding proofs for the proposed algorithms. Numerical experiments appear in Sect. 6, and we conclude in Sect. 7 with a summary of key findings and directions for future research. Finally, Sect. Appendix A offers background material on Korpelevich’s extragradient method, Sect. Appendix B presents the counterexamples mentioned earlier, and Sect. Appendix C contains a comparison to recent last-iterate convergence results for the extragradient method." }, { "section": "Conclusions", "element": "p", "matched_phrases": [ "future_work" ], "passage": "Future research directions include developing solution-independent Lyapunov functions for other related methods. In particular, the forward-reflected-backward method of Malitsky and Tam [41] is a relevant case. Relatedly, a solution-independent Lyapunov function is already available for Malitsky’s projected reflected gradient method [42] in [43, Section D]. Another promising direction is to broaden the scope of the analysis beyond the monotone setting to include cohypomonotone operators [44], and the more general class of problems characterized by the weak Minty condition [45,46,47]. Additionally, further exploration is warranted to adapt the approach to the mirror prox framework [48]." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-026-02325-5", "title": "Exponential Lower Bounds for Many Pivot Rules for the Simplex Method", "authors": [ "Alexander E. Black" ], "published_online": "2026-01-28", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-026-02325-5/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [ { "section": "Abstract", "element": "p", "matched_phrases": [ "open_problem" ], "passage": "The existence of a pivot rule for the simplex method that guarantees a polynomial run-time is a longstanding, fundamental open problem in the theory of linear programming. The most popular pivot rule for theoretical analysis is the shadow pivot rule, which solves a linear program by projecting the feasible region onto a polygon. It has been shown to perform in expected polynomial time on uniformly random instances and in smoothed analysis. In practice, the pivot rule of choice is the steepest edge rule, which normalizes the set of improving neighbors and then chooses a maximally improving normalized neighbor. Exponential lower bounds are known for both rules in worst-case analysis. However, for the shadow simplex method, all exponential examples were only proven for one choice of projection, and for the steepest edge rule, the lower bounds were only proven for the Euclidean norm. In this work, we construct linear programs for which any choice of projection for shadow rule variants will lead to an exponential run-time and exponential examples for any choice of norm for a steepest edge variant." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-026-02329-1", "title": "Bounding the optimal number of policies for robust K-Adaptability", "authors": [ "Jannis Kurtz" ], "published_online": "2026-01-29", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-026-02329-1/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Conclusion", "element": "p", "matched_phrases": [ "open_question" ], "passage": "There remain several open questions to be tackled in the future. First, it would be interesting if we can achieve better values for R by focusing on certain applications and the corresponding problem structures. Furthermore, it would be interesting if more general approximation bounds as for objective uncertainty could also be derived for the constraint uncertainty case. Finally, an important question is if the methodology derived in this work can contribute to the development of efficient solution methods." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-026-02331-7", "title": "A space-decoupling framework for optimization on bounded-rank matrices with orthogonally invariant constraints", "authors": [ "Yan Yang", "Bin Gao", "Ya-xiang Yuan" ], "published_online": "2026-02-04", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-026-02331-7/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-026-02328-2", "title": "Finding almost tight witness trees", "authors": [ "Dylan Hyatt-Denesik", "Afrouz Jabal Ameli", "Laura Sanitá" ], "published_online": "2026-02-05", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-026-02328-2/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Introduction", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "As an additional result, we also improve the approximation bound for Steiner Tree in the special case of Steiner-claw free instances. A Steiner-Claw Free instance is a Steiner-Tree instance where the subgraph \\(G[V\\setminus R]\\) induced by the Steiner nodes is claw-free (i.e., every node has degree at most 2). These instances were introduced in [20] in the context of studying the integrality gap of a famous LP relaxation for Steiner Tree, called the bidirected cut relaxation, that had long been conjectured to have integrality gap strictly smaller than 2, which has recently been shown to be true, see [21]." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-026-02330-8", "title": "Correction: McCormick envelopes in mixed-integer PDE-constrained optimization", "authors": [ "Sven Leyffer", "Paul Manns" ], "published_online": "2026-02-17", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-026-02330-8/fulltext.html", "article_kind": "non_research_item", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-026-02324-6", "title": "Complexity of polytope diameters via perfect matchings", "authors": [ "Christian Nöbel", "Raphael Steiner" ], "published_online": "2026-02-19", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-026-02324-6/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Abstract", "element": "p", "matched_phrases": [ "open_problem" ], "passage": "The (monotone) diameter of a polytope is a fundamental parameter with important connections to the efficiency of the simplex method. Despite the central role played by this parameter in discrete and linear optimization, determining the precise complexity of computing the diameter of an input polytope remains a long-standing open problem. In 1994 Frieze and Teng (Comput. Complex. 4(3), 207–219 (1994)) proved the first cornerstone result in this direction by establishing that computing the diameter of an input polytope is weakly \\(\\textsf{NP}\\)-hard. In a recent breakthrough-paper, Sanità (FOCS, IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS). pp. 910–921. (2018)) studied the diameter of a special class of graph-based polytopes, known as fractional matching polytopes, and showed that determining their diameters is \\(\\textsf{NP}\\)-hard, thus establishing strong \\(\\textsf{NP}\\)-hardness of computing the diameter of polytopes. As our first main result, we show that computing the diameter of perfect matching polytopes (of bipartite graphs) is \\(\\textsf{NP}\\)-hard, giving an alternative, short proof for the strong \\(\\textsf{NP}\\)-hardness of polytope diameters. In particular this shows that computing the diameter is \\(\\textsf{NP}\\)-hard even when restricting to 0/1-polytopes with totally unimodular constraint matrices. In our second main result, we give a precise graph-theoretic description of the monotone diameter of perfect matching polytopes and use this description to prove the novel result that computing the monotone diameter of an input polytope is strongly \\(\\textsf{NP}\\)-hard. As a consequence of these results, we solve an open problem posed and reiterated by Sanità (2020), Diameter of Polytopes: Algorithmic and Combinatorial Aspects. (2020); Kafer PhD thesis. University of Waterloo, (2022); and Borgwardt, Grewe, Kafer, Lee and Sanità On the Hardness of Short and Sign-Compatible Circuit Walks. (2024) by proving the strong \\(\\textsf{NP}\\)-hardness of computing the so-called circuit diameter of polytopes. The latter is a well-studied polytope invariant that has attracted a lot of attention in the last decade due to its connection with circuit-augmentation schemes for linear optimization. Finally, complementing these negative complexity results, we also present a positive algorithmic result for diameters of perfect matching polytopes: we prove that for every fixed integer k, the problem of deciding whether the perfect matching polytope of a bipartite graph has monotone diameter at most k can be solved in polynomial time. In other words, there is an XP-algorithm for computing the monotone diameter of bipartite perfect matching polytopes." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-026-02333-5", "title": "A Hessian-aware stochastic differential equation for modelling SGD", "authors": [ "Xiang Li", "Zebang Shen", "Liang Zhang", "Niao He" ], "published_online": "2026-02-23", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-026-02333-5/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-026-02336-2", "title": "A Reliability Theory of Compromise Decisions for Large-Scale Stochastic Programs", "authors": [ "Shuotao Diao", "Suvrajeet Sen" ], "published_online": "2026-03-02", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-026-02336-2/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Algorithms for Compromise Decision Problems", "element": "p", "matched_phrases": [ "unresolved" ], "passage": "In the previous section, we have discussed the reliability of the compromise decision using SAA in the replication step. There are two issues which remain unresolved: (1) How should we solve the SAA problems in the replication step? (2) How should we simplify the aggregated objective function estimate without losing reliability?" }, { "section": "Conclusion", "element": "p", "matched_phrases": [ "open_question" ], "passage": "There remain several open questions about extensions of the compromise decision framework. One open question is how to tighten the upper bounds on the reliability of the compromise decisions, especially the bounds of the variance of the pessimistic distance and the bounds of the reliability of cutting plane method/SAA for solving two-stage SQQP problems. Another open question is to design an optimal strategy of choosing (m, n) for a given total sample size N. A third open question is whether it is possible to connect our decision-based regularizer to a class of Distributionally Robust Optimization (DRO) problems. In connection with this question, we note that a study of DRO and robust statistics (Blanchet et al. [7]), reveals that \\(\\phi \\)-Divergence-Based DRO is related to the variance regularization problem and optimal transport-based DRO is related to norm regularization of the fitted parameters. In the context of SP, we observe that our decision-based regularizer shares some properties which are similar to norm regularization. Finally, a fourth open question is about how one might reformulate compromise decision problems for nonconvex SP problems and problems with constraint sets that are specified by nonlinear equations/inequalities such as those arising in Neural Networks (NN). The latter have a reputation for being reasonably good on average, and yet, they can be unreliable (e.g., resulting in so-called “hallucinations”). Perhaps, a study of reliability of solutions for non-convex stochastic optimization models will lead to greater reliability in algorithms for NN. In connection with these directions, one could also investigate the impact of different regularizers (e.g., kernel methods [54]) for compromise decisions." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-026-02334-4", "title": "Stochastic and incremental subgradient methods for convex optimization on Hadamard spaces", "authors": [ "Ariel Goodwin", "Adrian S. Lewis", "Genaro López-Acedo", "Adriana Nicolae" ], "published_online": "2026-03-04", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-026-02334-4/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-026-02337-1", "title": "Distributionally robust optimization with multimodal decision-dependent ambiguity sets", "authors": [ "Xian Yu", "Beste Basciftci" ], "published_online": "2026-03-09", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-026-02337-1/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [ { "section": "Solution Algorithm", "element": "p", "matched_phrases": [ "future_work" ], "passage": "We note that the master problem can be solved to optimality in many practical applications, such as the facility location problem studied in Section 7.1, where the master problem has a MILP reformulation and the separation problem is a linear program as support and first moment information are utilized to construct the corresponding ambiguity set. Our results in Section 7.1.6 over a polyhedral support set (see Table 7) demonstrate convergence to the optimal solution along with significant computational speed-ups over various instances. We note that our algorithm is also valid when a discrete support set is considered for each mode \\(l=1,\\cdots ,L\\), by solving the separation problem under each realization of the random variable \\(\\varvec{\\xi }\\), obtaining their optimum objective function values, selecting the maximum of these values as \\(\\Psi _l(\\varvec{y}^*,\\underline{\\varvec{\\beta }}_l^*,\\bar{\\varvec{\\beta }}_l^*)\\) and identifying the corresponding realization as \\(\\varvec{\\xi }^*\\) to be used in the subsequent cut generation process. We further note that as a potential future research direction, alternative decomposition algorithms can be extended to our setting by leveraging algorithms that solve general semi-infinite optimization problems, which can provide solutions with certain feasibility and optimality guarantees under specific assumptions [22, 26]." }, { "section": "Conclusion", "element": "p", "matched_phrases": [ "future_work" ], "passage": "Overall, our paper proposes a novel framework to address both multimodalities and decision-dependent uncertainties within a DRO problem while considering various forms of ambiguity sets, providing computationally tractable reformulations, and demonstrating its performance both analytically and computationally. As a future research direction, our proposed framework can be applied to various applications and extended by addressing potential non-convexities arising under specific forms of problem settings and ambiguity sets." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-026-02339-z", "title": "Distributional stability of sparse inverse covariance matrix estimators", "authors": [ "Renjie Chen", "Huifu Xu", "Henryk Zähle" ], "published_online": "2026-03-09", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-026-02339-z/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Introduction", "element": "p", "matched_phrases": [ "open_question", "future_work" ], "passage": "The rest of this article is organized as follows. In Section 2, we first introduce some basic notation. In Section 3, we introduce a criterion for distributional stability of general point estimators, which we use in Section 5 to prove our main results. For one of our main results, we also need information on the optimization problem underlying the estimator \\({\\widehat{S}}_N\\), which we provide in Section 4. Some applications and numerical experiments can be found in Section 6 (and Appendix D). All the proofs are given in Appendix B, some of which rely on the auxiliary results in Appendix A. Some open questions for future research are listed in Section 7." }, { "section": "Concluding remarks", "element": "p", "matched_phrases": [ "open_question", "we_leave", "future_work" ], "passage": "There are still a number of open questions that remain to be addressed. For instance, when in the penalty term in (1) the norm \\(\\Vert \\cdot \\Vert _1\\) is replaced by a different norm such as \\(\\Vert \\cdot \\Vert _{1-}\\), we are unable to assert Lipschitz continuity of the minimizer of the optimization problem underlying the estimator (in the spirit of (18)) and subsequently unable to obtain distributional stability (in the spirit of (6)). Likewise, we are unable to assert distributional stability of the empirical estimators of various ratios of interest in the context of portfolio theory, including the Sharpe ratio, the Omega ratio and the Rachev ratio. To address the issue, one may consider qualitative statistical robustness, which does not require the Lipschitz continuity, see [25, 31]. In that case, we can establish a weaker version of distributional stability of the statistical estimators where the distance between statistical estimators based on contaminated data and true data is implicitly related to the distance between the original probability distributions generating the data sets. We leave all these to future research." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-026-02340-6", "title": "Sparse approximation in lattices and semigroups", "authors": [ "Stefan Kuhlmann", "Timm Oertel", "Robert Weismantel" ], "published_online": "2026-03-09", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-026-02340-6/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-026-02348-y", "title": "An Improved Pseudopolynomial Time Algorithm for Subset Sum", "authors": [ "Lin Chen", "Jiayi Lian", "Yuchen Mao", "Guochuan Zhang" ], "published_online": "2026-03-10", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-026-02348-y/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Abstract", "element": "p", "matched_phrases": [ "open_question" ], "passage": "We investigate pseudo-polynomial time algorithms for Subset Sum. Given a multi-set X consisting of n positive integers and a target t, Subset Sum asks whether some subset of X sums to t. Bringmann proposed an \\(\\widetilde{O}(n + t)\\)-time algorithm [Bringmann SODA’17]. An open question has naturally arisen: can Subset Sum be solved in \\(O(n + w)\\) time? Here w is the largest integer in X. We make progress towards resolving the open question by proposing an \\(\\widetilde{O}(n + \\sqrt{wt})\\)-time algorithm." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-026-02345-1", "title": "Stochastic Halpern iteration in normed spaces and applications to reinforcement learning", "authors": [ "Mario Bravo", "Juan Pablo Contreras" ], "published_online": "2026-03-16", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-026-02345-1/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-026-02350-4", "title": "Kantorovich and Zalgaller (1951): the 0-th column generation algorithm", "authors": [ "Eduardo Uchoa", "Ruslan Sadykov" ], "published_online": "2026-03-16", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-026-02350-4/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-026-02332-6", "title": "Concrete convergence rates for common fixed point problems under Karamata regularity", "authors": [ "Tianxiang Liu", "Bruno F. Lourenço" ], "published_online": "2026-03-17", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-026-02332-6/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-026-02341-5", "title": "(Near)-Optimal algorithms for sparse separable convex integer programs", "authors": [ "Christoph Hunkenschröder", "Martin Koutecký", "Asaf Levin", "Tung Anh Vu" ], "published_online": "2026-03-18", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-026-02341-5/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [ { "section": "Abstract", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "We study the general integer programming (IP) problem of optimizing a separable convex function over the integer points of a polytope: \\(\\min \\{f({\\textbf {x}}) \\mid A{\\textbf {x}}= {\\textbf {b}}, \\, {\\textbf {l}}\\le {\\textbf {x}}\\le {\\textbf {u}}, \\, {\\textbf {x}}\\in \\mathbb {Z}^n\\}\\). The number of variables n is a variable part of the input, and we consider the regime where the constraint matrix A has small coefficients \\(\\Vert A\\Vert _\\infty \\) and small primal or dual treedepth \\({{\\,\\mathrm{\\textrm{td}}\\,}}_P(A)\\) or \\({{\\,\\mathrm{\\textrm{td}}\\,}}_D(A)\\), respectively. Equivalently, we consider block-structured matrices, in particular n-fold, tree-fold, 2-stage and multi-stage matrices.We ask about the possibility of near-linear time algorithms in the general case of (non-linear) separable convex functions. The techniques of previous works for the linear case are inherently limited to it; in fact, no strongly-polynomial algorithm may exist due to a simple unconditional information-theoretic lower bound of \\(n \\log \\Vert {\\textbf {u}}-{\\textbf {l}}\\Vert _\\infty \\), where \\({\\textbf {l}}, {\\textbf {u}}\\) are the vectors of lower and upper bounds. Our first result is that with parameters \\({{\\,\\mathrm{\\textrm{td}}\\,}}_P(A)\\) and \\(\\Vert A\\Vert _\\infty \\), this lower bound can be matched (up to dependency on the parameters). Second, with parameters \\({{\\,\\mathrm{\\textrm{td}}\\,}}_D(A)\\) and \\(\\Vert A\\Vert _\\infty \\), the situation is more involved, and we design an algorithm with time complexity \\(g({{\\,\\mathrm{\\textrm{td}}\\,}}_D(A), \\Vert A\\Vert _\\infty ) n \\log (n) \\log \\Vert {\\textbf {u}}-{\\textbf {l}}\\Vert _\\infty \\) where \\(g\\) is some computable function. We conjecture that a stronger lower bound is possible in this regime, and our algorithm is in fact optimal. Our algorithms combine ideas from scaling, proximity, and sensitivity of integer programs, together with a new dynamic data structure." }, { "section": "Dual algorithm", "element": "h3", "matched_phrases": [ "open_problem" ], "passage": "6.1 Open problems" } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-026-02349-x", "title": "On the number of degenerate simplex pivots", "authors": [ "Kirill Kukharenko", "Laura Sanità" ], "published_online": "2026-03-23", "volume": "216", "issue": "1-2", "pages": "757-791", "publisher_url": "https://link.springer.com/article/10.1007/s10107-026-02349-x/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-026-02326-4", "title": "New finite relaxation hierarchies for concavo-convex, disjoint bilinear programs, and facial disjunctions", "authors": [ "Mohit Tawarmalani" ], "published_online": "2026-03-26", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-026-02326-4/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [ { "section": "Conclusions", "element": "p", "matched_phrases": [ "we_leave", "future_work" ], "passage": "While we leave the computational exploration of the hierarchies for different problem classes to future work, we discuss several challenges and opportunities briefly. Although the hierarchy terminates finitely, the first challenge is that the degree of the resulting expressions can grow rapidly (see Proposition 8). A successful implementation will require careful selection of inequalities to execute Algorithm 1 while balancing the associated tradeoffs. Acceleration techniques for double description may provide useful insights in this regard [15, 28]. Second, we showed that by processing just a few constraints, inequalities that are not seen in RLT relaxations can be derived. These ideas can be extended to analyze simple structured sets. As discussed in Corollary 1 and Remark 3 when constraints are processed in different orders simultaneously, the algebraic relationships between the barycentric coordinates provide opportunities for strengthening the relaxations just as RLT strengthens lift-and-project relaxations. However, the computational burden increases as well since the number of constraint orders grows exponentially with the number of constraints. The technique detailed in Theorem 4 can be used after projecting polytopes P and \\(P^y\\) to small dimensions to derive algebraic proofs of validity. Finally, related work [26, 27] has leveraged insights from this paper, and specifically the double-description procedure to design a simplicial branch-and-bound algorithm for disjoint bilinear programs that terminates finitely." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-026-02344-2", "title": "Unboundedness in Bilevel Optimization", "authors": [ "Bárbara Rodrigues", "Margarida Carvalho", "Miguel F. Anjos", "Nagisa Sugishita" ], "published_online": "2026-03-26", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-026-02344-2/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Introduction", "element": "p", "matched_phrases": [ "open_question" ], "passage": "To sum up, studying the conclusions that can be drawn about the bilevel problem when its relaxation is unbounded is relevant, though much of the literature so far has focused on other aspects of the field. In this paper, we address this gap and close important open questions with our main contribution being from a theoretical computational complexity perspective. Jeroslow’s ground-breaking work [4] establishes that the complexity of determining if a multilevel model admits an optimal solution rises one level in the polynomial hierarchy for each additional level added. Complementing this seminal work, we show the hardness of deciding unboundedness in multilevel problems. Additionally, we present two algorithmic approaches for dealing with unboundedness from a practical perspective." }, { "section": "Introduction", "element": "p", "matched_phrases": [ "future_work" ], "passage": "The remainder of this paper is organised as follows. In Section 2, we show that the decision problem of whether a linear bilevel problem is unbounded is strongly NP-complete, and draw some parallels to the pessimistic bilevel formulation. More generally, we also show that checking unboundedness of a linear multilevel problem with k levels is \\(\\Sigma ^{p}_{k-1}\\)-hard in Section 3. In this section, we also show the \\(\\Sigma ^{p}_{k}\\)-hardness of deciding unboundedness of a mixed-integer k-level problem. In Section 4, we detail two possible algorithmic approaches for checking whether a bilevel problem is unbounded and, if so, computing a certificate of unboundedness. We also depict the potential of these algorithms for some example instances of interest, and present a brief computational comparison. Finally, in Section 5, we summarise our contributions, and propose directions for future research." }, { "section": "Conclusion", "element": "p", "matched_phrases": [ "future_work" ], "passage": "Future research could focus on deriving the computational complexity of deciding unboundedness of mixed-integer pessimistic multilevel problems, or of multi-follower bilevel problems. A more exhaustive computational experiment comparing the performance of the two algorithmic approaches proposed is also warranted. While the LPCC approach requires solving an NP-hard problem, Algorithm 1 solves a series of linear optimization problems. However, if the bilevel problem (B) is bounded (and its HPR unbounded), we must enumerate all vertices of the lower-level dual feasible region. Thus, in this case, it would be worth saving the best bilevel feasible solution along Algorithm 1, so that we are also able to return an incumbent solution. Furthermore, developing heuristics to prioritize the enumeration of vertices of the dual feasible space that correspond to primal lower-level faces which are unbounded for the HPR may enhance the algorithm’s efficiency in proving unboundedness." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-026-02347-z", "title": "Totally equimodular matrices: decomposition and triangulation", "authors": [ "Patrick Chervet", "Roland Grappe", "Mathieu Vallée" ], "published_online": "2026-03-26", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-026-02347-z/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Abstract", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "Totally equimodular matrices generalize totally unimodular matrices and arise in the context of box-totally dual integral polyhedra. This work further explores the parallels between these two classes and introduces foundational building blocks for constructing totally equimodular matrices. Consequently, we present a decomposition theorem for totally equimodular matrices of full row rank. Building on this decomposition theorem, we prove that simplicial cones whose generators form the rows of a totally equimodular matrix satisfy strong integrality decomposition properties. More precisely, we provide the Hilbert basis for these cones and construct regular unimodular Hilbert triangulations in most cases. We conjecture that cases not covered here do not exist." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-026-02353-1", "title": "A constraint-based approach to function interpolation, with application to performance estimation for weakly convex optimization.", "authors": [ "Anne Rubbens", "Julien M. Hendrickx" ], "published_online": "2026-04-02", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-026-02353-1/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-026-02352-2", "title": "Optimal and parameter-free gradient minimization methods for convex and nonconvex optimization", "authors": [ "Guanghui Lan", "Yuyuan Ouyang", "Zhe Zhang" ], "published_online": "2026-04-07", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-026-02352-2/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Introduction", "element": "p", "matched_phrases": [ "unresolved" ], "passage": "In spite of much progresses on optimal gradient minimization methods for convex optimization, there remain a few important unresolved issues. First, OGM-G requires one to supply the total number of iterations N in advance. As a consequence, we do not actually utilize the easily verifiable condition \\(\\Vert \\nabla f(\\hat{x})\\Vert \\le \\varepsilon \\) to terminate the algorithm. Second, to achieve the optimal complexity we need to run two different algorithms consecutively, where the first algorithm computes an approximate solution \\(\\tilde{x}\\) such that \\(f(\\tilde{x}) - f(x^*)\\le \\varepsilon \\) and the second one uses \\(\\tilde{x}\\) as the initial point to compute an approximate solution \\(\\hat{x}\\) with \\(\\Vert \\nabla f(\\hat{x})\\Vert \\le \\varepsilon \\). This two-step approach, although technically sound, appears to lack intuitive interpretation. Third, the aforementioned studies focus on the general convex case only, and the best possible complexity bound remains unknown when f is \\(\\mu \\)-strongly convex. Specifically, by using the accelerated gradient method [23] we can guarantee \\(\\Vert \\nabla f(\\hat{x})\\Vert \\le \\varepsilon \\) within \\(\\mathcal{O} (1)\\sqrt{L/\\mu }\\log (L/(\\mu \\varepsilon ))\\) gradient evaluations. It is unclear whether one could remove the logarithmic dependence on the condition number \\(L/\\mu \\) to obtain an optimal \\(\\mathcal{O} (1) \\sqrt{L/\\mu }\\log (1/\\varepsilon )\\) complexity bound. Fourth, it remains unclear what is the best possible complexity bound for nonconvex problems. Some results have been developed in the literature (see, e.g., [7, 12, 15] and the references therein) and the best-known complexity is given by \\(\\mathcal{O} (1)\\sqrt{L l}(f(x_0) - f(x^*)) (1+\\log (L/l))/\\varepsilon ^2\\) [12]. Note that if \\(l = L\\) then this bound is known to be optimal [2]. In fact, the optimal bound in the latter special case can be simply achieved by the gradient descent method. However, it is unclear if the above complexity can be improved for the more general case when \\(l \\in (0, L]\\)." }, { "section": "Concluding remarks", "element": "p", "matched_phrases": [ "remains_open", "future_work" ], "passage": "Three potential questions are still open for future research. First, the key strategy in our paper is accumulative regularization, which requires us to solve a series of proximal mapping subproblems consecutively. It will be interesting to study whether the algorithm framework can be further simplified, e.g., to a single-loop implementation. Similar questions are also open in the parameter-free implementation: in the strongly convex case, we need to call a series of the AR function to compute an approximate solution. In the nonconvex case, we need to call a series of the SCAR function. It would be interesting to study whether the implementation of parameter-free algorithms can be simplified. Second, the parameters of our algorithms depend on the accuracy threshold \\(\\varepsilon \\) of desired approximate solution. While the value of \\(\\varepsilon \\) is necessary in determining the termination of algorithms, it will be interesting to study whether an algorithm could be developed in which the parameters do not depend on the value of \\(\\varepsilon \\). Assuming that the target accuracy is unknown, one straightforward strategy is to set the target accuracy to be 1/2 of the current gradient norm when calling our algorithms, and repeat this procedure multiple times until the desired accuracy is reached. This strategy will allow us to be more adaptive to the target accuracy without impacting the order of the total gradient complexity. However, it would be an interesting research topic to study whether a better implementation than the aforementioned straightforward strategy can be developed. Third, for practical purpose, our proposed algorithm should likely be tuned, for example, to allow more flexible selection of the parameter \\(\\sigma _s\\) and \\(N_s\\), early termination, or warm start among different calls to subroutine \\(\\mathcal{A} \\). Due to the length of our paper, we defer some of the above research questions for future studies." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-026-02354-0", "title": "Fast finite-sum optimization via cyclically-sampled Hessian averaging methods", "authors": [ "Thomas O’Leary-Roseberry", "Raghu Bollapragada" ], "published_online": "2026-04-07", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-026-02354-0/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Convergence analysis", "element": "p", "matched_phrases": [ "future_work" ], "passage": "The three phases described above are introduced for analytical clarity and are not explicitly detected in practice. Instead, an Armijo backtracking line search is employed, which automatically adapts the step size across the different phases of the algorithm. In particular, once the iterates enter the locally strongly convex region and the curvature approximation becomes sufficiently accurate, these strategies naturally accept a unit step size. While our numerical experiments use exact (and therefore relatively expensive) function evaluations in the line search, developing computationally efficient variants based on inexact function evaluations is part of our future work." }, { "section": "Conclusion", "element": "p", "matched_phrases": [ "future_work" ], "passage": "This work is focused on the theoretical benefits of (1) Hessian averaging, (2) adaptive gradient sampling and (3) cyclic sampling without replacement. In future work, we will investigate practical algorithms for the deployment of these methods to challenging, high-dimensional nonconvex optimization problems. We will additionally extend this to a stochastic setting, where results are proven in expectation." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-026-02356-y", "title": "Strongly-polynomial time and validation analysis of policy gradient methods", "authors": [ "Caleb Ju", "Guanghui Lan" ], "published_online": "2026-04-07", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-026-02356-y/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Conclusion", "element": "p", "matched_phrases": [ "future_work" ], "passage": "We provide new convergence guarantees and validation analysis for policy mirror descent (PMD). For the deterministic case, we introduce a novel step size that allows PMD to obtain the stronger distribution-free linear convergence. Moreover, by incorporating our proposed advantage gap function into the step size, we improve PMD so it achieves, for the first time, strongly polynomial runtime to get the optimal solution. For the stochastic setting, we show the stochastic PMD can also achieve the stronger distribution-free sublinear convergence, and it does so with the same constant step size as in previous developments [23]. We also pair this convergence analysis with a novel validation analysis, which can be used to possibly terminate policy gradient methods sooner. This extends the validation analysis for stochastic convex optimization [24] to the challenging nonconvex landscape of policy optimization [1]. An important future work can be to extend the advantage gap function and its analysis to more realistic general state and action spaces that appear in RL [17]." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-026-02351-3", "title": "Preface", "authors": [ "Jarosław Byrka", "Jens Vygen" ], "published_online": "2026-04-08", "volume": "216", "issue": "1-2", "pages": "1-2", "publisher_url": "https://link.springer.com/article/10.1007/s10107-026-02351-3/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-026-02342-4", "title": "Quantitative Indicators for Strength of Inequalities with Respect to a Polyhedron", "authors": [ "David M. Warme" ], "published_online": "2026-04-13", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-026-02342-4/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-026-02343-3", "title": "A Randomized Linearly Convergent Frank-Wolfe-type Method for Smooth Convex Minimization over the Spectrahedron", "authors": [ "Dan Garber" ], "published_online": "2026-04-13", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-026-02343-3/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Introduction", "element": "p", "matched_phrases": [ "remains_open" ], "passage": "One may wonder if in order to obtain our results it is indeed required, in case the optimal solution is not unique, that all of them satisfy strict complementarity (which mandates that all solutions have the same rank). In previous analyzes of Frank-Wolfe methods with strict complementarity conditions [23, 26, 27] it was assumed that the optimal solution is unique (e.g., by assuming f is strongly convex). Our analysis, which does not assume uniqueness, indeed requires that all solutions satisfy this condition. It remains open whether this is mandatory." }, { "section": "Introduction", "element": "p", "matched_phrases": [ "open_question", "future_work" ], "passage": "Section 2 presents our algorithm and its efficient implementation. Section 3 then formally presents and proves the convergence rates of our proposed algorithm. Section 4 presents some numerical experiments. Finally, Section 5 outlines some open questions for future research." }, { "section": "Numerical Experiments", "element": "p", "matched_phrases": [ "unknown_whether" ], "passage": "We begin by comparing our Algorithm 1 (ALG 1 in the figures below) to the standard Frank-Wolfe with line-search method (FW in the figures below). Recall that FW has a worst case convergence rate of O(1/t) [5]. In [23] it was established that under strict complementarity and in the special case \\(r^*=1\\), it enjoys a linear convergence rate, while it remained unknown if this is also the case for \\(r^*\\ge 2\\). Figure 1, which plots for each method the value \\(\\log ({f({\\textbf{X}}_t) - f(\\widehat{{\\textbf{X}}^*})})\\) vs the number of iteration t, shows that indeed in case \\(r^*=1\\), in both the LS and Huber settings, both FW and our Algorithm 1 clearly exhibit a linear convergence rate. However, once we increase \\(r^*\\), we see that while our Algorithm 1 maintains a linear convergence rate, FW converges only with a sub-linear rate. Additionally, in the No SC setting, we see that even for \\(r^*=1\\), FW converges sub-linearly, while our algorithm exhibits a linear convergence rate. This suggests that, similarly to the state-of-affairs in the case of optimization over polytopes [27], the standard FW does not exhibit linear convergence once the optimal solution is not an extreme point of the feasible set." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-026-02357-x", "title": "High Probability Complexity Bounds of Trust-Region Stochastic Sequential Quadratic Programming with Heavy-Tailed Noise", "authors": [ "Yuchen Fang", "Javad Lavaei", "Sen Na" ], "published_online": "2026-04-13", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-026-02357-x/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Probabilistic oracles with irreducible and heavy-tailed noise", "element": "p", "matched_phrases": [ "open_question" ], "passage": "\\(\\bullet \\) The heavy-tailed condition in (iii.1) significantly relaxes the sub-exponential condition in (iii.2) by reducing the requirement from the existence of infinite-order moments to a finite \\(1+\\delta \\) moment. Leveraging the Burkholder-type inequality [41, 42] and the martingale Fuk-Nagaev inequality [43,44,45], we establish high-probability complexity bounds for any \\(\\delta > 0\\). By Theorem 4.17 and its subsequent discussion, we know our high-probability complexity bounds directly imply that, for any \\(\\delta > 0\\), the algorithm finds a (first- or second-order) \\(\\epsilon \\)-stationary point in a finite number of iterations almost surely. If we suppress the irreducible noise by setting \\(\\epsilon _h=\\epsilon _g=\\epsilon _f=0\\), this result further implies a liminf-type almost-sure convergence; that is, for any run of the method, there exists a subsequence of iterates for which the (first- or second-order) stationarity residuals vanish (cf. Definition 4.6). We note that although liminf-type almost-sure convergence for second-order stationarity with heavy-tailed oracles is common in the literature [30, 33], the above asymptotic result implied from non-asymptotic analysis has a gap compared to the convergence results directly from asymptotic analyses in [28, Theorem 4.18] and [30, Theorem 4], where the authors established lim-type convergence to first-order stationarity without imposing any moment conditions on the zeroth-order oracle. That being said, those methods do not incorporate irreducible noise in all orders of the oracles, and have a substantially different convergence analysis based on an ad-hoc random function (e.g., [30, (20)]). Overall, it remains an open question whether one can strengthen our high-probability complexity bounds by completely removing the moment conditions in (ii) and (iii.1)." }, { "section": "Probabilistic oracles with irreducible and heavy-tailed noise", "element": "p", "matched_phrases": [ "open_question" ], "passage": "Despite these similarities, the aforementioned approaches differ from our method in two respects. First, clipping and normalization in SGD operate on gradient estimates, while our analysis requires analogous control on objective value estimates. Second, their methods rely crucially on unbiased gradient estimates, while our model includes irreducible noise in objective value (and gradient) estimates, preventing a direct adaptation of their techniques. Whether clipping- or normalization-type ideas can be effectively applied to objective value estimates remains an open question." }, { "section": "High-probability complexity bounds", "element": "p", "matched_phrases": [ "open_question" ], "passage": "\\(\\bullet \\) Due to the heavy-tailed noise, \\(P\\{T_{\\epsilon } \\le T-1\\}\\) no longer grows to 1 at an exponential rate as in [22, 40], but is instead reduced to a polynomial rate of \\(\\mathcal {O}(T^{-\\delta })\\) for \\(\\delta > 0\\) (cf. (36)). Despite this slower rate, we still obtain for any \\(\\delta >0\\), \\(P[T_{\\epsilon }=\\infty ]=P[\\cap _{T=0}^\\infty \\{T_\\epsilon> T\\}] = \\lim \\limits _{T\\rightarrow \\infty }P[T_{\\epsilon }>T]=0\\), where the second equality is due to the monotonicity of the events \\(\\{T_\\epsilon>T+1\\}\\subseteq \\{T_\\epsilon >T\\}\\). Thus, for any \\(\\delta >0\\), our algorithm reaches a (first- or second-order) \\(\\epsilon \\)-stationary point in finite time almost surely. If we suppress the irreducible noise by setting \\(\\epsilon _f=\\epsilon _g=\\epsilon _h=0\\), this result further yields a liminf-type almost-sure convergence: for any realization of the algorithm, there exists a subsequence of iterates converging to a (first- or second-order) stationary point. This liminf-type second-order convergence with heavy-tailed oracles matches existing literature [30, 33]; however, this asymptotic result implied from our non-asymptotic analysis still leaves a gap compared with the stronger lim-type first-order convergence established in [28, Theorem 4.18] and [30, Theorem 4], which requires no moment conditions on the objective value estimates. Closing this gap remains an open question." }, { "section": "High-probability complexity bounds", "element": "p", "matched_phrases": [ "future_work" ], "passage": "Reducing the above sample complexity of adaptive SSQP methods is still a challenging problem. Some helpful insights can be drawn from well-studied stochastic methods such as stochastic gradient descent (SGD). Numerous variance-reduction techniques have been proposed for SGD in nonconvex settings and have demonstrated significant improvements over vanilla SGD. For example, for a d-dimensional problem, Stochastic Variance-Reduced Gradient (SVRG) [70, 71] for finite-sum optimization achieves an improved sample complexity of \\(\\mathcal {O}(N + N^{2/3}\\epsilon ^{-2})\\); STOchastic Recursive Momentum (STORM) [72], designed for the online setting, attains \\(\\mathcal {O}(\\epsilon ^{-3})\\) sample complexity; and Stochastic Path-Integrated Differential EstimatoR (SPIDER) [73] unifies the study and achieves \\(\\mathcal {O}(\\epsilon ^{-3})\\) in the online setting and \\(\\mathcal {O}(N + N^{1/2}\\epsilon ^{-2})\\) in the finite-sum setting. In constrained stochastic optimization, [27] applies both truncated recursive momentum and truncated Polyak momentum for penalty methods, attaining \\(\\mathcal {O}(\\epsilon ^{-3})\\) sample complexity. [16] integrates SVRG into SSQP for finite-sum problems, attaining \\(\\mathcal {O}(N + N^{2/3}\\epsilon ^{-2})\\) sample complexity and matching the result in [71] for unconstrained optimization. However, the SVRG technique requires periodically computing a full gradient as a “checkpoint”, which presents challenges for directly adapting it to the online setting. We leave the development of variance-reduced trust-region SSQP methods for improving sample complexity to future research." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02317-x", "title": "Quantitative indicators for strength of inequalities with respect to a polyhedron, Part II: Applications and computational evidence", "authors": [ "David M. Warme" ], "published_online": "2026-04-20", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02317-x/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-026-02335-3", "title": "Metric entropy-free sample complexity bounds for sample average approximation in convex stochastic programming", "authors": [ "Hongcheng Liu", "Jindong Tong" ], "published_online": "2026-04-20", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-026-02335-3/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-026-02346-0", "title": "Distributionally robust optimization with decision-dependent information discovery", "authors": [ "Qing Jin", "Angelos Georghiou", "Phebe Vayanos", "Grani A. Hanasusanto" ], "published_online": "2026-04-20", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-026-02346-0/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-026-02360-2", "title": "Online matching on 3-uniform hypergraphs", "authors": [ "Sander Borst", "Danish Kashaev", "Zhuan Khye Koh" ], "published_online": "2026-04-27", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-026-02360-2/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Introduction", "element": "p", "matched_phrases": [ "future_work" ], "passage": "Since every randomized algorithm for integral matching induces a deterministic algorithm for fractional matching, the upper bound of \\((e-1)/(e+1)\\approx 0.4621\\) in Theorem 1 also applies to the integral problem on 3-uniform hypergraphs. However, the best known lower bound is 1/3, given by the greedy algorithm. An interesting question for future research is whether there exists an integral algorithm better than greedy on 3-uniform hypergraphs." }, { "section": "Concluding remarks", "element": "p", "matched_phrases": [ "open_question" ], "passage": "Our main contribution was a tight result for the online fractional matching problem under vertex arrivals for 3-uniform hypergraphs, with matching upper and lower bounds of \\((e-1)/(e+1) \\approx 0.46\\). The biggest open question raised by this work is the integral setting, where the best known algorithm is still the 1/3-competitive simple greedy algorithm. Possible directions to get an improvement would be to develop new rounding techniques of fractional algorithms on hypergraphs, or by trying to find an analogue of the randomized algorithm for bipartite graphs. This hypergraph setting adds additional difficulties to the online matching problem, for which new techniques seem to be required. Another interesting direction would be to study this problem in different online arrival models, for instance edge-arrivals." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-026-02362-0", "title": "Beyond minimax optimality: A subgame perfect gradient method", "authors": [ "Benjamin Grimmer", "Kevin Shu", "Alex L. Wang" ], "published_online": "2026-04-27", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-026-02362-0/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-026-02338-0", "title": "A Low-rank augmented lagrangian method for large-scale semidefinite programming based on a hybrid convex-nonconvex approach", "authors": [ "Renato D. C. Monteiro", "Arnesh Sujanani", "Diego Cifuentes" ], "published_online": "2026-04-29", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-026-02338-0/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-026-02355-z", "title": "Level proximal subdifferential, variational convexity, and pointwise quadratic approximation", "authors": [ "Honglin Luo", "Xianfu Wang", "Ziyuan Wang", "Xinmin Yang" ], "published_online": "2026-05-07", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-026-02355-z/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-026-02359-9", "title": "Forecasting outside the Box: application-driven optimal pointwise forecasts for a class of two-stage stochastic programs", "authors": [ "Tito Homem-de-Mello", "Juan Valencia", "Felipe Lagos", "Guido Lagos" ], "published_online": "2026-05-07", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-026-02359-9/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-026-02367-9", "title": "An exact penalty approach for equality constrained optimization over a convex set", "authors": [ "Nachuan Xiao", "Tianyun Tang", "Shiwei Wang", "Kim-Chuan Toh" ], "published_online": "2026-05-15", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-026-02367-9/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-026-02372-y", "title": "On Opial’s Lemma", "authors": [ "Aleksandr Arakcheev", "Heinz H. Bauschke" ], "published_online": "2026-05-22", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-026-02372-y/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-025-02313-1", "title": "Exact worst-case convergence rates of gradient descent: a complete analysis for all constant stepsizes over nonconvex and convex functions", "authors": [ "Teodor Rotaru", "François Glineur", "Panagiotis Patrinos" ], "published_online": "2026-05-26", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-025-02313-1/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Abstract", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "We consider gradient descent with constant stepsizes and derive exact worst-case convergence rates on the minimum gradient norm of the iterates. Our analysis covers all possible stepsizes and arbitrary upper/lower bounds on the curvature of the objective function, thus including convex, strongly convex and weakly convex (hypoconvex) objective functions. Among the challenging parts of the analysis, we note the necessity to exploit dependencies between non-consecutive iterates. While this complicates the proofs to some extent, it enables us to achieve an exact full-range analysis of gradient descent for any constant stepsize (covering, in particular, normalized stepsizes greater than one), whereas the literature contained only conjectured rates of this type. In the nonconvex case, allowing arbitrary bounds on upper and lower curvatures extends existing partial results that are valid only for gradient Lipschitz functions (i.e., where lower and upper bounds on curvature are equal), leading to improved rates for weakly convex functions. From our exact worst-case performance bounds,we deduce the optimal constant stepsize for gradient descent. Leveraging our analysis, we also introduce a new variant of gradient descent based on a unique, fixed sequence of variable stepsizes, demonstrating its superiority in the worst-case over any constant stepsize schedule." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-026-02358-w", "title": "Computational guarantees for restarted PDHG for LP based on “Limiting Error Ratios” and LP sharpness", "authors": [ "Zikai Xiong", "Robert M. Freund" ], "published_online": "2026-05-26", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-026-02358-w/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-026-02361-1", "title": "Accelerating operator Sinkhorn iteration with overrelaxation", "authors": [ "Tasuku Soma", "André Uschmajew" ], "published_online": "2026-05-26", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-026-02361-1/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Numerical experiments", "element": "p", "matched_phrases": [ "we_do_not_know" ], "passage": "Choice of \\(\\omega \\). For our overrelaxation methods, we need to estimate \\(\\omega _{\\text {opt}}\\) based on Theorem 3.3, which in turn requires the asymptotic convergence rate \\(\\beta ^2\\) of the operator Sinkhorn iteration. However, we do not know \\(\\beta ^2\\) in advance. It is hence common to use an adaptive strategy by monitoring the convergence of the standard method. In particular we adopt the proposed strategy from [25, 27] (considered in similar form already in [20]) as follows. Let p be a positive integer parameter. Then we start by running our SOR methods with \\(\\omega =1\\) (which is equivalent to the operator Sinkhorn iteration) for the first p iterations, and obtain an estimate \\({{\\hat{\\beta }}}^2\\) of \\(\\beta ^2\\) by computing" }, { "section": "Numerical experiments", "element": "p", "matched_phrases": [ "future_work" ], "passage": "The degraded accuracy of the SOR methods may be explained as follows. The main step in the algorithms is the computation of Cholesky decompositions and inverses of intermediate matrices which are sums of Gramians. In OSI these are Gramians of \\(\\bar{A}_{i,t} = L_t A_i R_t^\\top \\). Upon convergence to solutions of the operator scaling problem, the sum of these Gramians converge to well conditioned (normalized) identity matrices. This gradually compensates (to a certain limit) for the bad condition of the initial \\(A_i\\). In contrast, in the SOR methods, which are based on either Algorithm 2 or 3, sums of Gramians of only “half-scaled” matrices such as \\(A_i R_t^\\top \\) or \\(L_{t+1} A_i\\) appear as intermediate matrices. Hence the ill-conditioning of the \\(A_i\\) is never fully compensated. This may also explain, why OSI approximately achieves the double numerical accuracy compared to the SOR methods (\\(10^{-11}\\) vs. \\(10^{-6}\\)). Note that Algorithm 2 and 3 without overrelaxation would suffer from the same limitation of the numerical accuracy in this example (results not shown). This suggests that it would be beneficial to combine OSI (Algorithm 1) directly with overrelaxation. This is left for future work." }, { "section": "Conclusion", "element": "p", "matched_phrases": [ "future_work" ], "passage": "We investigated different variants of overrelaxation for the operator Sinkhorn iteration based on its interpretation as an alternating fixed-point scheme on cones of symmetric positive definite matrices. The numerically most efficient version operates directly on the Cholesky factors (Algorithm 5), whereas the theoretically most appealing version uses overrelaxation along geodesics in the affine invariant metric on positive definite matrices (Algorithm 6). For this geodesic version we were able to establish global convergence in a limited (and presumably too pessimistic) range of relaxation parameters for operator scaling problems with a positivity improving CP map. However, near a fixed-point all proposed versions were shown to be equivalent to first-order, and the local convergence rate of all of them could hence be deduced by analyzing the linearized SOR iteration, taking their orbital equivariance into account. Numerical experiments illustrate that overrelaxation can significantly accelerate the difficult operator scaling task at practically no additional cost (at least for Algorithm 5) using an easy heuristic for choosing an almost optimal relaxation parameter. However, on ill-conditioned instances our current versions of overrelaxation may suffer from a limited numerical accuracy. Addressing this issue will be one direction for future work. Another one is to establish the global convergence for a larger range of relaxation parameters by exploiting the geodesic convexity of the underlying problem more explicitly." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-026-02363-z", "title": "Tightening convex relaxations of trained neural networks", "authors": [ "Pablo Carrasco", "Gonzalo Muñoz" ], "published_online": "2026-05-26", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-026-02363-z/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-026-02364-y", "title": "A second-order cone representable class of nonconvex quadratic programs", "authors": [ "Santanu S. Dey", "Aida Khajavirad" ], "published_online": "2026-05-26", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-026-02364-y/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-026-02369-7", "title": "A semidefinite program for least distortion embeddings of flat tori into Hilbert spaces", "authors": [ "Arne Heimendahl", "Moritz Lücke", "Frank Vallentin", "Marc Christian Zimmermann" ], "published_online": "2026-05-26", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-026-02369-7/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [ { "section": "Introduction", "element": "p", "matched_phrases": [ "open_question" ], "passage": "In the last sections we apply our new methodology. In Section 4 we prove that an n-dimensional flat torus always admits a finite dimensional least distortion embedding, a space of (complex) dimension \\(2^{n}-1\\) suffices. Section 5 contains a new and simple proof of our constant factor improvement of the lower bound given in (4). In Section 6 we show that the standard embedding (2) of the standard torus is indeed optimal and has distortion \\(\\pi /2\\). We give an optimal embedding of the lattices \\(D_n^*\\) in Section 7. In Section 8 we determine least distortion embeddings of all two-dimensional flat tori. Open questions are discussed in Section 9." }, { "section": "Least Euclidean distortion embeddings always have finite dimension", "element": "p", "matched_phrases": [ "we_do_not_know" ], "passage": "The lemma gives rise to a procedure that transforms a feasible solution (C, z) into a solution \\( (C,\\tilde{z}) \\) such that \\( \\tilde{z} \\) has only support on at most one primitive lattice element per coset \\( u+2L^* \\). Roughly speaking, start with any solution and apply the above lemma “as long as possible”, i.e. as long as there are pairs of vectors that satisfy (1) or (2) of the above lemma. We do not know how to turn this procedure into an algorithm that terminates after finitely many steps, but we can show that the procedure converges to a solution with the desired properties." }, { "section": "Least Euclidean distortion embedding of the lattice \\( D_n^* \\)", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "The argument cannot be straightforwardly carried over to the lattice \\( A_n^* \\). However, we conjecture that there is also an optimal solution for (12) which assigns uniform weights to the roots of \\( A_n \\). This has been proved by Moustrou and Vallentin [25] for \\(n = 2\\)." }, { "section": "Least distortion embeddings of two-dimensional flat tori", "element": "p", "matched_phrases": [ "we_do_not_know" ], "passage": "Except for the easiest case of the standard torus we do not know how to determine D explicitly. Unfortunately, it seems to be difficult to compute most contracted pairs (0, x), i.e. vectors \\(x \\in V(L)\\) that are maximizers of the right hand side of (25)." }, { "section": "Discussion and open questions", "element": "p", "matched_phrases": [ "we_do_not_know" ], "passage": "For this a characterization of the most contracted pairs is needed. We believe that the most contracted pairs are always of the form (0, y) and y is a center of a face of the Voronoi cell. However, we do not know whether such a y can only lie on the Voronoi cell’s boundary. We do not even know whether there are only finitely many most contracted pairs." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-026-02370-0", "title": "Eliciting Von Neumann–Morgenstern utility from discrete choices with response error", "authors": [ "Bo Chen", "Jia Liu" ], "published_online": "2026-05-27", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-026-02370-0/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Notes", "element": "li", "matched_phrases": [ "we_leave", "future_work" ], "passage": "We clarify that this adjustment is computationally equivalent for the MLE problem (10) when \\(0\\!<\\!\\sigma \\!<\\!\\infty \\). Moreover, u and \\(\\sigma \\) are not separately identifiable from the DM’s choice \\(Z_k\\), nor can they be separately distinguished in the statistical convergence analysis in Section 3.4. However, u and \\(\\sigma \\) are modeled as separate components for important reasons. First, u and \\(\\varepsilon \\) are assumed to be independent, and merging them may hinder future statistical testing (beyond the scope of this paper). Second, this separation maintains a general modeling framework that accommodates broader data types and preference models, as the adjustment step \\(\\tilde{u}=u/\\sigma \\) may not be valid for other data types (e.g., ratings), alternative risk measures (e.g., variance), or more general utility functions (e.g., S-shaped utility functions). Finally, from an interpretive perspective, allowing a separate variable \\(\\sigma \\) for the response error \\(\\varepsilon \\) corresponds to a DM-specific level of inconsistency in making choices. We leave these related issues for future research." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-026-02373-x", "title": "Performance estimation for smooth and strongly convex sets", "authors": [ "Alan Luner", "Benjamin Grimmer" ], "published_online": "2026-06-01", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-026-02373-x/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-026-02366-w", "title": "An adaptive proximal ADMM for nonconvex linearly constrained composite programs", "authors": [ "Leandro Farias Maia", "David H. Gutman", "Renato D. C. Monteiro", "Gilson N. Silva" ], "published_online": "2026-06-05", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-026-02366-w/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-026-02368-8", "title": "MDP modeling for multi-stage stochastic programs", "authors": [ "David P. Morton", "Oscar Dowson", "Bernardo K. Pagnoncelli" ], "published_online": "2026-06-05", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-026-02368-8/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [ { "section": "Computational examples", "element": "p", "matched_phrases": [ "we_leave", "future_work" ], "passage": "As a computational demonstration of our algorithms, we solve two examples in SDDP.jl, a free and open-source Julia [7] package for modeling policy graphs and solving them with stochastic dual dynamic programming [14]. SDDP.jl is based on the JuMP modeling language [37]. The code and data for our experiments are available at https://github.com/odow/SDDP.jl. Our intent is neither to demonstrate large-scale examples nor to implement the adaptive decision-dependent learning algorithm; we leave that to future work." }, { "section": "Conclusion", "element": "p", "matched_phrases": [ "future_work" ], "passage": "Although our focus has been on modeling and algorithmic development, we provide an open-source implementation of our ideas in SDDP.jl [14]. The code and documentation are freely available at https://sddp.dev. As future work, we will conduct larger-scale computational experiments and integrate our work with other features supported by SDDP.jl, such as risk aversion." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-026-02374-w", "title": "Characterization of regularity via variational stability of alternating projection sequences", "authors": [ "F. Battistoni", "A. Daniilidis", "C. A. De Bernardi", "E. Miglierina" ], "published_online": "2026-06-05", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-026-02374-w/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [ { "section": "Introduction", "element": "p", "matched_phrases": [ "open_problem" ], "passage": "The main result of [11] asserts that if the pair (A, B) is regular and the sets E and F are (nonempty and) bounded, then the pair (A, B) is d-stable. The assumption on the boundedness of the best approximation sets E and F is necessary for this statement, since Example 5.2 in [11] provides a regular pair (A, B) with \\(E=F=A\\cap B\\ne \\emptyset \\) being unbounded, which is not d-stable. In the same paper, the authors asked whether the inverse implication holds, stating the following open problem:" }, { "section": "Introduction", "element": "p", "matched_phrases": [ "open_question" ], "passage": "We resume this introduction with a brief description of the structure of the paper. In Sect. 2 we fix our notation and review preliminary notions. Definitions and properties of the Hausdorff and, respectively, the Attouch-Wets set convergence are therein recalled, together with relevant properties of metric projections and of alternating projections method in a Hilbert space. In particular, the relationship between regularity and d-stability obtained in [11] is recalled. In Sect. 3, we prove the main result of the paper, namely, the fact that d-stability implies regularity for a given convex feasibility problem. After two key technical lemmas (namely, Lemma 3.1 and Lemma 3.2), the proof of our main result will be divided into two cases: first, we consider the case where \\(A\\cap B\\) is the empty set, then we move to the case where \\(A\\cap B\\) is nonempty, where the proof becomes more involved. Finally, in Sect. 4 we present some additional remarks, conclusions and open questions." }, { "section": "Main results", "element": "p", "matched_phrases": [ "open_problem" ], "passage": "The above result provides a complete proof of Theorem A and positively answers the open problem considered in Problem 1.2." }, { "section": "Open problems and remarks", "element": "p", "matched_phrases": [ "unknown_whether" ], "passage": "A posteriori, the fact that d-stability does not hold for these instances of Moment Problem is not surprising since the involved lattice cone B has usually an unbounded basis (and the approximation of such cones is often problematic). For the time being, it remains unknown if the alternating projection sequences converges in norm for a general moment problem of codimension \\(k>1\\). Our results suggest that the variational approach does not seem to be the proper way to investigate the Moment Problem and different techniques related to the lattice structure of the cone are required, especially for higher codimension k, see [3]." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-026-02375-9", "title": "A Knowledge Compilation Take on Binary Polynomial Optimization", "authors": [ "Florent Capelli", "Alberto Del Pia", "Silvia Di Gregorio" ], "published_online": "2026-06-10", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-026-02375-9/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-026-02378-6", "title": "Balancing notions of equity: trade-offs between fair portfolio sizes and achievable guarantees", "authors": [ "Swati Gupta", "Jai Moondra", "Mohit Singh" ], "published_online": "2026-06-12", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-026-02378-6/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-026-02384-8", "title": "Preface", "authors": [ "Leo Liberti", "Sebastian Sager", "Angelika Wiegele" ], "published_online": "2026-06-12", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-026-02384-8/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-026-02377-7", "title": "Optimized methods for composite optimization: a reduction perspective", "authors": [ "Jinho Bok", "Jason M. Altschuler" ], "published_online": "2026-06-13", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-026-02377-7/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Introduction", "element": "p", "matched_phrases": [ "open_problem" ], "passage": "Our unified framework leads to convergence rates that are competitive with highly optimized algorithms and, in some cases, yields state-of-the-art complexity guarantees and answers open problems. As concrete examples, we apply our framework to establish the following:" }, { "section": "Introduction", "element": "li", "matched_phrases": [ "open_problem", "open_question", "unknown_whether" ], "passage": "Stepsize-based acceleration of proximal GD. We answer the open questions of [8, 9] by showing that stepsize-based acceleration is possible for proximal GD. That is, we show that proximal GD can achieve accelerated rates with a judicious choice of stepsizes—without any other modifications to the algorithm (e.g., momentum). Previous results for stepsize-based acceleration were limited to the setting of GD for unconstrained smooth convex optimization, and it was unknown whether this phenomenon was possible in constrained or composite settings. See [1, Section 1] for a comprehensive discussion of this open problem and the challenges. We show a rate of \\(O(1/n^{\\log _2(1+\\sqrt{2})}) \\approx O(1/n^{1.2716})\\), which improves over the classical O(1/n) guarantee that is tight for proximal GD with constant stepsizes [25]. This asymptotic rate is conjecturally optimal even for the simpler setting of vanilla GD for unconstrained optimization [9,10,11]; hence, our rate is also conjecturally optimal. Appealingly, our framework enables us to use the same stepsizes that were developed for accelerating vanilla GD—an approach that is natural, yet was previously unclear how to analyze beyond the original setting." }, { "section": "Introduction", "element": "p", "matched_phrases": [ "open_question", "future_work" ], "passage": "Our analysis is based on a crucial observation in this template (1.1): optimized methods have simple solutions. In particular, for unconstrained optimized algorithms, the sum-of-squares term is often just a single square—rather than the sum of multiple squares that cannot be collapsed into a single square. In other words, the corresponding quadratic form is of rank 1. This core property was observed in [23, Chapter 5] for many different types of optimized methods; conversely, such property is not known for “non-optimized” methods such as constant-stepsize GD or AGD. It is an interesting open question in itself to understand if this phenomenon hints at an underlying general theory for optimized methods; see also the discussion of future work in Section 7." }, { "section": "Introduction", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "An emerging area of study which uses PEP is stepsize-accelerated GD [7,8,9,10,11, 45,46,47,48,49,50], which seeks to improve the rate of GD by only changing the stepsizes. Classically, this was only known to be possible for minimizing convex quadratics [51]. The recent line of work extends this improvement beyond the quadratic setting by using time-varying stepsize schedules that are nonmonotone and use exceedingly large steps. In particular, [10] showed the “silver convergence rate” \\(O(1/n^{\\log _2 (1+\\sqrt{2})})\\) and conjectured it to be asymptotically optimal among all possible stepsize schedules. See [62] for a recent survey overview of this very active literature on stepsize-based acceleration." }, { "section": "Introduction", "element": "p", "matched_phrases": [ "open_problem", "unknown_whether" ], "passage": "Our framework provides a general way to design and analyze methods in this setting. This leads to new algorithms as well as new rates for existing algorithms. Here, we further contextualize the applications of our framework. Our composite extension of OGM recovers POGM, a method that appeared in the original paper on PEP for composite optimization [2]. However, that paper only presents numerical bounds from PEP; we provide rigorous convergence analyses here. We remark that while the design and analysis are simpler for POGM than OptISTA, it does not achieve the exactly optimal rate. This is not just an artefact of our technique and it was numerically observed that the POGM is suboptimal, though only by a small multiplicative factor of roughly 1.12 [2]. Our theoretical result achieves a rate for POGM with a mild multiplicative factor of roughly 1.29 compared to the exactly optimal rate. Furthermore, our composite extension of OGM-G achieves the state-of-the-art rate, improving over the result of [27, Section D.4] by a factor of roughly 9.30. For stepsize-based acceleration, previous results were known only for GD (in the unconstrained setting), and it was unknown if this phenomenon extends to projected GD (in the constrained setting) or proximal GD (in the composite setting). This was posed as an open problem in [8, 9]; see [1, Section 1] for a detailed discussion of the challenges. We resolve this question by showing that the stepsizes for accelerating vanilla GD enable the same asymptotic rates for proximal GD." }, { "section": "Discussion", "element": "p", "matched_phrases": [ "open_question" ], "passage": "Further understanding of optimized methods. The starting point for this work is the observation that optimized methods admit simple proof structures, namely the sum-of-squares term being of rank 0 or 1 (see the technical overview in Section 1.2). This phenomenon holds more broadly beyond the optimized methods we covered here [23, Chapter 5], and for specific choices of stepsizes for GD it admits an equivalent characterization in terms of worst-case functions [49, Proposition 4]. On the other hand, it is unclear whether proofs for “non-optimized” methods such as constant-stepsize GD or AGD admit such structures (see e.g., [5, 28]), and our framework accordingly does not directly apply. It is an interesting open question to understand the generality of this phenomenon and whether it hints at an underlying unified theory for optimized methods." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-026-02371-z", "title": "Projection onto hyperbolicity cones and beyond: a dual Frank-Wolfe approach", "authors": [ "Takayuki Nagano", "Bruno F. Lourenço", "Akiko Takeda" ], "published_online": "2026-06-19", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-026-02371-z/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-026-02383-9", "title": "Special issue: stochastic programming and distributionally robust optimization with decision-dependent uncertainty", "authors": [ "Miguel Lejeune", "Ward Romeijnders", "Pavlo Krokhmal" ], "published_online": "2026-06-20", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-026-02383-9/fulltext.html", "article_kind": "non_research_item", "fetch_status": "screened", "series": "B", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-026-02380-y", "title": "Sensor network localization has a benign landscape after low-dimensional relaxation", "authors": [ "Christopher Criscitiello", "Andrew D. McRae", "Quentin Rebjock", "Nicolas Boumal" ], "published_online": "2026-06-22", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-026-02380-y/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Introduction", "element": "p", "matched_phrases": [ "open_question" ], "passage": "For several years, it remained an open question whether (SNL) with a complete graph admits a benign optimization landscape when \\(k= \\ell \\) [56, 62]. This question was recently resolved in the negative by Song et al. [72, §5], who numerically identified spurious local minimizers for (SNL) with \\(n = 100\\) points, and then analytically checked that configuration is stable by appealing to the Kantorovich theorem. We add to this finding in Sect. 6, where we construct simple counterexamples with \\(n > 6\\) points and \\(\\ell =2\\) (see Fig. 2), showing there exists a set of ground truth configurations (of positive measure) for which the landscape of (SNL) is not benign, even when the graph \\(E\\) is complete.Footnote 2 We further prove that for \\(\\ell \\ge 5\\), the landscape can be non-benign even with only \\(n = \\ell + 2\\) points—which is the minimal number of points for such behavior." }, { "section": "Introduction", "element": "p", "matched_phrases": [ "open_question", "conjecture" ], "passage": "Numerical experiments in Sect. 7 motivate us to conjecture that relaxing to just \\(k= \\ell + 1\\) is sufficient for the landscape to be benign when all distances are known, for any ground truth configuration. We discuss the implications and several open questions arising from our observations in Sect. 8." }, { "section": "Mathematical background for SNL", "element": "p", "matched_phrases": [ "future_work" ], "passage": "Useful consequence of 2-criticality. Lemma 3.3 below is a consequence of (9) and property P5. It is used in the proofs of both Theorems 1.1 and 1.2. We emphasize that this lemma can easily be extended to incomplete graphs: the proofs are chosen to reflect this, in anticipation of future work on the landscape of SNL." }, { "section": "Landscape for all ground truths, \\(\\varvec{k\\approx \\sqrt{n}}\\)", "element": "p", "matched_phrases": [ "we_do_not_know" ], "passage": "Sketch of the remaining argument. Lemma 4.5 tells us that the \\(\\ell + 1\\) smallest eigenvalues of \\(P\\) lie in the interval \\([0, \\frac{-b + \\sqrt{b^2 + 4 a n}}{2n})\\). Intuitively, these eigenvalues should all be strictly positive. As P is transformed to \\(-aP + bP^2 + nP^3\\), the eigenvalues of P are transformed through the polynomial \\(t \\mapsto -at + bt^2 + nt^3\\). That polynomial is negative on the stated (open) interval (see Fig. 3). Thus, we expect the matrix \\(-a P + b P^2 + n P^3\\) to have at least \\(\\ell + 1\\) negative eigenvalues. Since \\(\\ker (V^\\top Y_*V)\\) is a subspace of codimension at most \\(\\ell \\), the restriction of \\(-a P + b P^2 + n P^3\\) to this subspace should, by the eigenvalue interlacing theorem (Lemma 4.6 below), have at least \\((\\ell + 1) - \\ell = 1\\) negative eigenvalues, which would complete the contradiction. However, to make this argument rigorous, a bit more care is required because we do not know whether the eigenvalues of \\(P\\) are strictly positive." }, { "section": "Perspectives", "element": "p", "matched_phrases": [ "open_question" ], "passage": "We conclude with a list of open questions." }, { "section": "Perspectives", "element": "li", "matched_phrases": [ "conjecture" ], "passage": "(Relaxing by +1) Based on numerical simulations, we conjecture that relaxing to \\(k= \\ell + 1\\) yields a benign landscape for all ground truth configurations in the complete graph case. Is this truly the case, or is there a small pathological set of ground truths—perhaps of measure zero—for which the landscape admits spurious local minimizers? A possible avenue toward sharpening Theorem 4.1 in this direction is to optimize the weights placed on the eigenmodes in the construction of S in (17). Another possible avenue is to obtain finer control on the spectrum of P in the spirit of Lemma 4.4.Footnote 13" }, { "section": "Perspectives", "element": "li", "matched_phrases": [ "open_problem" ], "passage": "(Incomplete graph) Numerical experiments indicate that relaxing to \\(k\\) only slightly more than \\(\\ell \\) improves the probability of recovery in some regimes. Can we provide a theoretical explanation—for example, for specific graph structures such as trilateration graphs, expander graphs, geometric graphs, or unit disk graphs? Theorem 1.2 is extended to (slightly) incomplete graphs by Criscitiello [26, §8.8]. In particular, it is shown that if \\(E\\) in (SNL) is missing m edges, n is sufficiently large, and \\(k\\gtrsim \\kappa (1+m^3)(\\ell + \\log n)\\), then (SNL) has benign landscape with high probability. Extending this result to substantially sparser graphs remains an open problem." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-026-02385-7", "title": "Recoverable robust optimization with commitment", "authors": [ "Felix Hommelsheim", "Nicole Megow", "Komal Muluk", "Britta Peis" ], "published_online": "2026-06-22", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-026-02385-7/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Robust matroid", "element": "p", "matched_phrases": [ "remains_open" ], "passage": "We observe that this result does not hold if \\(k > \\ell \\), which is highlighted in the following example. It remains open whether this problem admits a polynomial-time algorithm for \\(k > \\ell \\)." }, { "section": "Conclusion and future research", "element": "p", "matched_phrases": [ "open_question", "remains_open" ], "passage": "Furthermore, we often have restricted our considerations to problem settings, where the adversary is only allowed to delete a single element (i.e., \\(k=\\ell =1\\)), which seems most fundamental. It would be interesting to see how the computation complexity unravels for the general setting with arbitrary valued k and \\(\\ell \\). For \\((k,\\ell )\\)-Robust Matroid with \\(k\\le \\ell \\), we showed that the nominal optimal solution is also optimal for the robust counterpart, and gave an example illustrating that this property does not necessarily hold for \\(k>\\ell \\), or non-matroid structures. It still remains an open question whether or not \\((k,\\ell )\\)-RM is polynomial-time solvable if \\(k> \\ell \\). Similarly, even though we have efficient algorithms for (1, p)-Robust Interval Scheduling for a constant value p, the complexity status of \\((k,\\ell )\\)-Robust Interval Scheduling and also the \\((k,\\ell )\\)-Robust Stable Set remain open." }, { "section": "Conclusion and future research", "element": "p", "matched_phrases": [ "remains_open" ], "passage": "Finally, the bounded-regret problem was crucial for solving Robust Interval Scheduling. It remains open whether the bounded-regret version of other problems can be solved in polynomial time. Particularly interesting is Robust Matroid which we can solve efficiently for \\(k\\le \\ell \\), but the complexity of the bounded-regret version is open." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-026-02388-4", "title": "Totally $$\\Delta $$-modular IPs with two non-zeros in most rows", "authors": [ "Stefan Kober" ], "published_online": "2026-06-22", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-026-02388-4/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [ { "section": "Abstract", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "Integer programs (IPs) on constraint matrices with bounded subdeterminants are conjectured to be solvable in polynomial time. We give a strongly polynomial-time algorithm to solve IPs where the constraint matrix has bounded subdeterminants and at most two non-zeros per row after removing a constant number of rows and columns. This result extends the work by Fiorini, Joret, Weltge & Yuditsky (J. ACM 72(1), 1-50 (2025)) by allowing for additional, unifying constraints and variables. Further, we give a randomized polynomial-time algorithm for the natural transposed case, i.e., where the main part of the matrix has two non-zeros per column instead of per row." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-026-02392-8", "title": "Integer and unsplittable multiflows in series-parallel digraphs", "authors": [ "Mohammed Majthoub Almoghrabi", "Martin Skutella", "Philipp Warode" ], "published_online": "2026-06-29", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-026-02392-8/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [ { "section": "Abstract", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "An unsplittable multiflow routes the demand of each commodity along a single path from its source to its sink node. As our main result, we prove that in series-parallel digraphs, any given multiflow can be expressed as a convex combination of unsplittable multiflows, where the total flow on any arc deviates from the given flow by less than the maximum demand of any commodity. This result confirms a 25-year-old conjecture by Goemans for single-source unsplittable flows, as well as a stronger recent conjecture by Morell and Skutella, for series-parallel digraphs—even for general multiflow instances where commodities have distinct source and sink nodes. Previously, no non-trivial class of digraphs was known for which either conjecture holds. En route to proving this result, we also establish strong integrality results for multiflows on series-parallel digraphs, showing that their computation can be reduced to a simple single-commodity network flow problem." }, { "section": "Introduction", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "where \\(d_{\\max }:=\\max _{i\\in [k]}d_i\\). A famous conjecture of Goemans says that flow \\((x_e)_{e\\in E}\\) can even be expressed as a convex combination of unsplittable flows \\((y_e)_{e\\in E}\\) that satisfy (2).Footnote 1 Skutella [11], based on a flow augmentation method also used by Kolliopoulos and Stein [12], proves that Goemans’ Conjecture is valid when the demands of the commodities are multiples of one another. For acyclic digraphs, Morell and Skutella [13] show that any fractional flow \\((x_e)_{e\\in E}\\) can be turned into an unsplittable flow \\((y_e)_{e\\in E}\\) that satisfies the lower bound" }, { "section": "Introduction", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "Rohwedder and Svensson (personal communication, 2021) observed that this result can also be achieved by augmenting flow in the reverse direction along the designated cycles of Dinitz, Garg, and Goemans [9]. Furthermore, Morell and Skutella conjecture the existence of an unsplittable flow that satisfies both the upper bounds (2) and the lower bounds (3). Only recently, for the special case of acyclic and planar digraphs, this conjecture has been proved by Traub, Vargas Koch, and Zenklusen [14]. Morell and Skutella also propose the following strengthening to Goemans’ conjecture." }, { "section": "Introduction", "element": "h3", "matched_phrases": [ "conjecture" ], "passage": "Conjecture 1" }, { "section": "Introduction", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "Using techniques of Martens, Salazar, and Skutella [10], they show that their Conjecture 1 is valid when the demands of the commodities are multiples of one another. Moreover, for the special case of acyclic and planar digraphs, and under the slightly relaxed lower and upper bounds \\(x_e-2d_{\\max }\\le y_e\\le x_e+2d_{\\max }\\) for \\(e\\in E\\), Conjecture 1 is proved by Traub, Vargas Koch, and Zenklusen [14], based on a reduction to a well-structured discrepancy problem." }, { "section": "Introduction", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "This result implies, in particular, that Conjecture 1 holds for series-parallel digraphs, even for general multiflow instances where commodities have individual source and sink nodes. Even for the weaker conjecture of Goemans, Theorem 2 provides the first proof for a non-trivial class of digraphs." }, { "section": "Notes", "element": "li", "matched_phrases": [ "conjecture" ], "passage": "Goemans’ original conjecture asserts that, for any arc costs, an unsplittable flow can be found that satisfies (2) and whose cost does not exceed that of \\((x_e)_{e\\in E}\\). This is equivalent to the stated existence of a convex combination; see, e.g., [10]." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-026-02390-w", "title": "The complete edge relaxation", "authors": [ "Alberto Del Pia", "Aida Khajavirad" ], "published_online": "2026-06-30", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-026-02390-w/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-026-02391-9", "title": "Reducing the large set threshold for Oertel’s conjecture on the mixed-integer volume", "authors": [ "Andrés Cristi", "David Salas" ], "published_online": "2026-07-07", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-026-02391-9/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "B", "candidate_passages": [ { "section": "Abstract", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "In 1960, B. Grünbaum proved that, for any convex body \\(C\\subset \\mathbb {R}^d\\) and every halfspace H containing the centroid of C, the volume of \\(H\\cap C\\) is at least a \\(\\frac{1}{e}\\)-fraction of the volume of C. In 2014, Oertel conjectured that a similar result holds for mixed-integer convex sets. Concretely, he proposed that for any convex body \\(C\\subset \\mathbb {R}^{n+d}\\), there should exist a point \\(\\textbf{x} \\in S=C\\cap (\\mathbb {Z}^{n}\\times \\mathbb {R}^d)\\) such that every halfspace H containing \\(\\textbf{x}\\) satisfies" }, { "section": "Abstract", "element": "p", "matched_phrases": [ "remains_open", "conjecture" ], "passage": "where \\(\\mathcal {H}_d\\) denotes the d-dimensional Hausdorff measure. While the conjecture remains open, Basu and Oertel proved in 2017 that the above inequality holds for sets that are sufficiently large in terms of a measure known as the lattice width. In this work, we improve upon this result, substantially reducing the threshold at which a set can be considered large. We reduce this threshold from an exponential to a polynomial dependency on the dimension, thereby significantly enlarging the family of mixed-integer convex sets for which Oertel’s conjecture holds." }, { "section": "Inline Recommendations", "element": "h3", "matched_phrases": [ "conjecture" ], "passage": "Reducing the Large Set Threshold for Oertel’s Conjecture on the Mixed-Integer Volume" }, { "section": "Notes", "element": "li", "matched_phrases": [ "conjecture" ], "passage": "The works [6, 23] and the subsequent works [5, 7] are much richer than what we describe here. To streamline the exposition, we focus only on the elements relevant to Conjecture 1." }, { "section": "About this article", "element": "p", "matched_phrases": [ "conjecture" ], "passage": "Cristi, A., Salas, D. Reducing the large set threshold for Oertel’s conjecture on the mixed-integer volume. Math. Program. (2026). https://doi.org/10.1007/s10107-026-02391-9" } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-026-02394-6", "title": "On the convergence rates of moment-SOS hierarchies approximation of truncated moment sequences", "authors": [ "Hoang Anh Tran", "Kim-Chuan Toh" ], "published_online": "2026-07-07", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-026-02394-6/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-026-02376-8", "title": "Accelerating diagonal methods for bilevel optimization: Unified convergence via continuous-time dynamics", "authors": [ "Radu I. Boţ", "Enis Chenchene", "E. Robert Csetnek", "David A. Hulett" ], "published_online": "2026-07-08", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-026-02376-8/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Appendices", "element": "p", "matched_phrases": [ "we_do_not_know" ], "passage": "A special case (i.e., when \\(t_{k} = k\\)) of the following lemma can be found in the literature, for example, in [10, Lemma A.5]. However, that version requires the sequence \\((\\varphi _{k})_{k \\ge 0}\\) to be bounded, which we cannot assume. This is important, since when applying this lemma we do not know beforehand if \\((\\varphi _{k})_{k \\ge 0}\\) is bounded or not." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-026-02381-x", "title": "Strongly convex maximization via the Frank-Wolfe algorithm with the Kurdyka-Łojasiewicz inequality", "authors": [ "Fatih S. Aktaş", "Christian Kroer" ], "published_online": "2026-07-08", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-026-02381-x/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-026-02386-6", "title": "On the complexity of lower-order implementations of higher-order methods", "authors": [ "Nikita Doikov", "Geovani Nunes Grapiglia" ], "published_online": "2026-07-08", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-026-02386-6/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [ { "section": "Introduction", "element": "p", "matched_phrases": [ "open_problem", "future_work" ], "passage": "The paper is organized as follows. Section 2 introduces the problem and presents the necessary auxiliary results. In Section 3, we describe our new method and analyze its worst-case oracle complexity. Section 4 concludes with a discussion of open problems and directions for future research." }, { "section": "Discussion", "element": "p", "matched_phrases": [ "remains_open", "future_work" ], "passage": "still needs to be investigated. In principle, one could envision a recursive scheme in which qth-order derivatives are used to approximate derivatives of order \\(q+1\\), these approximations could then be used to approximate derivatives of order \\(q+2\\), and so forth, up to order p. However, in the multivariate setting it remains unclear to us how to systematically construct the corresponding finite-difference formulas together with explicit error bounds of the type required by our analysis (as in Lemma 4). Such bounds are essential for selecting the finite-difference parameter h in a way that rigorously controls the cumulative approximation error (as in Lemma 5). Even for the known bounds, outside the main diagonal (\\(q=p\\)) it is unclear whether they can be improved, as the question of their tightness remains open. From a practical perspective, it would be interesting to design a less lazy variant of Algorithm 1 that incorporates quasi-tensor updates [27], as in [7], while still preserving the improved complexity bound with respect to the problem dimension n. Additionally, exploring universal lower-order implementations of higher-order methods for convex and nonconvex functions with Hölder-continuous derivatives would also be of interest [6, 15, 20]. We plan to address these questions in future work." } ], "full_text_source": "publisher_html" }, { "doi": "10.1007/s10107-026-02395-5", "title": "A low-rank augmented Lagrangian method for polyhedral-SDP and moment-SOS relaxations of polynomial optimization", "authors": [ "Di Hou", "Tianyun Tang", "Kim-Chuan Toh" ], "published_online": "2026-07-08", "volume": null, "issue": null, "pages": null, "publisher_url": "https://link.springer.com/article/10.1007/s10107-026-02395-5/fulltext.html", "article_kind": "research_article", "fetch_status": "screened", "series": "A", "candidate_passages": [], "full_text_source": "publisher_html" } ] }