Mathematical Programming最新文献

{"title":"Tight lower bounds for block-structured integer programs.","authors":"Christoph Hunkenschröder, Kim-Manuel Klein, Martin Koutecký, Alexandra Lassota, Asaf Levin","doi":"10.1007/s10107-025-02296-z","DOIUrl":"10.1007/s10107-025-02296-z","url":null,"abstract":"<p><p>We study fundamental block-structured integer programs called tree-fold and multi-stage IPs. Tree-fold IPs have a constraint matrix with independent blocks linked together by few constraints in a recursive pattern. Transposing this constraint matrix yields the constraint matrix of multi-stage IPs. The state-of-the-art algorithms to solve these IPs have an exponential gap in their running times, making it natural to ask whether this gap is inherent. We answer this question in the affirmative. Assuming the Exponential Time Hypothesis, we prove lower bounds showing that the exponential difference is necessary. This also proves that the known algorithms are essentially optimal. Moreover, we prove unconditional lower bounds on the size of the Graver basis elements, a fundamental building block of all known algorithms to solve these IPs. This shows that none of the current approaches can be improved beyond this bound unconditionally.</p>","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":"216 1-2","pages":"521-538"},"PeriodicalIF":2.1,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13124973/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147817113","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"An FPTAS for Connectivity Interdiction.","authors":"Chien-Chung Huang, Nidia Obscura Acosta, Sorrachai Yingchareonthawornchai","doi":"10.1007/s10107-025-02312-2","DOIUrl":"10.1007/s10107-025-02312-2","url":null,"abstract":"<p><p>In the connectivity interdiction problem, we are asked to find a global graph cut and remove a subset of edges under a budget constraint, so that the total weight of the remaining edges in this cut is minimized. This problem easily includes the knapsack problem as a special case, hence it is NP-hard. For this problem, Zenklusen [Zenklusen'14] designed a polynomial-time approximation scheme (PTAS) and exact algorithms for the special case of unit edge costs. He posed the question of whether a fully polynomial-time approximation scheme (FPTAS) is possible for the general case. We give an affirmative answer. For the special case of unit edge costs, we also give faster exact and approximation algorithms. Our main technical contribution is to establish a connection with an intermediate graph cut problem, called the <i>normalized</i> min-cut, which, roughly speaking, penalizes the edge weights of the remaining edges more severely, when more edges are taken out for free.</p>","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":"216 1-2","pages":"605-626"},"PeriodicalIF":2.1,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13124762/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147817171","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A first order method for linear programming parameterized by circuit imbalance.","authors":"Richard Cole, Christoph Hertrich, Yixin Tao, László A Végh","doi":"10.1007/s10107-025-02264-7","DOIUrl":"https://doi.org/10.1007/s10107-025-02264-7","url":null,"abstract":"<p><p>Various first order approaches have been proposed in the literature to solve Linear Programming (LP) problems, recently leading to practically efficient solvers for large-scale LPs. From a theoretical perspective, linear convergence rates have been established for first order LP algorithms, despite the fact that the underlying formulations are not strongly convex. However, the convergence rate typically depends on the Hoffman constant of a large matrix that contains the constraint matrix, as well as the right hand side, cost, and capacity vectors. We introduce a first order approach for LP optimization with a convergence rate depending polynomially on the circuit imbalance measure, which is a geometric parameter of the constraint matrix, and depending logarithmically on the right hand side, capacity, and cost vectors. This provides much stronger convergence guarantees. For example, if the constraint matrix is totally unimodular, we obtain polynomial-time algorithms, whereas the convergence guarantees for approaches based on primal-dual formulations may have arbitrarily slow convergence rates for this class. Our approach is based on a fast gradient method due to Necoara, Nesterov, and Glineur (Math. Prog. 2019); this algorithm is called repeatedly in a framework that gradually fixes variables to the boundary. This technique is based on a new approximate version of Tardos's method, that was used to obtain a strongly polynomial algorithm for combinatorial LPs (Oper. Res. 1986).</p>","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":"216 1-2","pages":"339-377"},"PeriodicalIF":2.5,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13124851/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147817167","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A hierarchy of eigencomputations for polynomial optimization on the sphere.","authors":"Benjamin Lovitz, Nathaniel Johnston","doi":"10.1007/s10107-025-02251-y","DOIUrl":"https://doi.org/10.1007/s10107-025-02251-y","url":null,"abstract":"<p><p>We introduce a convergent hierarchy of lower bounds on the minimum value of a real form over the unit sphere. The main practical advantage of our hierarchy over the real sum-of-squares (RSOS) hierarchy is that the lower bound at each level of our hierarchy is obtained by a minimum eigenvalue computation, as opposed to the full semidefinite program (SDP) required at each level of RSOS. In practice, this allows us to compute bounds on much larger forms than are computationally feasible for RSOS. Our hierarchy outperforms previous alternatives to RSOS, both asymptotically and in numerical experiments. We obtain our hierarchy by proving a reduction from real optimization on the sphere to Hermitian optimization on the sphere, and invoking the Hermitian sum-of-squares (HSOS) hierarchy. This opens the door to using other Hermitian optimization techniques for real optimization, and gives a path towards developing spectral hierarchies for more general constrained real optimization problems. To this end, we use our techniques to develop a hierarchy of eigencomputations for computing the real tensor spectral norm.</p>","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":"219 1-2","pages":"113-156"},"PeriodicalIF":2.1,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13553728/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148920074","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Accelerated first-order optimization under nonlinear constraints.","authors":"Michael Muehlebach, Michael I Jordan","doi":"10.1007/s10107-025-02224-1","DOIUrl":"10.1007/s10107-025-02224-1","url":null,"abstract":"<p><p>We exploit analogies between first-order algorithms for constrained optimization and non-smooth dynamical systems to design a new class of accelerated first-order algorithms for constrained optimization. Unlike Frank-Wolfe or projected gradients, these algorithms avoid optimization over the entire feasible set at each iteration. We prove convergence to stationary points even in a nonconvex setting and we derive accelerated rates for the convex setting both in continuous time, as well as in discrete time. An important property of these algorithms is that constraints are expressed in terms of velocities instead of positions, which naturally leads to sparse, local and convex approximations of the feasible set (even if the feasible set is nonconvex). Thus, the complexity tends to grow mildly in the number of decision variables and in the number of constraints, which makes the algorithms suitable for machine learning applications. We apply our algorithms to a compressed sensing and a sparse regression problem, showing that we can treat nonconvex <math><msup><mi>ℓ</mi> <mi>p</mi></msup> </math> constraints ( <math><mrow><mi>p</mi> <mo><</mo> <mn>1</mn></mrow> </math> ) efficiently, while recovering state-of-the-art performance for <math><mrow><mi>p</mi> <mo>=</mo> <mn>1</mn></mrow> </math> .</p>","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":"215 1-2","pages":"407-452"},"PeriodicalIF":2.5,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12790563/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145959590","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"<ArticleTitle xmlns:ns0=\"http://www.w3.org/1998/Math/MathML\">A <ns0:math> <ns0:mrow><ns0:mfrac><ns0:mn>4</ns0:mn> <ns0:mn>3</ns0:mn></ns0:mfrac> </ns0:mrow> </ns0:math> -approximation for the maximum leaf spanning arborescence problem in DAGs.","authors":"Meike Neuwohner","doi":"10.1007/s10107-025-02233-0","DOIUrl":"https://doi.org/10.1007/s10107-025-02233-0","url":null,"abstract":"<p><p>The Maximum Leaf Spanning Arborescence problem (MLSA) in directed acyclic graphs (dags) is defined as follows: Given a directed acyclic graph <i>G</i> and a vertex <math><mrow><mi>r</mi> <mo>∈</mo> <mi>V</mi> <mo>(</mo> <mi>G</mi> <mo>)</mo></mrow> </math> from which every other vertex is reachable, find a spanning arborescence rooted at <i>r</i> maximizing the number of leaves (vertices with out-degree zero). The MLSA in dags is known to be APX-hard as reported by Nadine Schwartges, Spoerhase, and Wolff (Approximation and Online Algorithms, Springer, Berlin Heidelberg, 2012) and the best known approximation guarantee of <math><mfrac><mn>7</mn> <mn>5</mn></mfrac> </math> is due to Fernandes and Lintzmayer (J. Comput. Syst. Sci. 135: 158-174,2023): They prove that any <math><mi>α</mi></math> -approximation for the <i>hereditary</i> 3-<i>set packing problem</i>, a special case of weighted 3-set packing, yields a <math><mrow><mo>max</mo> <mo>{</mo> <mfrac><mn>4</mn> <mn>3</mn></mfrac> <mo>,</mo> <mi>α</mi> <mo>}</mo></mrow> </math> -approximation for the MLSA in dags, and provide a <math><mfrac><mn>7</mn> <mn>5</mn></mfrac> </math> -approximation for the hereditary 3-set packing problem. In this paper, we improve upon this result by providing a <math><mfrac><mn>4</mn> <mn>3</mn></mfrac> </math> -approximation for the hereditary 3-set packing problem, and, thus, the MLSA in dags. The algorithm that we study is a simple local search procedure considering swaps of size up to 10 and can be analyzed via a two-stage charging argument. We further provide a clear picture of the general connection between the MLSA in dags and set packing by rephrasing the MLSA in dags as a <i>hereditary set packing problem</i>. With a much simpler proof, we extend the reduction by Fernandes and Lintzmayer and show that an <math><mi>α</mi></math> -approximation for the <i>hereditary</i> <i>k</i>-<i>set packing problem</i> implies a <math><mrow><mo>max</mo> <mo>{</mo> <mfrac><mrow><mi>k</mi> <mo>+</mo> <mn>1</mn></mrow> <mi>k</mi></mfrac> <mo>,</mo> <mi>α</mi> <mo>}</mo></mrow> </math> -approximation for the MLSA dags. On the other hand, we provide lower bound examples proving that our approximation guarantee of <math><mfrac><mn>4</mn> <mn>3</mn></mfrac> </math> is best possible for local search algorithms with constant improvement size.</p>","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":"216 1-2","pages":"111-133"},"PeriodicalIF":2.5,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13124901/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147817115","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A better-than-1.6-approximation for prize-collecting TSP.","authors":"Jannis Blauth, Nathan Klein, Martin Nägele","doi":"10.1007/s10107-025-02221-4","DOIUrl":"https://doi.org/10.1007/s10107-025-02221-4","url":null,"abstract":"<p><p>Prize-Collecting TSP is a variant of the traveling salesperson problem where one may drop vertices from the tour at the cost of vertex-dependent penalties. The quality of a solution is then measured by adding the length of the tour and the sum of all penalties of vertices that are not visited. We present a polynomial-time approximation algorithm with an approximation guarantee slightly below 1.6, where the guarantee is with respect to the natural linear programming relaxation of the problem. This improves upon the previous best-known approximation ratio of 1.774. Our approach is based on a known decomposition for solutions of this linear relaxation into rooted trees. Our algorithm takes a tree from this decomposition and then performs a pruning step before doing parity correction on the remainder. Using a simple analysis, we bound the approximation guarantee of the proposed algorithm by <math><mrow><mo>(</mo> <mn>1</mn> <mo>+</mo> <msqrt><mn>5</mn></msqrt> <mo>)</mo> <mrow><mo>/</mo></mrow> <mn>2</mn> <mo>≈</mo> <mn>1.618</mn></mrow> </math> , the golden ratio. With some additional technical care we further improve the approximation guarantee to 1.599. Furthermore, we show that for the path version of Prize-Collecting TSP (known as Prize-Collecting Stroll) our approach yields an approximation guarantee of 1.6662, improving upon the previous best-known guarantee of 1.926.</p>","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":"216 1-2","pages":"87-109"},"PeriodicalIF":2.5,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13124855/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147817169","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"MDP modeling for multi-stage stochastic programs.","authors":"David P Morton, Oscar Dowson, Bernardo K Pagnoncelli","doi":"10.1007/s10107-026-02368-8","DOIUrl":"https://doi.org/10.1007/s10107-026-02368-8","url":null,"abstract":"<p><p>We study a class of multi-stage stochastic programs, which incorporate modeling features from Markov decision processes (MDPs). This class includes structured MDPs with continuous action and state spaces. We extend policy graphs to include decision-dependent uncertainty for one-step transition probabilities as well as a limited form of statistical learning. We focus on the expressiveness of our modeling approach, illustrating ideas with a series of examples of increasing complexity. As a solution method, we develop new variants of stochastic dual dynamic programming, including approximations to handle non-convexities.</p>","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":"217 1-2","pages":"43-78"},"PeriodicalIF":2.1,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13473183/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148761735","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"<i>P</i>-split formulations: a class of intermediate formulations between big-M and convex hull for disjunctive constraints.","authors":"Jan Kronqvist, Ruth Misener, Calvin Tsay","doi":"10.1007/s10107-025-02232-1","DOIUrl":"10.1007/s10107-025-02232-1","url":null,"abstract":"<p><p>We develop a class of mixed-integer formulations for disjunctive constraints intermediate to the big-M and convex hull formulations in terms of relaxation strength. The main idea is to capture the best of both the big-M and convex hull formulations: a computationally light formulation with a tight relaxation. The \"<i>P</i>-split\" formulations are based on a lifted transformation that splits convex additively separable constraints into <i>P</i> partitions and forms the convex hull of the linearized and partitioned disjunction. The \"<i>P</i>-split\" formulations are derived for disjunctive constraints with convex constraints within each disjunct, and we generalize the results for the case with nonconvex constraints within the disjuncts. We analyze the continuous relaxation of the <i>P</i>-split formulations and show that, under certain assumptions, the formulations form a hierarchy starting from a big-M equivalent and converging to the convex hull. We computationally compare the <i>P</i>-split formulations against big-M and convex hull formulations on 344 test instances. The test problems include K-means clustering, semi-supervised clustering, P_ball problems, and optimization over trained ReLU neural networks. The computational results show promising potential of the <i>P</i>-split formulations. For many of the test problems, <i>P</i>-split formulations are solved with a similar number of explored nodes as the convex hull formulation, while reducing the solution time by an order of magnitude and outperforming big-M both in time and number of explored nodes.</p>","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":"218 1-2","pages":"57-94"},"PeriodicalIF":2.1,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13498507/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148794998","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the number of degenerate simplex pivots.","authors":"Kirill Kukharenko, Laura Sanità","doi":"10.1007/s10107-026-02349-x","DOIUrl":"https://doi.org/10.1007/s10107-026-02349-x","url":null,"abstract":"<p><p>The simplex algorithm is one of the most popular algorithms to solve linear programs (LPs). Starting at an extreme point solution of an LP, it performs a sequence of basis exchanges (called pivots) that allows one to move to a better extreme point along an improving edge-direction of the underlying polyhedron. A key issue in the simplex algorithm's performance is <i>degeneracy</i>, which may lead to a (potentially long) sequence of basis exchanges which do not change the current extreme point solution. In this paper, we prove that one can employ any improving feasible direction at an extreme point to limit the number of consecutive degenerate pivots that the simplex algorithm performs to <math><mrow><mi>n</mi> <mo>-</mo> <mi>m</mi> <mo>-</mo> <mn>1</mn></mrow> </math> , where <i>n</i> is the number of variables and <i>m</i> is the number of equality constraints of a given LP in standard equality form.</p>","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":"216 1-2","pages":"757-791"},"PeriodicalIF":2.1,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13368946/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148456272","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}