Mathematical Programming最新文献_第5页

Polarized consensus-based dynamics for optimization and sampling 基于共识的极化动态优化和抽样

IF 2.7 2区数学

Mathematical Programming Pub Date : 2024-05-31 DOI: 10.1007/s10107-024-02095-y

Leon Bungert, Tim Roith, Philipp Wacker

{"title":"Polarized consensus-based dynamics for optimization and sampling","authors":"Leon Bungert, Tim Roith, Philipp Wacker","doi":"10.1007/s10107-024-02095-y","DOIUrl":"https://doi.org/10.1007/s10107-024-02095-y","url":null,"abstract":"In this paper we propose polarized consensus-based dynamics in order to make consensus-based optimization (CBO) and sampling (CBS) applicable for objective functions with several global minima or distributions with many modes, respectively. For this, we “polarize” the dynamics with a localizing kernel and the resulting model can be viewed as a bounded confidence model for opinion formation in the presence of common objective. Instead of being attracted to a common weighted mean as in the original consensus-based methods, which prevents the detection of more than one minimum or mode, in our method every particle is attracted to a weighted mean which gives more weight to nearby particles. We prove that in the mean-field regime the polarized CBS dynamics are unbiased for Gaussian targets. We also prove that in the zero temperature limit and for sufficiently well-behaved strongly convex objectives the solution of the Fokker–Planck equation converges in the Wasserstein-2 distance to a Dirac measure at the minimizer. Finally, we propose a computationally more efficient generalization which works with a predefined number of clusters and improves upon our polarized baseline method for high-dimensional optimization.\u0000","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":null,"pages":null},"PeriodicalIF":2.7,"publicationDate":"2024-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141189696","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Exploiting the polyhedral geometry of stochastic linear bilevel programming 利用随机线性双层程序设计的多面体几何学

IF 2.7 2区数学

Mathematical Programming Pub Date : 2024-05-27 DOI: 10.1007/s10107-024-02097-w

Gonzalo Muñoz, David Salas, Anton Svensson

{"title":"Exploiting the polyhedral geometry of stochastic linear bilevel programming","authors":"Gonzalo Muñoz, David Salas, Anton Svensson","doi":"10.1007/s10107-024-02097-w","DOIUrl":"https://doi.org/10.1007/s10107-024-02097-w","url":null,"abstract":"We study linear bilevel programming problems whose lower-level objective is given by a random cost vector with known distribution. We consider the case where this distribution is nonatomic, allowing to reformulate the problem of the leader using the Bayesian approach in the sense of Salas and Svensson (SIAM J Optim 33(3):2311–2340, 2023), with a decision-dependent distribution that concentrates on the vertices of the feasible set of the follower’s problem. We call this a vertex-supported belief. We prove that this formulation is piecewise affine over the so-called chamber complex of the feasible set of the high-point relaxation. We propose two algorithmic approaches to solve general problems enjoying this last property. The first one is based on enumerating the vertices of the chamber complex. This approach is not scalable, but we present it as a computational baseline and for its theoretical interest. The second one is a Monte-Carlo approximation scheme based on the fact that randomly drawn points of the domain lie, with probability 1, in the interior of full-dimensional chambers, where the problem (restricted to this chamber) can be reduced to a linear program. Finally, we evaluate these methods through computational experiments showing both approaches’ advantages and challenges.\u0000","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":null,"pages":null},"PeriodicalIF":2.7,"publicationDate":"2024-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141171635","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Information complexity of mixed-integer convex optimization 混合整数凸优化的信息复杂性

IF 2.7 2区数学

Mathematical Programming Pub Date : 2024-05-27 DOI: 10.1007/s10107-024-02099-8

Amitabh Basu, Hongyi Jiang, Phillip Kerger, Marco Molinaro

{"title":"Information complexity of mixed-integer convex optimization","authors":"Amitabh Basu, Hongyi Jiang, Phillip Kerger, Marco Molinaro","doi":"10.1007/s10107-024-02099-8","DOIUrl":"https://doi.org/10.1007/s10107-024-02099-8","url":null,"abstract":"We investigate the information complexity of mixed-integer convex optimization under different types of oracles. We establish new lower bounds for the standard first-order oracle, improving upon the previous best known lower bound. This leaves only a lower order linear term (in the dimension) as the gap between the lower and upper bounds. This is derived as a corollary of a more fundamental “transfer” result that shows how lower bounds on information complexity of continuous convex optimization under different oracles can be transferred to the mixed-integer setting in a black-box manner. Further, we (to the best of our knowledge) initiate the study of, and obtain the first set of results on, information complexity under oracles that only reveal partial first-order information, e.g., where one can only make a binary query over the function value or subgradient at a given point. We give algorithms for (mixed-integer) convex optimization that work under these less informative oracles. We also give lower bounds showing that, for some of these oracles, every algorithm requires more iterations to achieve a target error compared to when complete first-order information is available. That is, these oracles are provably less informative than full first-order oracles for the purpose of optimization.","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":null,"pages":null},"PeriodicalIF":2.7,"publicationDate":"2024-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141171634","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

A tight $$(1.5+epsilon )$$ -approximation for unsplittable capacitated vehicle routing on trees 树上不可分容车辆路由的$$(1.5+epsilon )$$ 精确近似值

IF 2.7 2区数学

Mathematical Programming Pub Date : 2024-05-25 DOI: 10.1007/s10107-024-02094-z

Claire Mathieu, Hang Zhou

{"title":"A tight $$(1.5+epsilon )$$ -approximation for unsplittable capacitated vehicle routing on trees","authors":"Claire Mathieu, Hang Zhou","doi":"10.1007/s10107-024-02094-z","DOIUrl":"https://doi.org/10.1007/s10107-024-02094-z","url":null,"abstract":"In the unsplittable capacitated vehicle routing problem (UCVRP) on trees, we are given a rooted tree with edge weights and a subset of vertices of the tree called terminals. Each terminal is associated with a positive demand between 0 and 1. The goal is to find a minimum length collection of tours starting and ending at the root of the tree such that the demand of each terminal is covered by a single tour (i.e., the demand cannot be split), and the total demand of the terminals in each tour does not exceed the capacity of 1.For the special case when all terminals have equal demands, a long line of research culminated in a quasi-polynomial time approximation scheme [Jayaprakash and Salavatipour, TALG 2023] and a polynomial time approximation scheme [Mathieu and Zhou, TALG 2023].In this work, we study the general case when the terminals have arbitrary demands. Our main contribution is a polynomial time ((1.5+epsilon ))-approximation algorithm for the UCVRP on trees. This is the first improvement upon the 2-approximation algorithm more than 30 years ago. Our approximation ratio is essentially best possible, since it is NP-hard to approximate the UCVRP on trees to better than a 1.5 factor.","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":null,"pages":null},"PeriodicalIF":2.7,"publicationDate":"2024-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141153885","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

ReLU neural networks of polynomial size for exact maximum flow computation 用于精确最大流量计算的多项式大小 ReLU 神经网络

IF 2.7 2区数学

Mathematical Programming Pub Date : 2024-05-25 DOI: 10.1007/s10107-024-02096-x

Christoph Hertrich, Leon Sering

引用次数: 0

Stable set polytopes with high lift-and-project ranks for the Lovász–Schrijver SDP operator 针对洛瓦兹-施里弗 SDP 算子的具有高升降级的稳定集合多面体

IF 2.7 2区数学

Mathematical Programming Pub Date : 2024-05-25 DOI: 10.1007/s10107-024-02093-0

Yu Hin Au, Levent Tunçel

引用次数: 0

Optimal general factor problem and jump system intersection 最优一般因素问题与跳跃系统交集

IF 2.7 2区数学

Mathematical Programming Pub Date : 2024-05-23 DOI: 10.1007/s10107-024-02098-9

Yusuke Kobayashi

引用次数: 0

A characterization of maximal homogeneous-quadratic-free sets 最大同质无二次型集合的表征

IF 2.7 2区数学

Mathematical Programming Pub Date : 2024-05-23 DOI: 10.1007/s10107-024-02092-1

Gonzalo Muñoz, Joseph Paat, Felipe Serrano

{"title":"A characterization of maximal homogeneous-quadratic-free sets","authors":"Gonzalo Muñoz, Joseph Paat, Felipe Serrano","doi":"10.1007/s10107-024-02092-1","DOIUrl":"https://doi.org/10.1007/s10107-024-02092-1","url":null,"abstract":"The intersection cut framework was introduced by Balas in 1971 as a method for generating cutting planes in integer optimization. In this framework, one uses a full-dimensional convex S-free set, where S is the feasible region of the integer program, to derive a cut separating S from a non-integral vertex of a linear relaxation of S. Among all S-free sets, it is the inclusion-wise maximal ones that yield the strongest cuts. Recently, this framework has been extended beyond the integer case in order to obtain cutting planes in non-linear settings. In this work, we consider the specific setting when S is defined by a homogeneous quadratic inequality. In this ‘quadratic-free’ setting, every function (Gamma : D^m rightarrow D^n), where (D^k) is the unit sphere in (mathbb {R}^k), generates a representation of a quadratic-free set. While not every (Gamma ) generates a maximal quadratic free set, it is the case that every full-dimensional maximal quadratic free set is generated by some (Gamma ). Our main result shows that the corresponding quadratic-free set is full-dimensional and maximal if and only if (Gamma ) is non-expansive and satisfies a technical condition. This result yields a broader class of maximal S-free sets than previously known. Our result stems from a new characterization of maximal S-free sets (for general S beyond the quadratic setting) based on sequences that ‘expose’ inequalities defining the S-free set.","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":null,"pages":null},"PeriodicalIF":2.7,"publicationDate":"2024-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141152037","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

A linear time algorithm for linearizing quadratic and higher-order shortest path problems 二次方和高阶最短路径问题线性化的线性时间算法

IF 2.7 2区数学

Mathematical Programming Pub Date : 2024-05-09 DOI: 10.1007/s10107-024-02086-z

Eranda Çela, Bettina Klinz, Stefan Lendl, Gerhard J. Woeginger, Lasse Wulf

引用次数: 0

Gaining or losing perspective for convex multivariate functions on box domains 盒域上凸多元函数的得失视角

IF 2.7 2区数学

Mathematical Programming Pub Date : 2024-05-08 DOI: 10.1007/s10107-024-02087-y

Luze Xu, Jon Lee

{"title":"Gaining or losing perspective for convex multivariate functions on box domains","authors":"Luze Xu, Jon Lee","doi":"10.1007/s10107-024-02087-y","DOIUrl":"https://doi.org/10.1007/s10107-024-02087-y","url":null,"abstract":"Mixed-integer nonlinear optimization formulations of the disjunction between the origin and a polytope via a binary indicator variable is broadly used in nonlinear combinatorial optimization for modeling a fixed cost associated with carrying out a group of activities and a convex cost function associated with the levels of the activities. The perspective relaxation of such models is often used to solve to global optimality in a branch-and-bound context, but it typically requires suitable conic solvers and is not compatible with general-purpose NLP software in the presence of other classes of constraints. This motivates the investigation of when simpler but weaker relaxations may be adequate. Comparing the volume (i.e., Lebesgue measure) of the relaxations as a measure of tightness, we lift some of the results related to the simplex case to the box case. In order to compare the volumes of different relaxations in the box case, it is necessary to find an appropriate concave upper bound that preserves the convexity and is minimal, which is more difficult than in the simplex case. To address the challenge beyond the simplex case, the triangulation approach is used.","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":null,"pages":null},"PeriodicalIF":2.7,"publicationDate":"2024-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140941064","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0