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Machine learning augmented branch and bound for mixed integer linear programming 混合整数线性规划的机器学习增强分支与约束
IF 2.7 2区 数学
Mathematical Programming Pub Date : 2024-08-22 DOI: 10.1007/s10107-024-02130-y
Lara Scavuzzo, Karen Aardal, Andrea Lodi, Neil Yorke-Smith
{"title":"Machine learning augmented branch and bound for mixed integer linear programming","authors":"Lara Scavuzzo, Karen Aardal, Andrea Lodi, Neil Yorke-Smith","doi":"10.1007/s10107-024-02130-y","DOIUrl":"https://doi.org/10.1007/s10107-024-02130-y","url":null,"abstract":"<p>Mixed Integer Linear Programming (MILP) is a pillar of mathematical optimization that offers a powerful modeling language for a wide range of applications. The main engine for solving MILPs is the branch-and-bound algorithm. Adding to the enormous algorithmic progress in MILP solving of the past decades, in more recent years there has been an explosive development in the use of machine learning for enhancing all main tasks involved in the branch-and-bound algorithm. These include primal heuristics, branching, cutting planes, node selection and solver configuration decisions. This article presents a survey of such approaches, addressing the vision of integration of machine learning and mathematical optimization as complementary technologies, and how this integration can benefit MILP solving. In particular, we give detailed attention to machine learning algorithms that automatically optimize some metric of branch-and-bound efficiency. We also address appropriate MILP representations, benchmarks and software tools used in the context of applying learning algorithms.</p>","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":"6 1","pages":""},"PeriodicalIF":2.7,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142183169","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
On the strength of Lagrangian duality in multiobjective integer programming 论多目标整数编程中拉格朗日对偶性的强度
IF 2.7 2区 数学
Mathematical Programming Pub Date : 2024-08-20 DOI: 10.1007/s10107-024-02121-z
Matthew Brun, Tyler Perini, Saumya Sinha, Andrew J. Schaefer
{"title":"On the strength of Lagrangian duality in multiobjective integer programming","authors":"Matthew Brun, Tyler Perini, Saumya Sinha, Andrew J. Schaefer","doi":"10.1007/s10107-024-02121-z","DOIUrl":"https://doi.org/10.1007/s10107-024-02121-z","url":null,"abstract":"<p>This paper investigates the potential of Lagrangian relaxations to generate quality bounds on non-dominated images of multiobjective integer programs (MOIPs). Under some conditions on the relaxed constraints, we show that a set of Lagrangian relaxations can provide bounds that coincide with every bound generated by the convex hull relaxation. We also provide a guarantee of the relative quality of the Lagrangian bound at unsupported solutions. These results imply that, if the relaxed feasible region is bounded, some Lagrangian bounds will be strictly better than some convex hull bounds. We demonstrate that there exist Lagrangian multipliers which are sparse, satisfy a complementary slackness property, and generate tight relaxations at supported solutions. However, if all constraints are dualized, a relaxation can never be tight at an unsupported solution. These results characterize the strength of the Lagrangian dual at efficient solutions of an MOIP.</p>","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":"23 1","pages":""},"PeriodicalIF":2.7,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142183168","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
Convexification techniques for fractional programs 分数程序的凸化技术
IF 2.7 2区 数学
Mathematical Programming Pub Date : 2024-08-16 DOI: 10.1007/s10107-024-02131-x
Taotao He, Siyue Liu, Mohit Tawarmalani
{"title":"Convexification techniques for fractional programs","authors":"Taotao He, Siyue Liu, Mohit Tawarmalani","doi":"10.1007/s10107-024-02131-x","DOIUrl":"https://doi.org/10.1007/s10107-024-02131-x","url":null,"abstract":"<p>This paper develops a correspondence relating convex hulls of fractional functions with those of polynomial functions over the same domain. Using this result, we develop a number of new reformulations and relaxations for fractional programming problems. First, we relate <span>(0mathord {-}1)</span> problems involving a ratio of affine functions with the boolean quadric polytope, and use inequalities for the latter to develop tighter formulations for the former. Second, we derive a new formulation to optimize a ratio of quadratic functions over a polytope using copositive programming. Third, we show that univariate fractional functions can be convexified using moment hulls. Fourth, we develop a new hierarchy of relaxations that converges finitely to the simultaneous convex hull of a collection of ratios of affine functions of <span>(0mathord {-}1)</span> variables. Finally, we demonstrate theoretically and computationally that our techniques close a significant gap relative to state-of-the-art relaxations, require much less computational effort, and can solve larger problem instances.</p>","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":"33 1","pages":""},"PeriodicalIF":2.7,"publicationDate":"2024-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142183167","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
On the correlation gap of matroids 关于矩阵的相关性差距
IF 2.7 2区 数学
Mathematical Programming Pub Date : 2024-08-08 DOI: 10.1007/s10107-024-02116-w
Edin Husić, Zhuan Khye Koh, Georg Loho, László A. Végh
{"title":"On the correlation gap of matroids","authors":"Edin Husić, Zhuan Khye Koh, Georg Loho, László A. Végh","doi":"10.1007/s10107-024-02116-w","DOIUrl":"https://doi.org/10.1007/s10107-024-02116-w","url":null,"abstract":"<p>A set function can be extended to the unit cube in various ways; the correlation gap measures the ratio between two natural extensions. This quantity has been identified as the performance guarantee in a range of approximation algorithms and mechanism design settings. It is known that the correlation gap of a monotone submodular function is at least <span>(1-1/e)</span>, and this is tight for simple matroid rank functions. We initiate a fine-grained study of the correlation gap of matroid rank functions. In particular, we present an improved lower bound on the correlation gap as parametrized by the rank and girth of the matroid. We also show that for any matroid, the correlation gap of its weighted rank function is minimized under uniform weights. Such improved lower bounds have direct applications for submodular maximization under matroid constraints, mechanism design, and contention resolution schemes.</p>","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":"24 1","pages":""},"PeriodicalIF":2.7,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141933211","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
Configuration balancing for stochastic requests 随机请求的配置平衡
IF 2.7 2区 数学
Mathematical Programming Pub Date : 2024-08-08 DOI: 10.1007/s10107-024-02132-w
Franziska Eberle, Anupam Gupta, Nicole Megow, Benjamin Moseley, Rudy Zhou
{"title":"Configuration balancing for stochastic requests","authors":"Franziska Eberle, Anupam Gupta, Nicole Megow, Benjamin Moseley, Rudy Zhou","doi":"10.1007/s10107-024-02132-w","DOIUrl":"https://doi.org/10.1007/s10107-024-02132-w","url":null,"abstract":"<p>The configuration balancing problem with stochastic requests generalizes well-studied resource allocation problems such as load balancing and virtual circuit routing. There are given <i>m</i> resources and <i>n</i> requests; each request has multiple possible <i>configurations</i>, each of which increases the load of each resource by some amount. The goal is to select one configuration for each request to minimize the <i>makespan</i>: the load of the most-loaded resource. In the stochastic setting, the amount by which a configuration increases the resource load is uncertain until the configuration is chosen, but we are given a probability distribution. We develop both offline and online algorithms for configuration balancing with stochastic requests. When the requests are known offline, we give a non-adaptive policy for configuration balancing with stochastic requests that <span>(O(frac{log m}{log log m}))</span>-approximates the optimal adaptive policy, which matches a known lower bound for the special case of load balancing on identical machines. When requests arrive online in a list, we give a non-adaptive policy that is <span>(O(log m))</span> competitive. Again, this result is asymptotically tight due to information-theoretic lower bounds for special cases (e.g., for load balancing on unrelated machines). Finally, we show how to leverage adaptivity in the special case of load balancing on <i>related</i> machines to obtain a constant-factor approximation offline and an <span>(O(log log m))</span>-approximation online. A crucial technical ingredient in all of our results is a new structural characterization of the optimal adaptive policy that allows us to limit the correlations between its decisions.\u0000</p>","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":"30 1","pages":""},"PeriodicalIF":2.7,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141933214","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
Nonsmooth convex–concave saddle point problems with cardinality penalties 有数量惩罚的非光滑凸凹鞍点问题
IF 2.7 2区 数学
Mathematical Programming Pub Date : 2024-08-08 DOI: 10.1007/s10107-024-02123-x
Wei Bian, Xiaojun Chen
{"title":"Nonsmooth convex–concave saddle point problems with cardinality penalties","authors":"Wei Bian, Xiaojun Chen","doi":"10.1007/s10107-024-02123-x","DOIUrl":"https://doi.org/10.1007/s10107-024-02123-x","url":null,"abstract":"<p>In this paper, we focus on a class of convexly constrained nonsmooth convex–concave saddle point problems with cardinality penalties. Although such nonsmooth nonconvex–nonconcave and discontinuous min–max problems may not have a saddle point, we show that they have a local saddle point and a global minimax point, and some local saddle points have the lower bound properties. We define a class of strong local saddle points based on the lower bound properties for stability of variable selection. Moreover, we give a framework to construct continuous relaxations of the discontinuous min–max problems based on convolution, such that they have the same saddle points with the original problem. We also establish the relations between the continuous relaxation problems and the original problems regarding local saddle points, global minimax points, local minimax points and stationary points. Finally, we illustrate our results with distributionally robust sparse convex regression, sparse robust bond portfolio construction and sparse convex–concave logistic regression saddle point problems.</p>","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":"85 1","pages":""},"PeriodicalIF":2.7,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141933210","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
Unified smoothing approach for best hyperparameter selection problem using a bilevel optimization strategy 使用双层优化策略解决最佳超参数选择问题的统一平滑法
IF 2.7 2区 数学
Mathematical Programming Pub Date : 2024-08-08 DOI: 10.1007/s10107-024-02113-z
Jan Harold Alcantara, Chieu Thanh Nguyen, Takayuki Okuno, Akiko Takeda, Jein-Shan Chen
{"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","doi":"10.1007/s10107-024-02113-z","DOIUrl":"https://doi.org/10.1007/s10107-024-02113-z","url":null,"abstract":"<p>Strongly motivated from applications in various fields including machine learning, the methodology of sparse optimization has been developed intensively so far. Especially, the advancement of algorithms for solving problems with nonsmooth regularizers has been remarkable. However, those algorithms suppose that weight parameters of regularizers, called hyperparameters hereafter, are pre-fixed, but it is a crucial matter how the best hyperparameter should be selected. In this paper, we focus on the hyperparameter selection of regularizers related to the <span>(ell _p)</span> function with <span>(0&lt;ple 1)</span> and apply a bilevel programming strategy, wherein we need to solve a bilevel problem, whose lower-level problem is nonsmooth, possibly nonconvex and non-Lipschitz. Recently, for solving a bilevel problem for hyperparameter selection of the pure <span>(ell _p (0&lt;p le 1))</span> regularizer Okuno et al. discovered new necessary optimality conditions, called SB(scaled bilevel)-KKT conditions, and further proposed a smoothing-type algorithm using a specific smoothing function. However, this optimality measure is loose in the sense that there could be many points that satisfy the SB-KKT conditions. In this work, we propose new bilevel KKT conditions, which are new necessary optimality conditions tighter than the ones proposed by Okuno et al. Moreover, we propose a unified smoothing approach using smoothing functions that belong to the Chen-Mangasarian class, and then prove that generated iteration points accumulate at bilevel KKT points under milder constraint qualifications. Another contribution is that our approach and analysis are applicable to a wider class of regularizers. Numerical comparisons demonstrate which smoothing functions work well for hyperparameter optimization via bilevel optimization approach.</p>","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":"1 1","pages":""},"PeriodicalIF":2.7,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141933212","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
Optimizing for strategy diversity in the design of video games 在设计电子游戏时优化策略多样性
IF 2.7 2区 数学
Mathematical Programming Pub Date : 2024-08-08 DOI: 10.1007/s10107-024-02126-8
Oussama Hanguir, Will Ma, Jiangze Han, Christopher Thomas Ryan
{"title":"Optimizing for strategy diversity in the design of video games","authors":"Oussama Hanguir, Will Ma, Jiangze Han, Christopher Thomas Ryan","doi":"10.1007/s10107-024-02126-8","DOIUrl":"https://doi.org/10.1007/s10107-024-02126-8","url":null,"abstract":"<p>We consider the problem of designing a linear program that has diverse solutions as the right-hand side varies. This problem arises in video game settings where designers aim to have players use different “weapons” or “tactics” as they progress. We model this design question as a choice over the constraint matrix <i>A</i> and cost vector <i>c</i> to maximize the number of possible <i>supports</i> of unique optimal solutions (what we call “loadouts”) of Linear Programs <span>(max {c^top x mid Ax le b, x ge 0})</span> with nonnegative data considered over all resource vectors <i>b</i>. We provide an upper bound on the optimal number of loadouts and provide a family of constructions that have an asymptotically optimal number of loadouts. The upper bound is based on a connection between our problem and the study of triangulations of point sets arising from polyhedral combinatorics, and specifically the combinatorics of the cyclic polytope. Our asymptotically optimal construction also draws inspiration from the properties of the cyclic polytope.</p>","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":"193 1","pages":""},"PeriodicalIF":2.7,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141933215","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 radial basis function method for noisy global optimisation 噪声全局优化的径向基函数方法
IF 2.7 2区 数学
Mathematical Programming Pub Date : 2024-08-08 DOI: 10.1007/s10107-024-02125-9
Dirk Banholzer, Jörg Fliege, Ralf Werner
{"title":"A radial basis function method for noisy global optimisation","authors":"Dirk Banholzer, Jörg Fliege, Ralf Werner","doi":"10.1007/s10107-024-02125-9","DOIUrl":"https://doi.org/10.1007/s10107-024-02125-9","url":null,"abstract":"<p>We present a novel response surface method for global optimisation of an expensive and noisy (black-box) objective function, where error bounds on the deviation of the observed noisy function values from their true counterparts are available. The method is based on Gutmann’s well-established RBF method for minimising an expensive and deterministic objective function, which has become popular both from a theoretical and practical perspective. To construct suitable radial basis function approximants to the objective function and to determine new sample points for successive evaluation of the expensive noisy objective, the method uses a regularised least-squares criterion. In particular, new points are defined by means of a target value, analogous to the original RBF method. We provide essential convergence results, and provide a numerical illustration of the method by means of a simple test problem.\u0000</p>","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":"7 1","pages":""},"PeriodicalIF":2.7,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141933213","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
Optimizing distortion riskmetrics with distributional uncertainty 优化具有分布不确定性的扭曲风险度量
IF 2.7 2区 数学
Mathematical Programming Pub Date : 2024-07-29 DOI: 10.1007/s10107-024-02128-6
Silvana M. Pesenti, Qiuqi Wang, Ruodu Wang
{"title":"Optimizing distortion riskmetrics with distributional uncertainty","authors":"Silvana M. Pesenti, Qiuqi Wang, Ruodu Wang","doi":"10.1007/s10107-024-02128-6","DOIUrl":"https://doi.org/10.1007/s10107-024-02128-6","url":null,"abstract":"<p>Optimization of distortion riskmetrics with distributional uncertainty has wide applications in finance and operations research. Distortion riskmetrics include many commonly applied risk measures and deviation measures, which are not necessarily monotone or convex. One of our central findings is a unifying result that allows to convert an optimization of a non-convex distortion riskmetric with distributional uncertainty to a convex one induced from the concave envelope of the distortion function, leading to practical tractability. A sufficient condition to the unifying equivalence result is the novel notion of closedness under concentration, a variation of which is also shown to be necessary for the equivalence. Our results include many special cases that are well studied in the optimization literature, including but not limited to optimizing probabilities, Value-at-Risk, Expected Shortfall, Yaari’s dual utility, and differences between distortion risk measures, under various forms of distributional uncertainty. We illustrate our theoretical results via applications to portfolio optimization, optimization under moment constraints, and preference robust optimization.</p>","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":"48 1","pages":""},"PeriodicalIF":2.7,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141866962","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
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