Mathematical Programming最新文献_第9页

Submodular maximization and its generalization through an intersection cut lens 次模态最大化及其通过交叉切分透镜的广义化

IF 2.7 2区数学

Mathematical Programming Pub Date : 2024-02-19 DOI: 10.1007/s10107-024-02059-2

Liding Xu, Leo Liberti

{"title":"Submodular maximization and its generalization through an intersection cut lens","authors":"Liding Xu, Leo Liberti","doi":"10.1007/s10107-024-02059-2","DOIUrl":"https://doi.org/10.1007/s10107-024-02059-2","url":null,"abstract":"We study a mixed-integer set (mathcal {S}:={(x,t) in {0,1}^n times mathbb {R}: f(x) ge t}) arising in the submodular maximization problem, where f is a submodular function defined over ({0,1}^n). We use intersection cuts to tighten a polyhedral outer approximation of (mathcal {S}). We construct a continuous extension (bar{textsf{F}}_f) of f, which is convex and defined over the entire space (mathbb {R}^n). We show that the epigraph ({{,textrm{epi},}}(bar{textsf{F}}_f)) of (bar{textsf{F}}_f) is an (mathcal {S})-free set, and characterize maximal (mathcal {S})-free sets containing ({{,textrm{epi},}}(bar{textsf{F}}_f)). We propose a hybrid discrete Newton algorithm to compute an intersection cut efficiently and exactly. Our results are generalized to the hypograph or the superlevel set of a submodular-supermodular function over the Boolean hypercube, which is a model for discrete nonconvexity. A consequence of these results is intersection cuts for Boolean multilinear constraints. We evaluate our techniques on max cut, pseudo Boolean maximization, and Bayesian D-optimal design problems within a MIP solver.","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":"111 1","pages":""},"PeriodicalIF":2.7,"publicationDate":"2024-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140886471","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

Deciding whether a lattice has an orthonormal basis is in co-NP 决定一个网格是否有正交基础属于共 NP

IF 2.7 2区数学

Mathematical Programming Pub Date : 2024-02-19 DOI: 10.1007/s10107-023-02052-1

Christoph Hunkenschröder

引用次数: 0

Convex hulls of monomial curves, and a sparse positivstellensatz 单项式曲线的凸壳和稀疏正定定理

IF 2.7 2区数学

Mathematical Programming Pub Date : 2024-02-16 DOI: 10.1007/s10107-024-02060-9

Gennadiy Averkov, Claus Scheiderer

{"title":"Convex hulls of monomial curves, and a sparse positivstellensatz","authors":"Gennadiy Averkov, Claus Scheiderer","doi":"10.1007/s10107-024-02060-9","DOIUrl":"https://doi.org/10.1007/s10107-024-02060-9","url":null,"abstract":"Consider the closed convex hull K of a monomial curve given parametrically as ((t^{m_1},ldots ,t^{m_n})), with the parameter t varying in an interval I. We show, using constructive arguments, that K admits a lifted semidefinite description by (mathcal {O}(d)) linear matrix inequalities (LMIs), each of size (leftlfloor frac{n}{2} rightrfloor +1), where (d= max {m_1,ldots ,m_n}) is the degree of the curve. On the dual side, we show that if a univariate polynomial p(t) of degree d with at most (2k+1) monomials is non-negative on ({mathbb {R}}_+), then p admits a representation (p = t^0 sigma _0 + cdots + t^{d-k} sigma _{d-k}), where the polynomials (sigma _0,ldots ,sigma _{d-k}) are sums of squares and (deg (sigma _i) le 2k). The latter is a univariate positivstellensatz for sparse polynomials, with non-negativity of p being certified by sos polynomials whose degree only depends on the sparsity of p. Our results fit into the general attempt of formulating polynomial optimization problems as semidefinite problems with LMIs of small size. Such small-size descriptions are much more tractable from a computational viewpoint.","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":"10 1","pages":""},"PeriodicalIF":2.7,"publicationDate":"2024-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140886077","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

Semiglobal exponential stability of the discrete-time Arrow-Hurwicz-Uzawa primal-dual algorithm for constrained optimization 约束优化离散时 Arrow-Hurwicz-Uzawa primal-dual 算法的半全局指数稳定性

IF 2.7 2区数学

Mathematical Programming Pub Date : 2024-02-12 DOI: 10.1007/s10107-023-02051-2

Michelangelo Bin, Ivano Notarnicola, Thomas Parisini

引用次数: 0

A general framework for multi-marginal optimal transport 多边际优化运输的总体框架

IF 2.7 2区数学

Mathematical Programming Pub Date : 2024-02-05 DOI: 10.1007/s10107-023-02032-5

Brendan Pass, Adolfo Vargas-Jiménez

{"title":"A general framework for multi-marginal optimal transport","authors":"Brendan Pass, Adolfo Vargas-Jiménez","doi":"10.1007/s10107-023-02032-5","DOIUrl":"https://doi.org/10.1007/s10107-023-02032-5","url":null,"abstract":"We establish a general condition on the cost function to obtain uniqueness and Monge solutions in the multi-marginal optimal transport problem, under the assumption that a given collection of the marginals are absolutely continuous with respect to local coordinates. When only the first marginal is assumed to be absolutely continuous, our condition is equivalent to the twist on splitting sets condition found in Kim and Pass (SIAM J Math Anal 46:1538–1550, 2014; SIAM J Math Anal 46:1538–1550, 2014). In addition, it is satisfied by the special cost functions in our earlier work (Pass and Vargas-Jiménez in SIAM J Math Anal 53:4386–4400, 2021; Monge solutions and uniqueness in multi-marginal optimal transport via graph theory. arXiv:2104.09488, 2021), when absolute continuity is imposed on certain other collections of marginals. We also present several new examples of cost functions which violate the twist on splitting sets condition but satisfy the new condition introduced here, including a class of examples arising in robust risk management problems; we therefore obtain Monge solution and uniqueness results for these cost functions, under regularity conditions on an appropriate subset of the marginals.\u0000","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":"9 1","pages":""},"PeriodicalIF":2.7,"publicationDate":"2024-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139760932","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

Frank–Wolfe-type methods for a class of nonconvex inequality-constrained problems 一类非凸不等式约束问题的 Frank-Wolfe 型方法

IF 2.7 2区数学

Mathematical Programming Pub Date : 2024-02-03 DOI: 10.1007/s10107-023-02055-y

{"title":"Frank–Wolfe-type methods for a class of nonconvex inequality-constrained problems","authors":"","doi":"10.1007/s10107-023-02055-y","DOIUrl":"https://doi.org/10.1007/s10107-023-02055-y","url":null,"abstract":"<h3>Abstract</h3> The Frank–Wolfe (FW) method, which implements efficient linear oracles that minimize linear approximations of the objective function over a fixed compact convex set, has recently received much attention in the optimization and machine learning literature. In this paper, we propose a new FW-type method for minimizing a smooth function over a compact set defined as the level set of a single difference-of-convex function, based on new generalized linear-optimization oracles (LO). We show that these LOs can be computed efficiently with closed-form solutions in some important optimization models that arise in compressed sensing and machine learning. In addition, under a mild strict feasibility condition, we establish the subsequential convergence of our nonconvex FW-type method. Since the feasible region of our generalized LO typically changes from iteration to iteration, our convergence analysis is completely different from those existing works in the literature on FW-type methods that deal with fixed feasible regions among subproblems. Finally, motivated by the away steps for accelerating FW-type methods for convex problems, we further design an away-step oracle to supplement our nonconvex FW-type method, and establish subsequential convergence of this variant. Numerical results on the matrix completion problem with standard datasets are presented to demonstrate the efficiency of the proposed FW-type method and its away-step variant. ","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":"10 1","pages":""},"PeriodicalIF":2.7,"publicationDate":"2024-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139677806","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

Designing tractable piecewise affine policies for multi-stage adjustable robust optimization 为多阶段可调鲁棒优化设计可操作的片断仿射策略

IF 2.7 2区数学

Mathematical Programming Pub Date : 2024-02-03 DOI: 10.1007/s10107-023-02053-0

Simon Thomä, Grit Walther, Maximilian Schiffer

{"title":"Designing tractable piecewise affine policies for multi-stage adjustable robust optimization","authors":"Simon Thomä, Grit Walther, Maximilian Schiffer","doi":"10.1007/s10107-023-02053-0","DOIUrl":"https://doi.org/10.1007/s10107-023-02053-0","url":null,"abstract":"We study piecewise affine policies for multi-stage adjustable robust optimization (ARO) problems with non-negative right-hand side uncertainty. First, we construct new dominating uncertainty sets and show how a multi-stage ARO problem can be solved efficiently with a linear program when uncertainty is replaced by these new sets. We then demonstrate how solutions for this alternative problem can be transformed into solutions for the original problem. By carefully choosing the dominating sets, we prove strong approximation bounds for our policies and extend many previously best-known bounds for the two-staged problem variant to its multi-stage counterpart. Moreover, the new bounds are—to the best of our knowledge—the first bounds shown for the general multi-stage ARO problem considered. We extensively compare our policies to other policies from the literature and prove relative performance guarantees. In two numerical experiments, we identify beneficial and disadvantageous properties for different policies and present effective adjustments to tackle the most critical disadvantages of our policies. Overall, the experiments show that our piecewise affine policies can be computed by orders of magnitude faster than affine policies, while often yielding comparable or even better results.","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":"12 1","pages":""},"PeriodicalIF":2.7,"publicationDate":"2024-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139677417","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 constant-factor approximation for generalized malleable scheduling under $$M ^{natural }$$ -concave processing speeds 在 $$M ^{natural }$$ -凹处理速度条件下的广义可延展调度的常系数近似值

IF 2.7 2区数学

Mathematical Programming Pub Date : 2024-01-29 DOI: 10.1007/s10107-023-02054-z

Dimitris Fotakis, Jannik Matuschke, Orestis Papadigenopoulos

引用次数: 0

Adjustability in robust linear optimization 稳健线性优化中的可调整性

IF 2.7 2区数学

Mathematical Programming Pub Date : 2024-01-27 DOI: 10.1007/s10107-023-02049-w

{"title":"Adjustability in robust linear optimization","authors":"","doi":"10.1007/s10107-023-02049-w","DOIUrl":"https://doi.org/10.1007/s10107-023-02049-w","url":null,"abstract":"<h3>Abstract</h3> We investigate the concept of adjustability—the difference in objective values between two types of dynamic robust optimization formulations: one where (static) decisions are made before uncertainty realization, and one where uncertainty is resolved before (adjustable) decisions. This difference reflects the value of information and decision timing in optimization under uncertainty, and is related to several other concepts such as the optimality of decision rules in robust optimization. We develop a theoretical framework to quantify adjustability based on the input data of a robust optimization problem with a linear objective, linear constraints, and fixed recourse. We make very few additional assumptions. In particular, we do not assume constraint-wise separability or parameter nonnegativity that are commonly imposed in the literature for the study of adjustability. This allows us to study important but previously under-investigated problems, such as formulations with equality constraints and problems with both upper and lower bound constraints. Based on the discovery of an interesting connection between the reformulations of the static and fully adjustable problems, our analysis gives a necessary and sufficient condition—in the form of a theorem-of-the-alternatives—for adjustability to be zero when the uncertainty set is polyhedral. Based on this sharp characterization, we provide two efficient mixed-integer optimization formulations to verify zero adjustability. Then, we develop a constructive approach to quantify adjustability when the uncertainty set is general, which results in an efficient and tight poly-time algorithm to bound adjustability. We demonstrate the efficiency and tightness via both theoretical and numerical analyses.","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":"32 1","pages":""},"PeriodicalIF":2.7,"publicationDate":"2024-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139583987","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

Constrained optimization of rank-one functions with indicator variables 带指标变量的秩一函数的约束优化

IF 2.7 2区数学

Mathematical Programming Pub Date : 2024-01-20 DOI: 10.1007/s10107-023-02047-y

Soroosh Shafiee, Fatma Kılınç-Karzan

{"title":"Constrained optimization of rank-one functions with indicator variables","authors":"Soroosh Shafiee, Fatma Kılınç-Karzan","doi":"10.1007/s10107-023-02047-y","DOIUrl":"https://doi.org/10.1007/s10107-023-02047-y","url":null,"abstract":"Optimization problems involving minimization of a rank-one convex function over constraints modeling restrictions on the support of the decision variables emerge in various machine learning applications. These problems are often modeled with indicator variables for identifying the support of the continuous variables. In this paper we investigate compact extended formulations for such problems through perspective reformulation techniques. In contrast to the majority of previous work that relies on support function arguments and disjunctive programming techniques to provide convex hull results, we propose a constructive approach that exploits a hidden conic structure induced by perspective functions. To this end, we first establish a convex hull result for a general conic mixed-binary set in which each conic constraint involves a linear function of independent continuous variables and a set of binary variables. We then demonstrate that extended representations of sets associated with epigraphs of rank-one convex functions over constraints modeling indicator relations naturally admit such a conic representation. This enables us to systematically give perspective formulations for the convex hull descriptions of these sets with nonlinear separable or non-separable objective functions, sign constraints on continuous variables, and combinatorial constraints on indicator variables. We illustrate the efficacy of our results on sparse nonnegative logistic regression problems.","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":"1 1","pages":""},"PeriodicalIF":2.7,"publicationDate":"2024-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139507788","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