Mathematical Programming最新文献_第4页

Matrix discrepancy and the log-rank conjecture 矩阵差异和对数秩猜想

IF 2.7 2区数学

Mathematical Programming Pub Date : 2024-07-05 DOI: 10.1007/s10107-024-02117-9

Benny Sudakov, István Tomon

引用次数: 0

On solving a rank regularized minimization problem via equivalent factorized column-sparse regularized models 通过等效因子化列稀疏正则化模型求解秩正则化最小化问题

IF 2.7 2区数学

Mathematical Programming Pub Date : 2024-07-03 DOI: 10.1007/s10107-024-02103-1

Wenjing Li, Wei Bian, Kim-Chuan Toh

{"title":"On solving a rank regularized minimization problem via equivalent factorized column-sparse regularized models","authors":"Wenjing Li, Wei Bian, Kim-Chuan Toh","doi":"10.1007/s10107-024-02103-1","DOIUrl":"https://doi.org/10.1007/s10107-024-02103-1","url":null,"abstract":"Rank regularized minimization problem is an ideal model for the low-rank matrix completion/recovery problem. The matrix factorization approach can transform the high-dimensional rank regularized problem to a low-dimensional factorized column-sparse regularized problem. The latter can greatly facilitate fast computations in applicable algorithms, but needs to overcome the simultaneous non-convexity of the loss and regularization functions. In this paper, we consider the factorized column-sparse regularized model. Firstly, we optimize this model with bound constraints, and establish a certain equivalence between the optimized factorization problem and rank regularized problem. Further, we strengthen the optimality condition for stationary points of the factorization problem and define the notion of strong stationary point. Moreover, we establish the equivalence between the factorization problem and its nonconvex relaxation in the sense of global minimizers and strong stationary points. To solve the factorization problem, we design two types of algorithms and give an adaptive method to reduce their computation. The first algorithm is from the relaxation point of view and its iterates own some properties from global minimizers of the factorization problem after finite iterations. We give some analysis on the convergence of its iterates to a strong stationary point. The second algorithm is designed for directly solving the factorization problem. We improve the PALM algorithm introduced by Bolte et al. (Math Program Ser A 146:459–494, 2014) for the factorization problem and give its improved convergence results. Finally, we conduct numerical experiments to show the promising performance of the proposed model and algorithms for low-rank matrix completion.","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":null,"pages":null},"PeriodicalIF":2.7,"publicationDate":"2024-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141526139","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 tightness of an SDP relaxation for homogeneous QCQP with three real or four complex homogeneous constraints 论具有三个实数或四个复数同质约束条件的同质 QCQP 的 SDP 松弛的紧密性

IF 2.7 2区数学

Mathematical Programming Pub Date : 2024-06-21 DOI: 10.1007/s10107-024-02105-z

Wenbao Ai, Wei Liang, Jianhua Yuan

{"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","doi":"10.1007/s10107-024-02105-z","DOIUrl":"https://doi.org/10.1007/s10107-024-02105-z","url":null,"abstract":"In this paper, we consider the problem of minimizing a general homogeneous quadratic function, subject to three real or four complex homogeneous quadratic inequality or equality constraints. For this problem, we present a sufficient and necessary test condition to detect whether its standard semi-definite programming (SDP) relaxation is tight or not. This test condition is based on only an optimal solution pair of the SDP relaxation and its dual. When the tightness is confirmed, a global optimal solution of the original problem is found simultaneously in polynomial-time. While the tightness does not hold, the SDP relaxation and its dual are proved to have the unique optimal solutions. Moreover, the Lagrangian version of such the test condition is specified for non-homogeneous cases. Based on the Lagrangian version, it is proved that several latest sufficient conditions to test the SDP tightness are contained by our test condition under the situation of two constraints. Thirdly, as an application of the test condition, S-lemma and Yuan’s lemma are generalized to three real and four complex quadratic forms first under certain exact conditions, which improves some classical results in literature. Finally, a counterexample is presented to show that the test condition cannot be simply extended to four real or five complex homogeneous quadratic constraints.","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":null,"pages":null},"PeriodicalIF":2.7,"publicationDate":"2024-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141503531","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 slope generalization of Attouch theorem 阿图什定理的斜率一般化

IF 2.7 2区数学

Mathematical Programming Pub Date : 2024-06-20 DOI: 10.1007/s10107-024-02108-w

Aris Daniilidis, David Salas, Sebastián Tapia-García

引用次数: 0

Generalized scaling for the constrained maximum-entropy sampling problem 受限最大熵抽样问题的广义缩放

IF 2.7 2区数学

Mathematical Programming Pub Date : 2024-06-20 DOI: 10.1007/s10107-024-02101-3

Zhongzhu Chen, Marcia Fampa, Jon Lee

引用次数: 0

A PTAS for the horizontal rectangle stabbing problem 水平矩形刺入问题的 PTAS

IF 2.7 2区数学

Mathematical Programming Pub Date : 2024-06-13 DOI: 10.1007/s10107-024-02106-y

Arindam Khan, Aditya Subramanian, Andreas Wiese

{"title":"A PTAS for the horizontal rectangle stabbing problem","authors":"Arindam Khan, Aditya Subramanian, Andreas Wiese","doi":"10.1007/s10107-024-02106-y","DOIUrl":"https://doi.org/10.1007/s10107-024-02106-y","url":null,"abstract":"We study rectangle stabbing problems in which we are given n axis-aligned rectangles in the plane that we want to stab, that is, we want to select line segments such that for each given rectangle there is a line segment that intersects two opposite edges of it. In the horizontal rectangle stabbing problem (Stabbing), the goal is to find a set of horizontal line segments of minimum total length such that all rectangles are stabbed. In the horizontal–vertical stabbing problem (HV-Stabbing), the goal is to find a set of rectilinear (that is, either vertical or horizontal) line segments of minimum total length such that all rectangles are stabbed. Both variants are NP-hard. Chan et al. (ISAAC, 2018) initiated the study of these problems by providing constant approximation algorithms. Recently, Eisenbrand et al. (A QPTAS for stabbing rectangles, 2021) have presented a QPTAS and a polynomial-time 8-approximation algorithm for Stabbing, but it was open whether the problem admits a PTAS. In this paper, we obtain a PTAS for Stabbing, settling this question. For HV-Stabbing, we obtain a ((2+varepsilon ))-approximation. We also obtain PTASs for special cases of HV-Stabbing: (i) when all rectangles are squares, (ii) when each rectangle’s width is at most its height, and (iii) when all rectangles are (delta )-large, that is, have at least one edge whose length is at least (delta ), while all edge lengths are at most 1. Our result also implies improved approximations for other problems such as generalized minimum Manhattan network.","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":null,"pages":null},"PeriodicalIF":2.7,"publicationDate":"2024-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141503476","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 unified framework for symmetry handling 处理对称性的统一框架

IF 2.7 2区数学

Mathematical Programming Pub Date : 2024-06-04 DOI: 10.1007/s10107-024-02102-2

Jasper van Doornmalen, Christopher Hojny

引用次数: 0

An asynchronous proximal bundle method 异步近端捆绑法

IF 2.7 2区数学

Mathematical Programming Pub Date : 2024-06-04 DOI: 10.1007/s10107-024-02088-x

Frank Fischer

引用次数: 0

Universal heavy-ball method for nonconvex optimization under Hölder continuous Hessians 赫尔德连续赫西亚条件下非凸优化的通用重球法

IF 2.7 2区数学

Mathematical Programming Pub Date : 2024-06-04 DOI: 10.1007/s10107-024-02100-4

Naoki Marumo, Akiko Takeda

{"title":"Universal heavy-ball method for nonconvex optimization under Hölder continuous Hessians","authors":"Naoki Marumo, Akiko Takeda","doi":"10.1007/s10107-024-02100-4","DOIUrl":"https://doi.org/10.1007/s10107-024-02100-4","url":null,"abstract":"We propose a new first-order method for minimizing nonconvex functions with Lipschitz continuous gradients and Hölder continuous Hessians. The proposed algorithm is a heavy-ball method equipped with two particular restart mechanisms. It finds a solution where the gradient norm is less than (varepsilon ) in (O(H_{nu }^{frac{1}{2 + 2 nu }} varepsilon ^{- frac{4 + 3 nu }{2 + 2 nu }})) function and gradient evaluations, where (nu in [0, 1]) and (H_{nu }) are the Hölder exponent and constant, respectively. This complexity result covers the classical bound of (O(varepsilon ^{-2})) for (nu = 0) and the state-of-the-art bound of (O(varepsilon ^{-7/4})) for (nu = 1). Our algorithm is (nu )-independent and thus universal; it automatically achieves the above complexity bound with the optimal (nu in [0, 1]) without knowledge of (H_{nu }). In addition, the algorithm does not require other problem-dependent parameters as input, including the gradient’s Lipschitz constant or the target accuracy (varepsilon ). Numerical results illustrate that the proposed method is promising.","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":null,"pages":null},"PeriodicalIF":2.7,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141251840","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

The complexity of first-order optimization methods from a metric perspective 从度量角度看一阶优化方法的复杂性

IF 2.7 2区数学

Mathematical Programming Pub Date : 2024-06-04 DOI: 10.1007/s10107-024-02091-2

A. S. Lewis, Tonghua Tian

引用次数: 0