Mathematical Programming最新文献

Fast convergence to non-isolated minima: four equivalent conditions for $${textrm{C}^{2}}$$ functions 非孤立极小值的快速收敛：$${textrm{C}^{2}}$函数的四个等价条件

IF 2.7 2区数学

Mathematical Programming Pub Date : 2024-09-19 DOI: 10.1007/s10107-024-02136-6

Quentin Rebjock, Nicolas Boumal

{"title":"Fast convergence to non-isolated minima: four equivalent conditions for $${textrm{C}^{2}}$$ functions","authors":"Quentin Rebjock, Nicolas Boumal","doi":"10.1007/s10107-024-02136-6","DOIUrl":"https://doi.org/10.1007/s10107-024-02136-6","url":null,"abstract":"Optimization algorithms can see their local convergence rates deteriorate when the Hessian at the optimum is singular. These singularities are inescapable when the optima are non-isolated. Yet, under the right circumstances, several algorithms preserve their favorable rates even when optima form a continuum (e.g., due to over-parameterization). This has been explained under various structural assumptions, including the Polyak–Łojasiewicz condition, Quadratic Growth and the Error Bound. We show that, for cost functions which are twice continuously differentiable ((textrm{C}^2)), those three (local) properties are equivalent. Moreover, we show they are equivalent to the Morse–Bott property, that is, local minima form differentiable submanifolds, and the Hessian of the cost function is positive definite along its normal directions. We leverage this insight to improve local convergence guarantees for safe-guarded Newton-type methods under any (hence all) of the above assumptions. First, for adaptive cubic regularization, we secure quadratic convergence even with inexact subproblem solvers. Second, for trust-region methods, we argue capture can fail with an exact subproblem solver, then proceed to show linear convergence with an inexact one (Cauchy steps).","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":null,"pages":null},"PeriodicalIF":2.7,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142258237","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

Complexity of chordal conversion for sparse semidefinite programs with small treewidth 小树宽稀疏半inite程序的和弦转换复杂性

IF 2.7 2区数学

Mathematical Programming Pub Date : 2024-09-17 DOI: 10.1007/s10107-024-02137-5

Richard Y. Zhang

{"title":"Complexity of chordal conversion for sparse semidefinite programs with small treewidth","authors":"Richard Y. Zhang","doi":"10.1007/s10107-024-02137-5","DOIUrl":"https://doi.org/10.1007/s10107-024-02137-5","url":null,"abstract":"If a sparse semidefinite program (SDP), specified over (ntimes n) matrices and subject to m linear constraints, has an aggregate sparsity graph G with small treewidth, then chordal conversion will sometimes allow an interior-point method to solve the SDP in just (O(m+n)) time per-iteration, which is a significant speedup over the (varOmega (n^{3})) time per-iteration for a direct application of the interior-point method. Unfortunately, the speedup is not guaranteed by an O(1) treewidth in G that is independent of m and n, as a diagonal SDP would have treewidth zero but can still necessitate up to (varOmega (n^{3})) time per-iteration. Instead, we construct an extended aggregate sparsity graph (overline{G}supseteq G) by forcing each constraint matrix (A_{i}) to be its own clique in G. We prove that a small treewidth in (overline{G}) does indeed guarantee that chordal conversion will solve the SDP in (O(m+n)) time per-iteration, to (epsilon )-accuracy in at most (O(sqrt{m+n}log (1/epsilon ))) iterations. This sufficient condition covers many successful applications of chordal conversion, including the MAX-k-CUT relaxation, the Lovász theta problem, sensor network localization, polynomial optimization, and the AC optimal power flow relaxation, thus allowing theory to match practical experience.","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":null,"pages":null},"PeriodicalIF":2.7,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142258110","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

Recycling valid inequalities for robust combinatorial optimization with budgeted uncertainty 具有预算不确定性的稳健组合优化的循环有效不等式

IF 2.7 2区数学

Mathematical Programming Pub Date : 2024-08-29 DOI: 10.1007/s10107-024-02135-7

Christina Büsing, Timo Gersing, Arie M. C. A. Koster

{"title":"Recycling valid inequalities for robust combinatorial optimization with budgeted uncertainty","authors":"Christina Büsing, Timo Gersing, Arie M. C. A. Koster","doi":"10.1007/s10107-024-02135-7","DOIUrl":"https://doi.org/10.1007/s10107-024-02135-7","url":null,"abstract":"Robust combinatorial optimization with budgeted uncertainty is one of the most popular approaches for integrating uncertainty into optimization problems. The existence of a compact reformulation for (mixed-integer) linear programs and positive complexity results give the impression that these problems are relatively easy to solve. However, the practical performance of the reformulation is quite poor when solving robust integer problems, in particular due to its weak linear relaxation. To overcome this issue, we propose procedures to derive new classes of valid inequalities for robust combinatorial optimization problems. For this, we recycle valid inequalities of the underlying deterministic problem such that the additional variables from the robust formulation are incorporated. The valid inequalities to be recycled may either be readily available model constraints or actual cutting planes, where we can benefit from decades of research on valid inequalities for classical optimization problems. We first demonstrate the strength of the inequalities theoretically, by proving that recycling yields a facet-defining inequality in many cases, even if the original valid inequality was not facet-defining. Afterwards, we show in an extensive computational study that using recycled inequalities can lead to a significant improvement of the computation time when solving robust optimization problems.","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":null,"pages":null},"PeriodicalIF":2.7,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142183155","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

Accelerated stochastic approximation with state-dependent noise 带有状态相关噪声的加速随机逼近

IF 2.7 2区数学

Mathematical Programming Pub Date : 2024-08-27 DOI: 10.1007/s10107-024-02138-4

Sasila Ilandarideva, Anatoli Juditsky, Guanghui Lan, Tianjiao Li

{"title":"Accelerated stochastic approximation with state-dependent noise","authors":"Sasila Ilandarideva, Anatoli Juditsky, Guanghui Lan, Tianjiao Li","doi":"10.1007/s10107-024-02138-4","DOIUrl":"https://doi.org/10.1007/s10107-024-02138-4","url":null,"abstract":"We consider a class of stochastic smooth convex optimization problems under rather general assumptions on the noise in the stochastic gradient observation. As opposed to the classical problem setting in which the variance of noise is assumed to be uniformly bounded, herein we assume that the variance of stochastic gradients is related to the “sub-optimality” of the approximate solutions delivered by the algorithm. Such problems naturally arise in a variety of applications, in particular, in the well-known generalized linear regression problem in statistics. However, to the best of our knowledge, none of the existing stochastic approximation algorithms for solving this class of problems attain optimality in terms of the dependence on accuracy, problem parameters, and mini-batch size. We discuss two non-Euclidean accelerated stochastic approximation routines—stochastic accelerated gradient descent (SAGD) and stochastic gradient extrapolation (SGE)—which carry a particular duality relationship. We show that both SAGD and SGE, under appropriate conditions, achieve the optimal convergence rate, attaining the optimal iteration and sample complexities simultaneously. However, corresponding assumptions for the SGE algorithm are more general; they allow, for instance, for efficient application of the SGE to statistical estimation problems under heavy tail noises and discontinuous score functions. We also discuss the application of the SGE to problems satisfying quadratic growth conditions, and show how it can be used to recover sparse solutions. Finally, we report on some simulation experiments to illustrate numerical performance of our proposed algorithms in high-dimensional settings.","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":null,"pages":null},"PeriodicalIF":2.7,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142183165","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 fast combinatorial algorithm for the bilevel knapsack problem with interdiction constraints 带拦截约束的双层knapsack问题的快速组合算法

IF 2.7 2区数学

Mathematical Programming Pub Date : 2024-08-22 DOI: 10.1007/s10107-024-02133-9

Noah Weninger, Ricardo Fukasawa

{"title":"A fast combinatorial algorithm for the bilevel knapsack problem with interdiction constraints","authors":"Noah Weninger, Ricardo Fukasawa","doi":"10.1007/s10107-024-02133-9","DOIUrl":"https://doi.org/10.1007/s10107-024-02133-9","url":null,"abstract":"We consider the bilevel knapsack problem with interdiction constraints, a fundamental bilevel integer programming problem which generalizes the 0–1 knapsack problem. In this problem, there are two knapsacks and n items. The objective is to select some items to pack into the first knapsack such that the maximum profit attainable from packing some of the remaining items into the second knapsack is minimized. We present a combinatorial branch-and-bound algorithm which outperforms the current state-of-the-art solution method in computational experiments for 99% of the instances reported in the literature. On many of the harder instances, our algorithm is orders of magnitude faster, which enabled it to solve 53 of the 72 previously unsolved instances. Our result relies fundamentally on a new dynamic programming algorithm which computes very strong lower bounds. This dynamic program solves a relaxation of the problem from bilevel to 2n-level where the items are processed in an online fashion. The relaxation is easier to solve but approximates the original problem surprisingly well in practice. We believe that this same technique may be useful for other interdiction problems.","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":null,"pages":null},"PeriodicalIF":2.7,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142183166","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

Nonlinear conjugate gradient methods: worst-case convergence rates via computer-assisted analyses 非线性共轭梯度方法：通过计算机辅助分析实现最坏情况收敛率

IF 2.7 2区数学

Mathematical Programming Pub Date : 2024-08-22 DOI: 10.1007/s10107-024-02127-7

Shuvomoy Das Gupta, Robert M. Freund, Xu Andy Sun, Adrien Taylor

引用次数: 0

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

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

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

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

引用次数: 0