Mathematical Programming最新文献_第8页

Multiplicative auction algorithm for approximate maximum weight bipartite matching 近似最大权重双网匹配的乘法拍卖算法

IF 2.7 2区数学

Mathematical Programming Pub Date : 2024-03-06 DOI: 10.1007/s10107-024-02066-3

引用次数: 0

Matroid-based TSP rounding for half-integral solutions 基于矩阵的 TSP 四舍五入半积分解法

IF 2.7 2区数学

Mathematical Programming Pub Date : 2024-03-06 DOI: 10.1007/s10107-024-02065-4

引用次数: 0

The effect of smooth parametrizations on nonconvex optimization landscapes 平滑参数化对非凸优化景观的影响

IF 2.7 2区数学

Mathematical Programming Pub Date : 2024-03-04 DOI: 10.1007/s10107-024-02058-3

Eitan Levin, Joe Kileel, Nicolas Boumal

{"title":"The effect of smooth parametrizations on nonconvex optimization landscapes","authors":"Eitan Levin, Joe Kileel, Nicolas Boumal","doi":"10.1007/s10107-024-02058-3","DOIUrl":"https://doi.org/10.1007/s10107-024-02058-3","url":null,"abstract":"We develop new tools to study landscapes in nonconvex optimization. Given one optimization problem, we pair it with another by smoothly parametrizing the domain. This is either for practical purposes (e.g., to use smooth optimization algorithms with good guarantees) or for theoretical purposes (e.g., to reveal that the landscape satisfies a strict saddle property). In both cases, the central question is: how do the landscapes of the two problems relate? More precisely: how do desirable points such as local minima and critical points in one problem relate to those in the other problem? A key finding in this paper is that these relations are often determined by the parametrization itself, and are almost entirely independent of the cost function. Accordingly, we introduce a general framework to study parametrizations by their effect on landscapes. The framework enables us to obtain new guarantees for an array of problems, some of which were previously treated on a case-by-case basis in the literature. Applications include: optimizing low-rank matrices and tensors through factorizations; solving semidefinite programs via the Burer–Monteiro approach; training neural networks by optimizing their weights and biases; and quotienting out symmetries.","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":"43 1","pages":""},"PeriodicalIF":2.7,"publicationDate":"2024-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140036042","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

Hessian barrier algorithms for non-convex conic optimization 非凸圆锥优化的黑森障碍算法

IF 2.7 2区数学

Mathematical Programming Pub Date : 2024-03-04 DOI: 10.1007/s10107-024-02062-7

Pavel Dvurechensky, Mathias Staudigl

{"title":"Hessian barrier algorithms for non-convex conic optimization","authors":"Pavel Dvurechensky, Mathias Staudigl","doi":"10.1007/s10107-024-02062-7","DOIUrl":"https://doi.org/10.1007/s10107-024-02062-7","url":null,"abstract":"A key problem in mathematical imaging, signal processing and computational statistics is the minimization of non-convex objective functions that may be non-differentiable at the relative boundary of the feasible set. This paper proposes a new family of first- and second-order interior-point methods for non-convex optimization problems with linear and conic constraints, combining logarithmically homogeneous barriers with quadratic and cubic regularization respectively. Our approach is based on a potential-reduction mechanism and, under the Lipschitz continuity of the corresponding derivative with respect to the local barrier-induced norm, attains a suitably defined class of approximate first- or second-order KKT points with worst-case iteration complexity (O(varepsilon ^{-2})) (first-order) and (O(varepsilon ^{-3/2})) (second-order), respectively. Based on these findings, we develop new path-following schemes attaining the same complexity, modulo adjusting constants. These complexity bounds are known to be optimal in the unconstrained case, and our work shows that they are upper bounds in the case with complicated constraints as well. To the best of our knowledge, this work is the first which achieves these worst-case complexity bounds under such weak conditions for general conic constrained non-convex optimization problems.","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":"4 1","pages":""},"PeriodicalIF":2.7,"publicationDate":"2024-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140036039","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

Automated tight Lyapunov analysis for first-order methods 一阶方法的自动紧 Lyapunov 分析

IF 2.7 2区数学

Mathematical Programming Pub Date : 2024-02-26 DOI: 10.1007/s10107-024-02061-8

Manu Upadhyaya, Sebastian Banert, Adrien B. Taylor, Pontus Giselsson

引用次数: 0

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