Mathematical Programming最新文献_第8页

The Chvátal–Gomory procedure for integer SDPs with applications in combinatorial optimization 整数 SDP 的 Chvátal-Gomory 程序及其在组合优化中的应用

IF 2.7 2区数学

Mathematical Programming Pub Date : 2024-03-13 DOI: 10.1007/s10107-024-02069-0

{"title":"The Chvátal–Gomory procedure for integer SDPs with applications in combinatorial optimization","authors":"","doi":"10.1007/s10107-024-02069-0","DOIUrl":"https://doi.org/10.1007/s10107-024-02069-0","url":null,"abstract":"<h3>Abstract</h3> In this paper we study the well-known Chvátal–Gomory (CG) procedure for the class of integer semidefinite programs (ISDPs). We prove several results regarding the hierarchy of relaxations obtained by iterating this procedure. We also study different formulations of the elementary closure of spectrahedra. A polyhedral description of the elementary closure for a specific type of spectrahedra is derived by exploiting total dual integrality for SDPs. Moreover, we show how to exploit (strengthened) CG cuts in a branch-and-cut framework for ISDPs. Different from existing algorithms in the literature, the separation routine in our approach exploits both the semidefinite and the integrality constraints. We provide separation routines for several common classes of binary SDPs resulting from combinatorial optimization problems. In the second part of the paper we present a comprehensive application of our approach to the quadratic traveling salesman problem (QTSP). Based on the algebraic connectivity of the directed Hamiltonian cycle, two ISDPs that model the QTSP are introduced. We show that the CG cuts resulting from these formulations contain several well-known families of cutting planes. Numerical results illustrate the practical strength of the CG cuts in our branch-and-cut algorithm, which outperforms alternative ISDP solvers and is able to solve large QTSP instances to optimality.","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":"2017 1","pages":""},"PeriodicalIF":2.7,"publicationDate":"2024-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140127998","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

Generalized minimum 0-extension problem and discrete convexity 广义最小 0-扩展问题和离散凸性

IF 2.7 2区数学

Mathematical Programming Pub Date : 2024-03-07 DOI: 10.1007/s10107-024-02064-5

Martin Dvorak, Vladimir Kolmogorov

{"title":"Generalized minimum 0-extension problem and discrete convexity","authors":"Martin Dvorak, Vladimir Kolmogorov","doi":"10.1007/s10107-024-02064-5","DOIUrl":"https://doi.org/10.1007/s10107-024-02064-5","url":null,"abstract":"Given a fixed finite metric space ((V,mu )), the minimum 0-extension problem, denoted as (mathtt{0hbox {-}Ext}[{mu }]), is equivalent to the following optimization problem: minimize function of the form (min nolimits _{xin V^n} sum _i f_i(x_i) + sum _{ij} c_{ij}hspace{0.5pt}mu (x_i,x_j)) where (f_i:Vrightarrow mathbb {R}) are functions given by (f_i(x_i)=sum _{vin V} c_{vi}hspace{0.5pt}mu (x_i,v)) and (c_{ij},c_{vi}) are given nonnegative costs. The computational complexity of (mathtt{0hbox {-}Ext}[{mu }]) has been recently established by Karzanov and by Hirai: if metric (mu ) is orientable modular then (mathtt{0hbox {-}Ext}[{mu }]) can be solved in polynomial time, otherwise (mathtt{0hbox {-}Ext}[{mu }]) is NP-hard. To prove the tractability part, Hirai developed a theory of discrete convex functions on orientable modular graphs generalizing several known classes of functions in discrete convex analysis, such as (L^natural )-convex functions. We consider a more general version of the problem in which unary functions (f_i(x_i)) can additionally have terms of the form (c_{uv;i}hspace{0.5pt}mu (x_i,{u,v})) for ({u,!hspace{0.5pt}hspace{0.5pt}v}in F), where set (Fsubseteq left( {begin{array}{c}V 2end{array}}right) ) is fixed. We extend the complexity classification above by providing an explicit condition on ((mu ,F)) for the problem to be tractable. In order to prove the tractability part, we generalize Hirai’s theory and define a larger class of discrete convex functions. It covers, in particular, another well-known class of functions, namely submodular functions on an integer lattice. Finally, we improve the complexity of Hirai’s algorithm for solving (mathtt{0hbox {-}Ext}[{mu }]) on orientable modular graphs.","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":"87 1","pages":""},"PeriodicalIF":2.7,"publicationDate":"2024-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140074954","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

Level constrained first order methods for function constrained optimization 函数约束优化的水平约束一阶方法

IF 2.7 2区数学

Mathematical Programming Pub Date : 2024-03-06 DOI: 10.1007/s10107-024-02057-4

Digvijay Boob, Qi Deng, Guanghui Lan

{"title":"Level constrained first order methods for function constrained optimization","authors":"Digvijay Boob, Qi Deng, Guanghui Lan","doi":"10.1007/s10107-024-02057-4","DOIUrl":"https://doi.org/10.1007/s10107-024-02057-4","url":null,"abstract":"We present a new feasible proximal gradient method for constrained optimization where both the objective and constraint functions are given by summation of a smooth, possibly nonconvex function and a convex simple function. The algorithm converts the original problem into a sequence of convex subproblems. Formulating those subproblems requires the evaluation of at most one gradient-value of the original objective and constraint functions. Either exact or approximate subproblems solutions can be computed efficiently in many cases. An important feature of the algorithm is the constraint level parameter. By carefully increasing this level for each subproblem, we provide a simple solution to overcome the challenge of bounding the Lagrangian multipliers and show that the algorithm follows a strictly feasible solution path till convergence to the stationary point. We develop a simple, proximal gradient descent type analysis, showing that the complexity bound of this new algorithm is comparable to gradient descent for the unconstrained setting which is new in the literature. Exploiting this new design and analysis technique, we extend our algorithms to some more challenging constrained optimization problems where (1) the objective is a stochastic or finite-sum function, and (2) structured nonsmooth functions replace smooth components of both objective and constraint functions. Complexity results for these problems also seem to be new in the literature. Finally, our method can also be applied to convex function constrained problems where we show complexities similar to the proximal gradient method.","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":"43 1","pages":""},"PeriodicalIF":2.7,"publicationDate":"2024-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140054873","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

Polyhedral properties of RLT relaxations of nonconvex quadratic programs and their implications on exact relaxations 非凸二次方程程 RLT 松弛的多面体特性及其对精确松弛的影响

IF 2.7 2区数学

Mathematical Programming Pub Date : 2024-03-06 DOI: 10.1007/s10107-024-02070-7

Yuzhou Qiu, E. Alper Yıldırım

引用次数: 0

On convergence of iterative thresholding algorithms to approximate sparse solution for composite nonconvex optimization 论复合非凸优化近似稀疏解的迭代阈值算法的收敛性

IF 2.7 2区数学

Mathematical Programming Pub Date : 2024-03-06 DOI: 10.1007/s10107-024-02068-1

Yaohua Hu, Xinlin Hu, Xiaoqi Yang

{"title":"On convergence of iterative thresholding algorithms to approximate sparse solution for composite nonconvex optimization","authors":"Yaohua Hu, Xinlin Hu, Xiaoqi Yang","doi":"10.1007/s10107-024-02068-1","DOIUrl":"https://doi.org/10.1007/s10107-024-02068-1","url":null,"abstract":"This paper aims to find an approximate true sparse solution of an underdetermined linear system. For this purpose, we propose two types of iterative thresholding algorithms with the continuation technique and the truncation technique respectively. We introduce a notion of limited shrinkage thresholding operator and apply it, together with the restricted isometry property, to show that the proposed algorithms converge to an approximate true sparse solution within a tolerance relevant to the noise level and the limited shrinkage magnitude. Applying the obtained results to nonconvex regularization problems with SCAD, MCP and (ell _p) penalty ((0le p le 1)) and utilizing the recovery bound theory, we establish the convergence of their proximal gradient algorithms to an approximate global solution of nonconvex regularization problems. The established results include the existing convergence theory for (ell _1) or (ell _0) regularization problems for finding a true sparse solution as special cases. Preliminary numerical results show that our proposed algorithms can find approximate true sparse solutions that are much better than stationary solutions that are found by using the standard proximal gradient algorithm.","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":"105 1","pages":""},"PeriodicalIF":2.7,"publicationDate":"2024-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140054988","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

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