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Local convergence analysis of an inexact trust-region method for nonsmooth optimization 非平滑优化非精确信任区域方法的局部收敛分析
IF 1.6 4区 数学
Optimization Letters Pub Date : 2024-02-21 DOI: 10.1007/s11590-023-02092-8
Robert J. Baraldi, Drew P. Kouri
{"title":"Local convergence analysis of an inexact trust-region method for nonsmooth optimization","authors":"Robert J. Baraldi, Drew P. Kouri","doi":"10.1007/s11590-023-02092-8","DOIUrl":"https://doi.org/10.1007/s11590-023-02092-8","url":null,"abstract":"<p>In Baraldi (Math Program 20:1–40, 2022), we introduced an inexact trust-region algorithm for minimizing the sum of a smooth nonconvex function and a nonsmooth convex function in Hilbert space—a class of problems that is ubiquitous in data science, learning, optimal control, and inverse problems. This algorithm has demonstrated excellent performance and scalability with problem size. In this paper, we enrich the convergence analysis for this algorithm, proving strong convergence of the iterates with guaranteed rates. In particular, we demonstrate that the trust-region algorithm recovers superlinear, even quadratic, convergence rates when using a second-order Taylor approximation of the smooth objective function term.</p>","PeriodicalId":49720,"journal":{"name":"Optimization Letters","volume":"8 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139923801","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
Optimality conditions for robust weakly efficient solutions in uncertain optimization 不确定优化中稳健弱效率解的最优条件
IF 1.6 4区 数学
Optimization Letters Pub Date : 2024-02-13 DOI: 10.1007/s11590-023-02085-7
Yuwen Zhai, Qilin Wang, Tian Tang, Maoyuan Lv
{"title":"Optimality conditions for robust weakly efficient solutions in uncertain optimization","authors":"Yuwen Zhai, Qilin Wang, Tian Tang, Maoyuan Lv","doi":"10.1007/s11590-023-02085-7","DOIUrl":"https://doi.org/10.1007/s11590-023-02085-7","url":null,"abstract":"<p>In this paper, we find the flimsily robust weakly efficient solution to the uncertain vector optimization problem by means of the weighted sum scalarization method and strictly robust counterpart. In addition, we introduce a higher-order weak upper inner Studniarski epiderivative of set-valued maps, and obtain two properties of the new notion under the assumption of the star-shaped set. Finally, by applying the higher-order weak upper inner Studniarski epiderivative, we obtain a sufficient and necessary optimality condition of the vector-based robust weakly efficient solution to an uncertain vector optimization problem under the condition of the higher-order strictly generalized cone convexity. As applications, the corresponding optimality conditions of the robust (weakly) Pareto solutions are obtained by the current methods.</p>","PeriodicalId":49720,"journal":{"name":"Optimization Letters","volume":"88 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139760741","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
Characterization of unique solvability of absolute value equations: an overview, extensions, and future directions 绝对值方程唯一可解性的特征：概述、扩展和未来方向
IF 1.6 4区 数学
Optimization Letters Pub Date : 2024-02-09 DOI: 10.1007/s11590-024-02094-0
Shubham Kumar, Deepmala, Milan Hladík, Hossein Moosaei
{"title":"Characterization of unique solvability of absolute value equations: an overview, extensions, and future directions","authors":"Shubham Kumar, Deepmala, Milan Hladík, Hossein Moosaei","doi":"10.1007/s11590-024-02094-0","DOIUrl":"https://doi.org/10.1007/s11590-024-02094-0","url":null,"abstract":"<p>This paper provides an overview of the necessary and sufficient conditions for guaranteeing the unique solvability of absolute value equations. In addition to discussing the basic form of these equations, we also address several generalizations, including generalized absolute value equations and matrix absolute value equations. Our survey encompasses known results as well as novel characterizations proposed in this study.</p>","PeriodicalId":49720,"journal":{"name":"Optimization Letters","volume":"42 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139760737","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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 primal-dual algorithm via dynamical system with variable mass for linearly constrained convex optimization 通过具有可变质量的动态系统实现线性约束凸优化的快速原始双算法
IF 1.6 4区 数学
Optimization Letters Pub Date : 2024-01-28 DOI: 10.1007/s11590-023-02091-9
Ziyi Jiang, Dan Wang, Xinwei Liu
{"title":"A fast primal-dual algorithm via dynamical system with variable mass for linearly constrained convex optimization","authors":"Ziyi Jiang, Dan Wang, Xinwei Liu","doi":"10.1007/s11590-023-02091-9","DOIUrl":"https://doi.org/10.1007/s11590-023-02091-9","url":null,"abstract":"<p>We aim to solve the linearly constrained convex optimization problem whose objective function is the sum of a differentiable function and a non-differentiable function. We first propose an inertial continuous primal-dual dynamical system with variable mass for linearly constrained convex optimization problems with differentiable objective functions. The dynamical system is composed of a second-order differential equation with variable mass for the primal variable and a first-order differential equation for the dual variable. The fast convergence properties of the proposed dynamical system are proved by constructing a proper energy function. We then extend the results to the case where the objective function is non-differentiable, and a new accelerated primal-dual algorithm is presented. When both variable mass and time scaling satisfy certain conditions, it is proved that our new algorithm owns fast convergence rates for the objective function residual and the feasibility violation. Some preliminary numerical results on the <span>(ell _{1})</span>–<span>(ell _{2})</span> minimization problem demonstrate the validity of our algorithm.</p>","PeriodicalId":49720,"journal":{"name":"Optimization Letters","volume":"20 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139583223","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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 many-to-many point matching in two dimensions 二维受限多对多点匹配
IF 1.6 4区 数学
Optimization Letters Pub Date : 2024-01-26 DOI: 10.1007/s11590-023-02089-3
L. E. Caraballo, R. A. Castro, J. M. Díaz-Báñez, M. A. Heredia, J. Urrutia, I. Ventura, F. J. Zaragoza
{"title":"Constrained many-to-many point matching in two dimensions","authors":"L. E. Caraballo, R. A. Castro, J. M. Díaz-Báñez, M. A. Heredia, J. Urrutia, I. Ventura, F. J. Zaragoza","doi":"10.1007/s11590-023-02089-3","DOIUrl":"https://doi.org/10.1007/s11590-023-02089-3","url":null,"abstract":"<p>In the minimum-weight many-to-many point matching problem, we are given a set <i>R</i> of red points and a set <i>B</i> of blue points in the plane, of total size <i>N</i>, and we want to pair up each point in <i>R</i> to one or more points in <i>B</i> and vice versa so that the sum of distances between the paired points is minimized. This problem can be solved in <span>(O(N^3))</span> time by using a reduction to the minimum-weight perfect matching problem, and thus, it is not fast enough to be used for on-line systems where a large number of tunes need to be compared. Motivated by similarity problems in music theory, in this paper we study several constrained minimum-weight many-to-many point matching problems in which the allowed pairings are given by geometric restrictions, i.e., a bichromatic pair can be matched if and only if the corresponding points satisfy a specific condition of closeness. We provide algorithms to solve these constrained versions in <i>O</i>(<i>N</i>) time when the sets <i>R</i> and <i>B</i> are given ordered by abscissa.</p>","PeriodicalId":49720,"journal":{"name":"Optimization Letters","volume":"136 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139583286","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
Duality in the problems of optimal control described by Darboux-type differential inclusions 达尔布型微分结论描述的最优控制问题中的对偶性
IF 1.6 4区 数学
Optimization Letters Pub Date : 2024-01-25 DOI: 10.1007/s11590-023-02088-4
Sevilay Demir Sağlam
{"title":"Duality in the problems of optimal control described by Darboux-type differential inclusions","authors":"Sevilay Demir Sağlam","doi":"10.1007/s11590-023-02088-4","DOIUrl":"https://doi.org/10.1007/s11590-023-02088-4","url":null,"abstract":"<p>This paper is devoted to the optimization of the Mayer problem with hyperbolic differential inclusions of the Darboux type and duality. We use the discrete approximation method to get sufficient conditions of optimality for the convex problem given by Darboux differential inclusions and the polyhedral problem for a hyperbolic differential inclusion with state constraint. We formulate the adjoint inclusions in the Euler-Lagrange inclusion and Hamiltonian forms. Then, we construct the dual problem to optimal control problem given by Darboux differential inclusions with state constraint and prove so-called duality results. Moreover, we show that each pair of primal and dual problem solutions satisfy duality relations and that the optimal values in the primal convex and dual concave problems are equal. Finally, we establish the dual problem to the polyhedral Darboux problem and provide an example to demonstrate the main constructions of our approach.</p>","PeriodicalId":49720,"journal":{"name":"Optimization Letters","volume":"138 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139583678","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
Differentially private k-center problems 差分私有 K 中心问题
IF 1.6 4区 数学
Optimization Letters Pub Date : 2024-01-25 DOI: 10.1007/s11590-023-02090-w
Fan Yuan, Dachuan Xu, Donglei Du, Min Li
{"title":"Differentially private k-center problems","authors":"Fan Yuan, Dachuan Xu, Donglei Du, Min Li","doi":"10.1007/s11590-023-02090-w","DOIUrl":"https://doi.org/10.1007/s11590-023-02090-w","url":null,"abstract":"<p>Data privacy has become one of the most important concerns in the big data era. Because of its broad applications in machine learning and data analysis, many algorithms and theoretical results have been established for privacy clustering problems, such as <i>k</i>-means and <i>k</i>-median problems with privacy protection. However, there is little work on privacy protection in <i>k</i>-center clustering. Our research focuses on the <i>k</i>-center problem, its distributed variant, and the distributed <i>k</i>-center problem under differential privacy constraints. These problems model the concept of safeguarding the privacy of individual input elements, with the integration of differential privacy aimed at ensuring the security of individual information during data processing and analysis. We propose three approximation algorithms for these problems, respectively, and achieve a constant factor approximation ratio.</p>","PeriodicalId":49720,"journal":{"name":"Optimization Letters","volume":"5 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139583288","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
Representation of positive polynomials on a generalized strip and its application to polynomial optimization 正多项式在广义条带上的表示及其在多项式优化中的应用
IF 1.6 4区 数学
Optimization Letters Pub Date : 2024-01-11 DOI: 10.1007/s11590-023-02087-5
{"title":"Representation of positive polynomials on a generalized strip and its application to polynomial optimization","authors":"","doi":"10.1007/s11590-023-02087-5","DOIUrl":"https://doi.org/10.1007/s11590-023-02087-5","url":null,"abstract":"<h3>Abstract</h3> <p>We study the representation of nonnegative polynomials in two variables on a certain class of unbounded closed basic semi-algebraic sets (which are called generalized strips). This class includes the strip <span> <span>([a,b] times {mathbb {R}})</span> </span> which was studied by Marshall in (Proc Am Math Soc 138(5):1559–1567, 2010). A denominator-free Nichtnegativstellensätz holds true on a generalized strip when the width of the generalized strip is constant and fails otherwise. As a consequence, we confirm that the standard hierarchy of semidefinite programming relaxations defined for the compact case can indeed be adapted to the generalized strip with constant width. For polynomial optimization problems on the generalized strip with non-constant width, we follow Ha-Pham’s work: Solving polynomial optimization problems via the truncated tangency variety and sums of squares. </p>","PeriodicalId":49720,"journal":{"name":"Optimization Letters","volume":"84 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139458753","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
Determining optimal channel partition for 2:4 fine grained structured sparsity 确定 2:4 细粒度结构稀疏性的最佳信道分区
IF 1.6 4区 数学
Optimization Letters Pub Date : 2024-01-11 DOI: 10.1007/s11590-023-02084-8
{"title":"Determining optimal channel partition for 2:4 fine grained structured sparsity","authors":"","doi":"10.1007/s11590-023-02084-8","DOIUrl":"https://doi.org/10.1007/s11590-023-02084-8","url":null,"abstract":"<h3>Abstract</h3> <p>Deep Neural Networks (DNNs) have demonstrated tremendous success in many applications, but incur high computational burden on the inference side. The 2:4 sparsity pruning method has recently been developed to effectively compress and accelerate DNNs with little to no loss in performance. The method comprises a training phase followed by a pruning step where 2 out of 4 consecutive weights are eliminated to obtain a pruned matrix, which is then retrained to fine-tune the remaining weights. The accuracy of the resultant sparse network is maximized by permuting the matrix along the channel dimension in a way that maximizes the total magnitude of weights preserved during pruning. While earlier works have proposed heuristic methods to generate good permutations, we formalized the problem as a discrete optimization problem. In this paper, we propose four different mathematical programs to determine the optimal permutations and compare their performance for small-sized instances using a standard solver. Further, we develop a complementary column generation scheme to solve DNNs with realistic number of channels. </p>","PeriodicalId":49720,"journal":{"name":"Optimization Letters","volume":"57 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139465305","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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 reduced Jacobian method with full convergence property 具有完全收敛特性的减雅各布方法
IF 1.6 4区 数学
Optimization Letters Pub Date : 2024-01-07 DOI: 10.1007/s11590-023-02083-9
M. El Maghri, Y. Elboulqe
{"title":"A reduced Jacobian method with full convergence property","authors":"M. El Maghri, Y. Elboulqe","doi":"10.1007/s11590-023-02083-9","DOIUrl":"https://doi.org/10.1007/s11590-023-02083-9","url":null,"abstract":"<p>In this paper, we propose a variant of the reduced Jacobian method (RJM) introduced by El Maghri and Elboulqe (J Optim Theory Appl 179:917–943, 2018) for multicriteria optimization under linear constraints. Motivation is that, contrarily to RJM which has only global convergence to Pareto KKT-stationary points in the classical sense of accumulation points, this new variant possesses the full convergence property in the sense that the entire sequence converges whenever the objectives are quasiconvex. Simulations are reported showing the performance of this variant compared to RJM and the nondominated sorting genetic algorithm (NSGA-II).</p>","PeriodicalId":49720,"journal":{"name":"Optimization Letters","volume":"7 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139396430","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
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