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Subdifferentials of convex matrix-valued functions 凸矩阵值函数的次微分
IF 1.6 4区 数学
Optimization Letters Pub Date : 2024-03-21 DOI: 10.1007/s11590-024-02105-0
{"title":"Subdifferentials of convex matrix-valued functions","authors":"","doi":"10.1007/s11590-024-02105-0","DOIUrl":"https://doi.org/10.1007/s11590-024-02105-0","url":null,"abstract":"<h3>Abstract</h3> <p>Subdifferentials (in the sense of convex analysis) of matrix-valued functions defined on <span> <span>(mathbb {R}^d)</span> </span> that are convex with respect to the Löwner partial order can have a complicated structure and might be very difficult to compute even in simple cases. The aim of this paper is to study subdifferential calculus for such functions and properties of their subdifferentials. We show that many standard results from convex analysis no longer hold true in the matrix-valued case. For example, in this case the subdifferential of the sum is not equal to the sum of subdifferentials, the Clarke subdifferential is not equal to the subdifferential in the sense of convex analysis, etc. Nonetheless, it is possible to provide simple rules for computing nonempty subsets of subdifferentials (in particular, individual subgradients) of convex matrix-valued functions in the general case and to completely describe subdifferentials of such functions defined on the real line. As a by-product of our analysis, we derive some interesting properties of convex matrix-valued functions, e.g. we show that if such function is nonsmooth, then its diagonal elements must be nonsmooth as well.</p>","PeriodicalId":49720,"journal":{"name":"Optimization Letters","volume":"9 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140199501","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
Learning to project in a criterion space search algorithm: an application to multi-objective binary linear programming 在标准空间搜索算法中学习预测：多目标二元线性规划的应用
IF 1.6 4区 数学
Optimization Letters Pub Date : 2024-03-18 DOI: 10.1007/s11590-024-02100-5
Alvaro Sierra-Altamiranda, Hadi Charkhgard, Iman Dayarian, Ali Eshragh, Sorna Javadi
{"title":"Learning to project in a criterion space search algorithm: an application to multi-objective binary linear programming","authors":"Alvaro Sierra-Altamiranda, Hadi Charkhgard, Iman Dayarian, Ali Eshragh, Sorna Javadi","doi":"10.1007/s11590-024-02100-5","DOIUrl":"https://doi.org/10.1007/s11590-024-02100-5","url":null,"abstract":"<p>In this paper, we investigate the possibility of improving the performance of multi-objective optimization solution approaches using machine learning techniques. Specifically, we focus on multi-objective binary linear programs and employ one of the most effective and recently developed criterion space search algorithms, the so-called KSA, during our study. This algorithm computes all nondominated points of a problem with <i>p</i> objectives by searching on a projected criterion space, i.e., a <span>((p-1))</span>-dimensional criterion apace. We present an effective and fast learning approach to identify on which projected space the KSA should work. We also present several generic features/variables that can be used in machine learning techniques for identifying the best projected space. Finally, we present an effective bi-objective optimization-based heuristic for selecting the subset of the features to overcome the issue of overfitting in learning. Through an extensive computational study over 2000 instances of tri-objective knapsack and assignment problems, we demonstrate that an improvement of up to 18% in time can be achieved by the proposed learning method compared to a random selection of the projected space. To show that the performance of our algorithm is not limited to instances of knapsack and assignment problems with three objective functions, we also report similar performance results when the proposed learning approach is used for solving random binary integer program instances with four objective functions.</p>","PeriodicalId":49720,"journal":{"name":"Optimization Letters","volume":"16 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140171236","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
Convergence analysis of the DFP algorithm for unconstrained optimization problems on Riemannian manifolds 黎曼流形上无约束优化问题的 DFP 算法收敛性分析
IF 1.6 4区 数学
Optimization Letters Pub Date : 2024-03-12 DOI: 10.1007/s11590-024-02103-2
Xiao-bo Li, Kai Tu, Jian Lu
{"title":"Convergence analysis of the DFP algorithm for unconstrained optimization problems on Riemannian manifolds","authors":"Xiao-bo Li, Kai Tu, Jian Lu","doi":"10.1007/s11590-024-02103-2","DOIUrl":"https://doi.org/10.1007/s11590-024-02103-2","url":null,"abstract":"<p>In this paper, we propose the DFP algorithm with inexact line search for unconstrained optimization problems on Riemannian manifolds. Under some reasonable conditions, the global convergence result is established and the superlinear local convergence rate of the DFP algorithm is proved on Riemannian manifolds. The preliminary computational experiment is also reported to illustrate the effectiveness of the DFP algorithm.</p>","PeriodicalId":49720,"journal":{"name":"Optimization Letters","volume":"12 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140114921","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
Stochastic single-machine scheduling with workload-dependent maintenance activities 随工作量而定的维护活动的单机随机调度
IF 1.6 4区 数学
Optimization Letters Pub Date : 2024-03-11 DOI: 10.1007/s11590-024-02102-3
Manzhan Gu, Weitao Yang, Peihai Liu
{"title":"Stochastic single-machine scheduling with workload-dependent maintenance activities","authors":"Manzhan Gu, Weitao Yang, Peihai Liu","doi":"10.1007/s11590-024-02102-3","DOIUrl":"https://doi.org/10.1007/s11590-024-02102-3","url":null,"abstract":"<p>This paper studies the stochastic single-machine scheduling problem with workload-dependent maintenance activities, in which the processing times of all jobs are independently subject to a common discrete distribution, and the aim is to find the optimal policy so as to minimize the expected total discounted holding cost. Based on the definition of Markov process, for each of the two cases with the discount rate being zero or a positive number, we present two dynamic programming algorithms to produce the optimal static policy and the optimal dynamic policy, respectively.</p>","PeriodicalId":49720,"journal":{"name":"Optimization Letters","volume":"285 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140099891","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
On the relation between affinely adjustable robust linear complementarity and mixed-integer linear feasibility problems 关于仿射可调鲁棒线性互补与混合整数线性可行性问题之间的关系
IF 1.6 4区 数学
Optimization Letters Pub Date : 2024-03-05 DOI: 10.1007/s11590-023-02093-7
{"title":"On the relation between affinely adjustable robust linear complementarity and mixed-integer linear feasibility problems","authors":"","doi":"10.1007/s11590-023-02093-7","DOIUrl":"https://doi.org/10.1007/s11590-023-02093-7","url":null,"abstract":"<h3>Abstract</h3> <p>We consider adjustable robust linear complementarity problems and extend the results of Biefel et al. (SIAM J Optim 32:152–172, 2022) towards convex and compact uncertainty sets. Moreover, for the case of polyhedral uncertainty sets, we prove that computing an adjustable robust solution of a given linear complementarity problem is equivalent to solving a properly chosen mixed-integer linear feasibility problem.</p>","PeriodicalId":49720,"journal":{"name":"Optimization Letters","volume":"46 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140035192","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 distributional Farkas’ lemma and moment optimization problems with no-gap dual semi-definite programs 分布式法卡斯定理和无差距对偶半有限程序的矩优化问题
IF 1.6 4区 数学
Optimization Letters Pub Date : 2024-03-04 DOI: 10.1007/s11590-024-02097-x
Queenie Yingkun Huang, Vaithilingam Jeyakumar
{"title":"A distributional Farkas’ lemma and moment optimization problems with no-gap dual semi-definite programs","authors":"Queenie Yingkun Huang, Vaithilingam Jeyakumar","doi":"10.1007/s11590-024-02097-x","DOIUrl":"https://doi.org/10.1007/s11590-024-02097-x","url":null,"abstract":"<p>We present a generalized Farkas’ Lemma for an inequality system involving distributions. This lemma establishes an equivalence between an infinite-dimensional system of moment inequalities and a semi-definite system, assuming that the support for the distributions is a spectrahedron. To the best of our knowledge, it is the first extension of Farkas’ Lemma to the distributional paradigm. Applying the new Lemma, we then establish no-gap duality results between a class of moment optimization problems and numerically tractable semi-definite programs.</p>","PeriodicalId":49720,"journal":{"name":"Optimization Letters","volume":"35 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140034923","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 new approach to the multiple obnoxious facility location problem based on combinatorial and continuous tools 基于组合工具和连续工具的多讨厌设施定位问题新方法
IF 1.6 4区 数学
Optimization Letters Pub Date : 2024-03-03 DOI: 10.1007/s11590-024-02096-y
M. Locatelli
{"title":"A new approach to the multiple obnoxious facility location problem based on combinatorial and continuous tools","authors":"M. Locatelli","doi":"10.1007/s11590-024-02096-y","DOIUrl":"https://doi.org/10.1007/s11590-024-02096-y","url":null,"abstract":"<p>In this paper we address the multiple obnoxious facility location problem. In this problem <i>p</i> facilities need to be spread within the unit square in such a way that they are far enough from each other and that their minimal distance from <i>n</i> communities, with known positions within the unit square, is maximized. The problem has a combinatorial component, related to the key observation made in Drezner (Omega 87:105–116, 2019) about the role played by Voronoi points. We propose a new approach, which exploits both the combinatorial component of the problem and, through continuous local optimizations, also its continuous component. We also propose techniques to limit the impact on computation times of the number <i>n</i> of communities. The approach turns out to be quite competitive and is able to return 24 new best known solutions with respect to the best results reported in Kalczynski (Optim Lett 16:1153–1166, 2022).</p>","PeriodicalId":49720,"journal":{"name":"Optimization Letters","volume":"4 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140034857","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
Hausdorff continuity of solution maps to equilibrium problems via the oriented distance function 通过定向距离函数实现平衡问题解映射的豪斯多夫连续性
IF 1.6 4区 数学
Optimization Letters Pub Date : 2024-02-27 DOI: 10.1007/s11590-024-02098-w
Lam Quoc Anh, Nguyen Huu Danh, Pham Thanh Duoc
{"title":"Hausdorff continuity of solution maps to equilibrium problems via the oriented distance function","authors":"Lam Quoc Anh, Nguyen Huu Danh, Pham Thanh Duoc","doi":"10.1007/s11590-024-02098-w","DOIUrl":"https://doi.org/10.1007/s11590-024-02098-w","url":null,"abstract":"<p>This paper aims to study the stability in the sense of Hausdorff continuity of solution maps to equilibrium problems without assuming the solid condition of ordered cones. We first propose a generalized concavity of set-valued maps and discuss its relation with the existing concepts. Then, by using the above property and the continuity of the objective function, sufficient conditions for the Hausdorff continuity of solution maps to scalar equilibrium problems are established. Finally, we utilize the oriented distance function to obtain the Hausdorff continuity of solution maps to set-valued equilibrium problems via the corresponding results of the scalar equilibrium problems.</p>","PeriodicalId":49720,"journal":{"name":"Optimization Letters","volume":"80 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139980291","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
Integrated optimization of design and production process with personalization level of products 综合优化设计和生产流程，提高产品的个性化水平
IF 1.6 4区 数学
Optimization Letters Pub Date : 2024-02-27 DOI: 10.1007/s11590-024-02099-9
Ba-Yi Cheng, Jie Duan, Xin-Yan Shi, Mi Zhou
{"title":"Integrated optimization of design and production process with personalization level of products","authors":"Ba-Yi Cheng, Jie Duan, Xin-Yan Shi, Mi Zhou","doi":"10.1007/s11590-024-02099-9","DOIUrl":"https://doi.org/10.1007/s11590-024-02099-9","url":null,"abstract":"<p>We study the design and production process of manufacturers who provide customers with personalized products. Each customer’s order needs to go through two stages: design and production. For this problem, we consider the two scheduling problems with the objective of minimizing the total weighted completion time. Then we consider two models of manufacturing at a personalized level. In the first model, personalized products have the same personalization level, which is proved to have an optimal solution. In the second model, we propose an approximate algorithm with an absolute worst-case ratio of no more than two for personalized products with arbitrary personalization levels, which is proved to be NP-hard in the strong sense.</p>","PeriodicalId":49720,"journal":{"name":"Optimization Letters","volume":"20 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140010655","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
Exact QR factorizations of rectangular matrices 矩形矩阵的精确 QR 因式分解
IF 1.6 4区 数学
Optimization Letters Pub Date : 2024-02-22 DOI: 10.1007/s11590-024-02095-z
Christopher Lourenco, Erick Moreno-Centeno
{"title":"Exact QR factorizations of rectangular matrices","authors":"Christopher Lourenco, Erick Moreno-Centeno","doi":"10.1007/s11590-024-02095-z","DOIUrl":"https://doi.org/10.1007/s11590-024-02095-z","url":null,"abstract":"<p>QR factorization is a key tool in mathematics, computer science, operations research, and engineering. This paper presents the roundoff-error-free (REF) QR factorization framework comprising integer-preserving versions of the standard and the thin QR factorizations and associated algorithms to compute them. Specifically, the standard REF QR factorization factors a given matrix <span>(Ain {mathbb {Z}}^{mtimes n})</span> as <span>(A=QDR)</span>, where <span>(Qin {mathbb {Z}}^{mtimes m})</span> has pairwise orthogonal columns, <i>D</i> is a diagonal matrix, and <span>(Rin {mathbb {Z}}^{mtimes n})</span> is an upper trapezoidal matrix; notably, the entries of <i>Q</i> and <i>R</i> are integral, while the entries of <i>D</i> are reciprocals of integers. In the thin REF QR factorization, <span>(Qin {mathbb {Z}}^{mtimes n})</span> also has pairwise orthogonal columns, and <span>(Rin {mathbb {Z}}^{ntimes n})</span> is also an upper triangular matrix. In contrast to traditional (i.e., floating-point) QR factorizations, every operation used to compute these factors is integral; thus, REF QR is guaranteed to be an exact orthogonal decomposition. Importantly, the bit-length of every entry in the REF QR factorizations (and within the algorithms to compute them) is bounded polynomially. Notable applications of our REF QR factorizations include finding exact least squares or exact basic solutions, <span>({textbf{x}}in {mathbb {Q}}^n)</span>, to any given full column rank or rank deficient linear system <span>(A {textbf{x}}= {textbf{b}})</span>, respectively. In addition, our exact factorizations can be used as a subroutine within exact and/or high-precision quadratic programming. Altogether, REF QR provides a framework to obtain exact orthogonal factorizations of any rational matrix (as any rational/decimal matrix can be easily transformed into an integral matrix).</p>","PeriodicalId":49720,"journal":{"name":"Optimization Letters","volume":"40 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139923998","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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