{"title":"Sensitivity Analysis in Parametric Convex Vector Optimization","authors":"Duong Thi Viet An, Le Thanh Tung","doi":"10.1007/s11228-024-00733-3","DOIUrl":"https://doi.org/10.1007/s11228-024-00733-3","url":null,"abstract":"<p>In this paper, sensitivity analysis of the efficient sets in parametric convex vector optimization is considered. Namely, the perturbation, weak perturbation, and proper perturbation maps are defined as set-valued maps. We establish the formulas for computing the Fréchet coderivative of the profile of the above three kinds of perturbation maps. Because of the convexity assumptions, the conditions set are fairly simple if compared to those in the general case. In addition, our conditions are stated directly on the data of the problem. It is worth emphasizing that our approach is based on convex analysis tools which are different from those in the general case.</p>","PeriodicalId":49537,"journal":{"name":"Set-Valued and Variational Analysis","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142263036","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Steepest Geometric Descent for Regularized Quasiconvex Functions","authors":"Aris Daniilidis, David Salas","doi":"10.1007/s11228-024-00731-5","DOIUrl":"https://doi.org/10.1007/s11228-024-00731-5","url":null,"abstract":"<p>We establish existence of steepest descent curves emanating from almost every point of a regular locally Lipschitz quasiconvex functions, where regularity means that the sweeping process flow induced by the sublevel sets is reversible. We then use max-convolution to regularize general quasiconvex functions and obtain a result of the same nature in a more general setting.</p>","PeriodicalId":49537,"journal":{"name":"Set-Valued and Variational Analysis","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142179534","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On New Generalized Differentials with Respect to a Set and Their Applications","authors":"Xiaolong Qin, Vo Duc Thinh, Jen-Chih Yao","doi":"10.1007/s11228-024-00729-z","DOIUrl":"https://doi.org/10.1007/s11228-024-00729-z","url":null,"abstract":"<p>The notions and certain fundamental characteristics of the proximal and limiting normal cones with respect to a set are first presented in this paper. We present the ideas of the limiting coderivative and subdifferential with respect to a set of multifunctions and singleton mappings, respectively, based on these normal cones. The necessary and sufficient conditions for the Aubin property with respect to a set of multifunctions are then described by using the limiting coderivative with respect to a set. As a result of the limiting subdifferential with respect to a set, we offer the requisite optimality criteria for local solutions to optimization problems. In addition, we also provide examples to demonstrate the outcomes.</p>","PeriodicalId":49537,"journal":{"name":"Set-Valued and Variational Analysis","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142179536","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Two New Splitting Methods for Three-Operator Monotone Inclusions in Hilbert Spaces","authors":"Van Dung Nguyen, Nguyen The Vinh","doi":"10.1007/s11228-024-00730-6","DOIUrl":"https://doi.org/10.1007/s11228-024-00730-6","url":null,"abstract":"<p>In this paper, we propose two new unified splitting methods for monotone inclusion problems with three operators in real Hilbert spaces. These methods are based on the combination of Douglas-Rachford method and other methods, forward-backward-forward method and reflected-forward-backward method. The weak convergence of new algorithms under standard assumptions is established. We also give some numerical examples to demonstrate the efficiency of the proposed methods.</p>","PeriodicalId":49537,"journal":{"name":"Set-Valued and Variational Analysis","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141933782","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Correction to: Calmness of a Perturbed Cournot Oligopoly Game with Nonsmooth Cost Functions","authors":"Matthieu Maréchal","doi":"10.1007/s11228-024-00726-2","DOIUrl":"https://doi.org/10.1007/s11228-024-00726-2","url":null,"abstract":"","PeriodicalId":49537,"journal":{"name":"Set-Valued and Variational Analysis","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141643250","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Lower Semicontinuity of Intersections of Set-Valued Maps and Applications on Bilevel Games","authors":"Anton Svensson","doi":"10.1007/s11228-024-00728-0","DOIUrl":"https://doi.org/10.1007/s11228-024-00728-0","url":null,"abstract":"","PeriodicalId":49537,"journal":{"name":"Set-Valued and Variational Analysis","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141640638","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Subtransversality and Strong CHIP of Closed Sets in Asplund Spaces","authors":"Zhou Wei, M. Théra, Jen-Chih Yao","doi":"10.1007/s11228-024-00727-1","DOIUrl":"https://doi.org/10.1007/s11228-024-00727-1","url":null,"abstract":"","PeriodicalId":49537,"journal":{"name":"Set-Valued and Variational Analysis","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141647283","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Sequential M-Stationarity Conditions for General Optimization Problems","authors":"Nooshin Movahedian, Fatemeh Pourahmad","doi":"10.1007/s11228-024-00724-4","DOIUrl":"https://doi.org/10.1007/s11228-024-00724-4","url":null,"abstract":"<p>In this paper, we investigate sequential M-stationarity conditions for a class of nonsmooth nonconvex general optimization problems. We introduce various types of such conditions and compare them with previously established conditions in smooth or convex cases. The application of the derived results is demonstrated in the context of nonsmooth sparsity-constrained optimization problems. Additionally, we devise a Lagrangian-type algorithm for a specific case of smooth sparsity problems. Several examples are presented throughout the paper to illustrate the results.</p>","PeriodicalId":49537,"journal":{"name":"Set-Valued and Variational Analysis","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141546529","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Approximation of Chattering Arcs in Optimal Control Problems Governed by Mono-Input Affine Control Systems","authors":"Térence Bayen, Francis Mairet","doi":"10.1007/s11228-024-00725-3","DOIUrl":"https://doi.org/10.1007/s11228-024-00725-3","url":null,"abstract":"<p>In this paper, we consider a general Mayer optimal control problem governed by a mono-input affine control system whose optimal solution involves a second-order singular arc (leading to chattering). The objective of the paper is to present a numerical scheme to approach the chattering control by controls with a simpler structure (concatenation of bang-bang controls with a finite number of switching times and first-order singular arcs). Doing so, we consider a sequence of vector fields converging to the drift such that the associated optimal control problems involve only first-order singular arcs (and thus, optimal controls necessarily have a finite number of bang arcs). Up to a subsequence, we prove convergence of the sequence of extremals to an extremal of the original optimal control problem as well as convergence of the value functions. Next, we consider several examples of problems involving chattering. For each of them, we give an explicit family of approximated optimal control problems whose solutions involve bang arcs and first-order singular arcs. This allows us to approximate numerically solutions (with chattering) to these original optimal control problems.</p>","PeriodicalId":49537,"journal":{"name":"Set-Valued and Variational Analysis","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141546530","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Generalized Nash Equilibrium from a Robustness Perspective in Variational Analysis","authors":"R. T. Rockafellar","doi":"10.1007/s11228-024-00722-6","DOIUrl":"https://doi.org/10.1007/s11228-024-00722-6","url":null,"abstract":"","PeriodicalId":49537,"journal":{"name":"Set-Valued and Variational Analysis","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141400873","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}