{"title":"Probability Functions Generated by Set-Valued Mappings: A Study of First Order Information","authors":"Wim van Ackooij, Pedro Pérez-Aros, Claudia Soto","doi":"10.1007/s11228-024-00709-3","DOIUrl":null,"url":null,"abstract":"<p>Probability functions appear in constraints of many optimization problems in practice and have become quite popular. Understanding their first-order properties has proven useful, not only theoretically but also in implementable algorithms, giving rise to competitive algorithms in several situations. Probability functions are built up from a random vector belonging to some parameter-dependent subset of the range of that given random vector. In this paper, we investigate first order information of probability functions specified through a convex-valued set-valued application. We provide conditions under which the resulting probability function is indeed locally Lipschitzian. We also provide subgradient formulæ. The resulting formulæ are made concrete in a classic optimization setting and put to work in an illustrative example coming from an energy application.</p>","PeriodicalId":49537,"journal":{"name":"Set-Valued and Variational Analysis","volume":"21 1","pages":""},"PeriodicalIF":1.3000,"publicationDate":"2024-02-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Set-Valued and Variational Analysis","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s11228-024-00709-3","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
Probability functions appear in constraints of many optimization problems in practice and have become quite popular. Understanding their first-order properties has proven useful, not only theoretically but also in implementable algorithms, giving rise to competitive algorithms in several situations. Probability functions are built up from a random vector belonging to some parameter-dependent subset of the range of that given random vector. In this paper, we investigate first order information of probability functions specified through a convex-valued set-valued application. We provide conditions under which the resulting probability function is indeed locally Lipschitzian. We also provide subgradient formulæ. The resulting formulæ are made concrete in a classic optimization setting and put to work in an illustrative example coming from an energy application.
期刊介绍：
The scope of the journal includes variational analysis and its applications to mathematics, economics, and engineering; set-valued analysis and generalized differential calculus; numerical and computational aspects of set-valued and variational analysis; variational and set-valued techniques in the presence of uncertainty; equilibrium problems; variational principles and calculus of variations; optimal control; viability theory; variational inequalities and variational convergence; fixed points of set-valued mappings; differential, integral, and operator inclusions; methods of variational and set-valued analysis in models of mechanics, systems control, economics, computer vision, finance, and applied sciences. High quality papers dealing with any other theoretical aspect of control and optimization are also considered for publication.
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