{"title":"On Some Problems with Multivalued Mappings","authors":"M. V. Balashov, K. Z. Biglov, A. A. Tremba","doi":"10.1134/s0005117924050035","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>We consider some problems with a set-valued mapping, which can be reduced to minimization of a homogeneous Lipschitz function on the unit sphere. Latter problem can be solved in some cases with a first order algorithm—the gradient projection method. As one of the examples, the case when set-valued mapping is the reachable set of a linear autonomous controlled system is considered. In several settings, the linear convergence is proven. The methods used in proofs follow those introduced by B.T. Polyak for the case where Lezanski–Polyak–Lojasiewicz condition holds. Unlike algorithms that use approximation of the reachable set, the proposed algorithms depend far less on dimension and other parameters of the problem. Efficient error estimation is possible. Numerical experiments confirm the effectiveness of the considered approach. This approach can also be applied to various set-theoretical problems with general set-valued mappings.</p>","PeriodicalId":55411,"journal":{"name":"Automation and Remote Control","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Automation and Remote Control","FirstCategoryId":"94","ListUrlMain":"https://doi.org/10.1134/s0005117924050035","RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"AUTOMATION & CONTROL SYSTEMS","Score":null,"Total":0}
引用次数: 0
Abstract
We consider some problems with a set-valued mapping, which can be reduced to minimization of a homogeneous Lipschitz function on the unit sphere. Latter problem can be solved in some cases with a first order algorithm—the gradient projection method. As one of the examples, the case when set-valued mapping is the reachable set of a linear autonomous controlled system is considered. In several settings, the linear convergence is proven. The methods used in proofs follow those introduced by B.T. Polyak for the case where Lezanski–Polyak–Lojasiewicz condition holds. Unlike algorithms that use approximation of the reachable set, the proposed algorithms depend far less on dimension and other parameters of the problem. Efficient error estimation is possible. Numerical experiments confirm the effectiveness of the considered approach. This approach can also be applied to various set-theoretical problems with general set-valued mappings.
期刊介绍：
Automation and Remote Control is one of the first journals on control theory. The scope of the journal is control theory problems and applications. The journal publishes reviews, original articles, and short communications (deterministic, stochastic, adaptive, and robust formulations) and its applications (computer control, components and instruments, process control, social and economy control, etc.).
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