{"title":"Regularization methods for hierarchical variational inequality problems on Hadamard manifolds","authors":"Qamrul Hasan Ansari, Feeroz Babu, Moin Uddin","doi":"10.1007/s40065-022-00395-9","DOIUrl":null,"url":null,"abstract":"<div><p>This article proposes two regularized iterative algorithms for solving variational inequality problems defined over a solution set of a variational inclusion problem, known as hierarchical variational inequality problems, in the setting of Hadamard manifolds. Instead of regularizing the variational inequality problem or an iterative method for solving it, we first regularize the considered variational inclusion problem, and then prove that the solution of the regularized problem converges to a solution of the hierarchical variational inequality problem. Using such a result, we prove the convergence of the sequences generated by the proposed algorithms to a solution of the considered hierarchical variational inequality problem. A computational experiment is provided to see the validity and effectiveness of the proposed algorithms.</p></div>","PeriodicalId":54135,"journal":{"name":"Arabian Journal of Mathematics","volume":"12 2","pages":"309 - 330"},"PeriodicalIF":0.9000,"publicationDate":"2022-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40065-022-00395-9.pdf","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Arabian Journal of Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://link.springer.com/article/10.1007/s40065-022-00395-9","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 1
Abstract
This article proposes two regularized iterative algorithms for solving variational inequality problems defined over a solution set of a variational inclusion problem, known as hierarchical variational inequality problems, in the setting of Hadamard manifolds. Instead of regularizing the variational inequality problem or an iterative method for solving it, we first regularize the considered variational inclusion problem, and then prove that the solution of the regularized problem converges to a solution of the hierarchical variational inequality problem. Using such a result, we prove the convergence of the sequences generated by the proposed algorithms to a solution of the considered hierarchical variational inequality problem. A computational experiment is provided to see the validity and effectiveness of the proposed algorithms.
期刊介绍：
The Arabian Journal of Mathematics is a quarterly, peer-reviewed open access journal published under the SpringerOpen brand, covering all mainstream branches of pure and applied mathematics.
Owned by King Fahd University of Petroleum and Minerals, AJM publishes carefully refereed research papers in all main-stream branches of pure and applied mathematics. Survey papers may be submitted for publication by invitation only.To be published in AJM, a paper should be a significant contribution to the mathematics literature, well-written, and of interest to a wide audience. All manuscripts will undergo a strict refereeing process; acceptance for publication is based on two positive reviews from experts in the field.Submission of a manuscript acknowledges that the manuscript is original and is not, in whole or in part, published or submitted for publication elsewhere. A copyright agreement is required before the publication of the paper.Manuscripts must be written in English. It is the author''s responsibility to make sure her/his manuscript is written in clear, unambiguous and grammatically correct language.
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