{"title":"Analytic formulas for alternating projection sequences for the positive semidefinite cone and an application to convergence analysis","authors":"Hiroyuki Ochiai , Yoshiyuki Sekiguchi , Hayato Waki","doi":"10.1016/j.jmaa.2024.129070","DOIUrl":null,"url":null,"abstract":"<div><div>We derive analytic formulas for the alternating projection method applied to the cone <span><math><msubsup><mrow><mi>S</mi></mrow><mrow><mo>+</mo></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> of positive semidefinite matrices and an affine subspace. More precisely, we find recursive relations on parameters representing a sequence constructed by the alternating projection method. By applying these formulas, we analyze the alternating projection method in detail and show that the upper bound given by the singularity degree is actually tight when the alternating projection method is applied to <span><math><msubsup><mrow><mi>S</mi></mrow><mrow><mo>+</mo></mrow><mrow><mn>3</mn></mrow></msubsup></math></span> and a 3-plane whose intersection is a singleton with singularity degree 2.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"544 1","pages":"Article 129070"},"PeriodicalIF":1.2000,"publicationDate":"2024-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Mathematical Analysis and Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0022247X24009922","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We derive analytic formulas for the alternating projection method applied to the cone of positive semidefinite matrices and an affine subspace. More precisely, we find recursive relations on parameters representing a sequence constructed by the alternating projection method. By applying these formulas, we analyze the alternating projection method in detail and show that the upper bound given by the singularity degree is actually tight when the alternating projection method is applied to and a 3-plane whose intersection is a singleton with singularity degree 2.
期刊介绍:
The Journal of Mathematical Analysis and Applications presents papers that treat mathematical analysis and its numerous applications. The journal emphasizes articles devoted to the mathematical treatment of questions arising in physics, chemistry, biology, and engineering, particularly those that stress analytical aspects and novel problems and their solutions.
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