{"title":"Dynamic System for Solving Saddle Point Problems in Hilbert Spaces and Its Application to Neural Computing","authors":"Xisheng Shen (沈喜生) , Xiaofang Wang (王晓芳) , Yueting Chai (柴跃廷)","doi":"10.1016/S1007-0214(11)70046-3","DOIUrl":null,"url":null,"abstract":"<div><p>This paper studies the existence and uniqueness of solutions and the stability and convergence of a dynamic system for solving saddle point problems (SPP) in Hilbert spaces. The analysis first converts the SPP into a problem of searching for equilibriums of a dynamic system using a criterion for solutions of the SPP, then shows the existence and uniqueness of the solutions by creating a positive function whose Fréchet derivative is decreasing along any solution. The construction of positively invariant subsets gives the global stability and convergence of this dynamic system, that is, the dynamic system globally converges to some exact solution of the SPP. Finally, the paper also shows that the obtained results can be applied to neural computing for solving SPP.</p></div>","PeriodicalId":60306,"journal":{"name":"Tsinghua Science and Technology","volume":null,"pages":null},"PeriodicalIF":5.2000,"publicationDate":"2011-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S1007-0214(11)70046-3","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Tsinghua Science and Technology","FirstCategoryId":"1093","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S1007021411700463","RegionNum":1,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"COMPUTER SCIENCE, INFORMATION SYSTEMS","Score":null,"Total":0}
引用次数: 0
Abstract
This paper studies the existence and uniqueness of solutions and the stability and convergence of a dynamic system for solving saddle point problems (SPP) in Hilbert spaces. The analysis first converts the SPP into a problem of searching for equilibriums of a dynamic system using a criterion for solutions of the SPP, then shows the existence and uniqueness of the solutions by creating a positive function whose Fréchet derivative is decreasing along any solution. The construction of positively invariant subsets gives the global stability and convergence of this dynamic system, that is, the dynamic system globally converges to some exact solution of the SPP. Finally, the paper also shows that the obtained results can be applied to neural computing for solving SPP.