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An existence-uniqueness theorem and alternating contraction projection methods for inverse variational inequalities
He,Songnian; Dong,Qiao-Li
2018-12-18
Source PublicationJournal of Inequalities and Applications
ISSN1029-242X
Volume2018Issue:1
AbstractAbstractLet C be a nonempty closed convex subset of a real Hilbert space H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{H}$\end{document} with inner product ??,??\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\langle \cdot , \cdot \rangle $\end{document}, and let f:H→H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f: \mathcal{H}\rightarrow \mathcal{H}$\end{document} be a nonlinear operator. Consider the inverse variational inequality (in short, IVI(C,f)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\operatorname{IVI}(C,f)$\end{document}) problem of finding a point ξ?∈H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\xi ^{*}\in \mathcal{H}$\end{document} such that f(ξ?)∈C,?ξ?,v?f(ξ?)?≥0,?v∈C.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ f\bigl(\xi ^{*}\bigr)\in C, \quad \bigl\langle \xi ^{*}, v-f \bigl(\xi ^{*}\bigr)\bigr\rangle \geq 0, \quad \forall v\in C. $$\end{document} In this paper, we prove that IVI(C,f)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\operatorname{IVI}(C,f)$\end{document} has a unique solution if f is Lipschitz continuous and strongly monotone, which essentially improves the relevant result in (Luo and Yang in Optim. Lett. 8:1261–1272, 2014). Based on this result, an iterative algorithm, named the alternating contraction projection method (ACPM), is proposed for solving Lipschitz continuous and strongly monotone inverse variational inequalities. The strong convergence of the ACPM is proved and the convergence rate estimate is obtained. Furthermore, for the case that the structure of C is very complex and the projection operator PC\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$P_{C}$\end{document} is difficult to calculate, we introduce the alternating contraction relaxation projection method (ACRPM) and prove its strong convergence. Some numerical experiments are provided to show the practicability and effectiveness of our algorithms. Our results in this paper extend and improve the related existing results.
KeywordInverse variational inequality Variational inequality Lipschitz continuous Strongly monotone 47J20 90C25 90C30 90C52
DOI10.1186/s13660-018-1943-0
Language英语
WOS IDBMC:10.1186/s13660-018-1943-0
PublisherSpringer International Publishing
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Document Type期刊论文
Identifierhttp://ir.amss.ac.cn/handle/2S8OKBNM/31503
Collection中国科学院数学与系统科学研究院
Affiliation
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He,Songnian,Dong,Qiao-Li. An existence-uniqueness theorem and alternating contraction projection methods for inverse variational inequalities[J]. Journal of Inequalities and Applications,2018,2018(1).
APA He,Songnian,&Dong,Qiao-Li.(2018).An existence-uniqueness theorem and alternating contraction projection methods for inverse variational inequalities.Journal of Inequalities and Applications,2018(1).
MLA He,Songnian,et al."An existence-uniqueness theorem and alternating contraction projection methods for inverse variational inequalities".Journal of Inequalities and Applications 2018.1(2018).
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