An existence-uniqueness theorem and alternating contraction projection methods for inverse variational inequalities

doi:10.1186/s13660-018-1943-0

CSpace

	An existence-uniqueness theorem and alternating contraction projection methods for inverse variational inequalities
	He,Songnian; Dong,Qiao-Li
	2018-12-18
发表期刊	Journal of Inequalities and Applications
ISSN	1029-242X
卷号	2018 期号:1
摘要	AbstractLet 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.
关键词	Inverse variational inequality Variational inequality Lipschitz continuous Strongly monotone 47J20 90C25 90C30 90C52
DOI	10.1186/s13660-018-1943-0
语种	英语
WOS记录号	BMC:10.1186/s13660-018-1943-0
出版者	Springer International Publishing
引用统计
文献类型	期刊论文
条目标识符	http://ir.amss.ac.cn/handle/2S8OKBNM/31503
专题	中国科学院数学与系统科学研究院
通讯作者	Dong,Qiao-Li
作者单位
推荐引用方式 GB/T 7714	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).