Lipschitz连续强单调逆变分不等式的迭代算法
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摘要
假设H是一个实的Hilbert空间,C是H的一个非空闭凸子集,f:H→H是一Lipschitz连续强单调算子。考虑逆变分不等式(简记为IVI(C,f)):即寻求ξ∈H满足f(ξ)∈C,〈ξ,v-f(ξ)〉≥0,坌v∈C。证明了IVI(C,f)解的一个存在唯一性定理,给出了解的两个迭代算法,改进了以往的相关结果。
Let C be a nonempty closed convex subset of a real Hilbert space H, f:H →H be a Lipschitz continuous and strongly monotone mapping. Then, inverse variational inequality is considered(in short, IVI(C, f)): find ξ∈H such that f(ξ)∈C, 〈ξ, v- f(ξ)〉≥0, 坌v∈C. A new existence and uniqueness theorem for inverse variational inequalities is proved and two iterative algorithms are introduced to improve the previous relevant results.
Let C be a nonempty closed convex subset of a real Hilbert space H, f:H →H be a Lipschitz continuous and strongly monotone mapping. Then, inverse variational inequality is considered(in short, IVI(C, f)): find ξ∈H such that f(ξ)∈C, 〈ξ, v- f(ξ)〉≥0, 坌v∈C. A new existence and uniqueness theorem for inverse variational inequalities is proved and two iterative algorithms are introduced to improve the previous relevant results.
引文
[1]HE B S,LIU H X.Inverse Variational Inequalities in the Economic Field:Applications and Algorithms[EB/OL].(2006-09-18)[2015-05-10].http://www.paper.edu.cn/html/release paper/2006/09/260/.
[2]HE B S,LIU H,LI X,et al.PPA-Base Methods for Monotone Inverse Variational Inequalities[EB/OL].(2006-06-21)[2015-05-10].http://www.paper.edu.cn/html/release paper/2006/06/219/.
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[5]MARINO G,XU H K.Weak and strong convergence theorems for strict pseudocontractions in Hilbert spaces[J].J Math Anal Appl,2007,329:336-346.
[6]XU H K.Iterative algorithm for nonlinear operators[J].J Lond Math Soc,2002,66(1):240-256.
[7]HARTMAN P,STAMPACCHIA G.On some non-linear elliptic differential-functional equations[J].J Acta Mathematica,1966,115:153-188.
[8]HALPERN B.Fixed points of nonexpanding maps[J].Bull Amer Math Soc,1967,73:957-961.
[2]HE B S,LIU H,LI X,et al.PPA-Base Methods for Monotone Inverse Variational Inequalities[EB/OL].(2006-06-21)[2015-05-10].http://www.paper.edu.cn/html/release paper/2006/06/219/.
[3]LUO X P.Tikhonov regularization methods for inverse variational inequalities[J].Optim Lett,2014,8:877-887.
[4]LUO X P,YANG J.Regularization and iterative methods for monotone inverse variational inequalities[J].Optim Lett,2014,8:1261-1272.
[5]MARINO G,XU H K.Weak and strong convergence theorems for strict pseudocontractions in Hilbert spaces[J].J Math Anal Appl,2007,329:336-346.
[6]XU H K.Iterative algorithm for nonlinear operators[J].J Lond Math Soc,2002,66(1):240-256.
[7]HARTMAN P,STAMPACCHIA G.On some non-linear elliptic differential-functional equations[J].J Acta Mathematica,1966,115:153-188.
[8]HALPERN B.Fixed points of nonexpanding maps[J].Bull Amer Math Soc,1967,73:957-961.