Title of Paper:An algorithm based on resolvent operators for solving variational inequalities in Hilbert spaces

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Date of Publication:2008-11-15

Journal:NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS

Included Journals:SCIE、EI、Scopus

Volume:69

Issue:10

Page Number:3344-3357

ISSN No.:0362-546X

Key Words:Hilbert space; Cone; M-Monotone operator; Resolvent operator; Variational inequality; Convergence property

Abstract:In this paper, a new monotonicity, M-monotonicity, is introduced, and the resolvent operator of an M-monotone operator is proved to be single valued and Lipschitz continuous. With the help of the resolvent operator, an equivalence between the variational inequality VI(C, F + G) and the fixed point problem of a nonexpansive mapping is established. A proximal point algorithm is constructed to solve the fixed point problem, which is proved to have a global convergence under the condition that F in the VI problem is strongly monotone and Lipschitz continuous. Furthermore, a convergent path Newton method, which is based on the assumption that the projection mapping Pi(C)(.) is semismooth, is given for calculating epsilon-solutions to the sequence of fixed point problems, enabling the proximal point algorithm to be implementable. (c) 2007 Elsevier Ltd. All rights reserved.