Approximate proximal algorithms for generalized variational inequalities with pseudomonotone multifunctions

• Published in: Journal of computational and applied mathematics Vol. 213; no. 2; pp. 423 - 438
• Main Authors: Ceng, L.C., Yao, J.C.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.04.2008; Elsevier
• Subjects: Approximate proximal algorithms; Calculus of variations and optimal control; Exact sciences and technology; Functional analysis; Generalized variational inequalities; Global analysis, analysis on manifolds; Hilbert space; Mathematical analysis; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Pseudomonotone multifunctions; Sciences and techniques of general use; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds; Weak accumulation points; Weak convergence; Approximate proximal algorithms; Weak convergence; Generalized variational inequalities; 65J05; Weak accumulation points; 65K05; 90C25; 49M27; Pseudomonotone multifunctions; Hilbert space; Variational problem; Numerical analysis; Variational inequality; Applied mathematics; 49M27; 65J05; 65K05; 90C25; Generalized variational inequalities; Pseudomonotone multifunctions; Approximate proximal algorithms; Weak accumulation points; Weak convergence; Hilbert space; Algorithm
• ISSN: 0377-0427; 1879-1778
• DOI: 10.1016/j.cam.2007.01.034

The purpose of this paper is to investigate the convergence of general approximate proximal algorithm (resp. general Bregman-function-based approximate proximal algorithm) for solving the generalized variational inequality problem (for short, GVI( T , Ω ) where T is a multifunction). The general approximate proximal algorithm (resp. general Bregman-function-based approximate proximal algorithm) is to define new approximating subproblems on the domains Ω n ⊃ Ω , n = 1 , 2 , … , which form a general approximate proximate point scheme (resp. a general Bregman-function-based approximate proximate point scheme) for solving GVI ( T , Ω ) . It is shown that if T is either relaxed α -pseudomonotone or pseudomonotone, then the general approximate proximal point scheme (resp. general Bregman-function-based approximate proximal point scheme) generates a sequence which converges weakly to a solution of GVI ( T , Ω ) under quite mild conditions.