Approximate proximal algorithms for generalized variational inequalities with pseudomonotone multifunctions

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, Omega) 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 0,1 D Omega, 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, Omega). It is shown that if T is either relaxed alpha-pseudo monotone 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, Omega) under quite mild conditions. (c) 2007 Elsevier B.V. All rights reserved.