Approximate proximal algorithms for generalized variational inequalities with pseudomonotone multifunctions

Abstract

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.