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,ΩT,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 Ωn⊃ΩΩn⊃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)(T,Ω). It is shown that if T   is either relaxed αα-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)(T,Ω) under quite mild conditions.