Communications in Optimization Theory

Abstract

We present an approximate algorithm for solving generalized variational inequality problem (P) by combining the bundle technique with the auxiliary problem method. For subproblems we con-struct a new lower approximation to the involved function phi, as in the bundle method for nonsmoothoptimization, which only requires the approximate function values and approximate subgradients insteadof the exact ones. This makes the subproblem easier to solve and more tractable. Besides that, a newstopping criterion is given to determine whether the current approximation is good enough, it not onlytakes into account the contribution of the operator F in problem (P), but also utilizes the inexact in-formation of the involved function phi in sub problem. Finally, we study the convergence of the proposedalgorithm for the case when the operator F is paramonotone, possibly multivalued and the stepsizes arechosen going to zero.