A simple projection method for solving quasimonotone variational inequality problems

Abstract

Projection methods have been one of the popular methods for solving variational inequality problems in the literature. Several projection methods have been studied for variational inequality problems and most of these methods require at least two evaluations of the cost operator and (or) two projections per iteration. One of the major interests of researchers in variational inequalities is to study projection methods with less computational complexity in terms of reduced number of evaluations of the cost operator and reduced number of projections onto feasible set per iteration especially when the cost operator is non-monotone. In this paper, we propose a simple projection method which requires only one evaluation of the cost operator and one projection onto the feasible set per iteration. We obtain weak convergence results when the cost operator is quasimonotone and Lipschitz continuous with self-adaptive line search strategy. We also give weak convergence analysis when the cost operator is non-monotone and Lipschitz continuous. Some numerical implementations using standard tests are given to illustrate our method and compare with other recently related methods.