A Modified Projected Gradient Method for Monotone Variational Inequalities

Abstract

In this paper, we investigate and analyze classical variational inequalities with Lipschitz continuous and monotone mapping in real Hilbert space. The projected reflected gradient method, with varying step size, requires at most two projections onto the feasible set and one value of the mapping per iteration. We modify the method with a simple structure; a weak convergence theorem for our algorithm is proved without any requirement of additional projections and the knowledge of the Lipschitz constant of the mapping. Meanwhile, R-linear convergence rate is obtained under strong monotonicity assumption of the mapping. Preliminary results from numerical experiments are performed.