An explicit algorithm for solving monotone variational inequalities

In this paper, we are interested in the generalized variational inequality problem in real Hilbert spaces. We propose an explicit proximal method which requires only one proximal step and one mapping evaluation per iteration and also uses an adaptive step-size rule that enables to avoid the prior knowledge of the Lipschitz constant of the involved mapping. Weak convergence of the proposed scheme is established under standard assumptions. Under strong monotonicity, we present the R-linear convergence rate of our new method. Intensive numerical experiments illustrate the advantages and the applicability of our scheme. Moreover, our work generalizes theoretically several recent results in this field.