A perturbed and inexact version of the auxiliary problem method for solving general variational inequalities with a multivalued operator

We consider general variational inequalities with a multivalued maximal monotone operator in a Hilbert space. For solving these problems, Cohen developed several years ago the auxiliary problem method. Perturbed versions of this method have been already studied in the literature for the single-valued case. They allow to consider for example, barrier functions and interior approximations of the feasible domain. In this paper, we present a relaxation of these perturbation methods by using the concept of E-enlargement of a maximal monotone operator. We prove that, under classical assumptions, the sequence generated by this scheme is bounded and weakly convergent to a solution of the problem. Strong convergence is also obtained under additional conditions. In the particular case of nondifferentiable convex optimization, the E-subdifferential will take place of the E-enlargement and some assumptions for convergence will be weakened. In the nonperturbed situation, our scheme reduces to the projected inexact subgradient procedure.