A Perturbed Auxiliary Problem Method for Paramonotone Multivalued Mappings

Abstract

We consider the auxiliary problem method to solve general variational inequalities with a multivalued mapping in a Hilbert space. Perturbed versions of this method have already been studied in the literature. They allow to consider for example barrier functions and interior ap­proximations of the feasible domain. In a preceding paper, convergence is ensured assuming strong monotonicity of the operator defining the problem. In this paper, we present convergence results under weaker monotonicity assumptions. First, when the operator is paramonotone and satisfies a kind of continuity property, we obtain that at least one weak limit point of the sequence generated by the algorithm is a solution of the original problem. In a second part, we prove weak convergence under a condition satisfied for example if the operator is compact-valued and paramonotone, or is strongly monotone, or is the subdifferential of a lower semicontinuous proper convex function. Finally, we present a strong convergence result. In the particular case of non differentiable convex optimization, our scheme is a generalization of the projected subgradient procedure.