CONVERGENCE AND STABILITY OF A REGULARIZATION METHOD FOR MAXIMAL MONOTONE INCLUSIONS AND ITS APPLICATIONS TO CONVEX OPTIMIZATION

Abstract: In this paper we study the stability and convergence of a regularization method for solving inclusions f Ax∈ k, where A is a maximal monotone point-to-set operator from a reflexive smooth Banach space X with the Kadec-Klee property to its dual. We assume that the data A and f involved in the inclusion are given by approximations A and kf converging to A and f, respectively, in the sense of Mosco type topologies. We prove that the sequence 1 ()k k k kx A J µα − = + f which results from the regularization process converges weakly and, under some conditions, converges strongly to the minimum norm solution of the inclusion f Ax ∈ , provided that the inclusion is consistent. These results lead to a regularization procedure for perturbed convex optimization problems whose objective functions and feasibility sets are given by approximations. In particular, we obtain a strongly convergent version of the generalized proximal point optimization algorithm which is applicable to problems whose feasibility sets are given by Mosco approximations Key words: Maximal monotone inclusion, Mosco convergence of sets, regularization method, convex optimization problem, generalized proximal point method for optimization