Convergence and stability of a regularization method for maximal monotone inclusions and its applications to convex optimization

In this paper we study the stability and convergence of a regularization method for solving inclusions f is an element of Ax, 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(k) and f(k) converging to A and f, respectively, in the sense of Mosco type topologies. We prove that the sequence x(k) = (A(k) + alpha(k)J(mu))(-1)f(k) which results from the regularization process converges weakly and, under some conditions, converges strongly to the minimum norm solution of the inclusion f is an element of 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.