A generalization of the proximal point algorithm

The problem that we consider in this paper is to find a solution to the generalized equation 0 ?? T(x,y), where T is a maximal monotone operator on the product H1 ?? H2 of two Hilbert spaces H1 and H2. We give a generalization of the proximal map and the proximal point algorithm in which the proposed iterative procedure is based on just one variable. Applying to convex programming problems, instead of adding a quadratic term for all variables as in the proximal point algorithm, we add a quadratic term for a subset of variables. We prove that under a mild assumption our algorithm has the same convergence properties as the regular proximal point algorithm.