A Generalization of the Proximal Point Algorithm

The problem considered 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, a quadratic term for a subset of variables is added. This paper proves that under a mild assumption our algorithm has the same convergence properties as the regular proximal point algorithm.