Forcing strong convergence of proximal point iterations in a Hilbert space

This paper concerns with convergence properties of the classical proximal p oint algorithm for finding zeroes of maximal monotone operators in an infin ite-dimensional Hilbert space. It is well known that the proximal point alg orithm converges weakly to a solution under very mild assumptions: However, it was shown by Guler [11] that the iterates may fail to converge strongly in the infinite-dirnensional case. We propose a new proximal-type algorith m which does converge strongly, provided the problem has a solution. Moreov er, our algorithm solves proximal point subproblems inexactly, with a const ructive stopping criterion introduced in [31]. Strong convergence is forced by combining proximal point iterations! with simple projection steps onto intersection of two halfspaces containing the solution set. Additional cost of this extra projection step is essentially negligible since it amounts, at most, to solving a linear system of two equations in two unknowns.