Mathematical Programming Manuscript No. Forcing Strong Convergence of Proximal Point Iterations in a Hilbert Space

This paper concerns with convergence properties of the classical proximal point algorithm for nding zeroes of maximal monotone operators in an innnite-dimensional Hilbert space. It is well known that the proximal point algorithm converges weakly to a solution under very mild assumptions. However, it was shown by G uler 11] that the iterates may fail to converge strongly in the innnite-dimensional case. We propose a new proximal-type algorithm which does converge strongly, provided the problem has a solution. Moreover, our algorithm solves proximal point subproblems inexactly, with a constructive 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.