A variational approach to the alternating projections method

Abstract

The 2-sets convex feasibility problem aims at finding a point in the nonempty intersection of two closed convex sets A and B in a Hilbert space H. The method of alternating projections is the simplest iterative procedure for finding a solution and it goes back to von Neumann. In the present paper, we study some stability properties for this method in the following sense: we consider two sequences of closed convex sets { An} and { Bn} , each of them converging, with respect to the Attouch-Wets variational convergence, respectively, to A and B. Given a starting point a, we consider the sequences of points obtained by projecting on the “perturbed” sets, i.e., the sequences { an} and { bn} given by bn=PBn(an-1) and an=PAn(bn). Under appropriate geometrical and topological assumptions on the intersection of the limit sets, we ensure that the sequences { an} and { bn} converge in norm to a point in the intersection of A and B. In particular, we consider both when the intersection A∩ B reduces to a singleton and when the interior of A∩ B is nonempty. Finally we consider the case in which the limit sets A and B are subspaces.