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 {A_n} and {B_n}, each of them converging, with respect to the Attouch-Wets variational convergence, respectively, to A and B. Given a starting point a_0, we consider the sequences of points obtained by projecting on the “perturbed” sets, i.e., the sequences {a_n} and {b_n} given by b_n=P_{B_n}(a_{n-1}) and a_n=P_{A_n}(b_n). Under appropriate geometrical and topological assumptions on the intersection of the limit sets, we ensure that the sequences {a_n} and {b_n} converge in norm to a point in the intersection of A and B. In particular, we consider both when the intersection Acap B reduces to a singleton and when the interior of A cap B is nonempty. Finally we consider the case in which the limit sets A and B are subspaces.