Relaxed Alternating Projection Methods

In this paper we deal with the von Neumann alternating projection method xk+1 = PAPBxk and with its generalization of the form xk+1 = PA(xk + k(PAPBxk xk)), where A;B are closed and convex subsets of a Hilbert space H and FixPAPB 6= ?. We do not suppose that A \ B 6= ?. We give su¢ cient conditions for the weak convergence of the sequence (xk) to FixPAPB in the general case and in the case A is a closed a¢ ne subspace. We present also the results of preliminary numerical experiments. Key words: alternating projection method, Fejér monotonicity, weak convergence AMS Subject Classi…cation: 65K05 1. Introduction Let H be a real Hilbert space equipped with a scalar product h ; i and with the norm k k induced by h ; i. Further, let A;B H be nonempty, convex and closed subsets. In the practical considerations one often needs to …nd an element of the intersection A \ B or, more general, to solve the following problem …nd a 2 A and b 2 B such that ka b k = inf a2A;b2B ka bk: (1)