Variable target value relaxed alternating projection method

Abstract

In this paper we propose a modification of the von Neumann method of alternating projection x k+1=P A P B x k where A,B are closed and convex subsets of a real Hilbert space ?. If Fix?P A P B ?? then any sequence generated by the classical method converges weakly to a fixed point of the operator T=P A P B . If the distance ?=inf? x?A,y?B ? x?y ? is known then one can efficiently apply a modification of the von Neumann method, which has the form x k+1=P A (x k +? k (P A P B x k ?x k )) for ? k >0 depending on x k (for details see: Cegielski and Suchocka, SIAM J. Optim. 19:1093---1106, 2008). Our paper contains a generalization of this modification, where we do not suppose that we know the value ?. Instead of ? we apply its approximation which is updated in each iteration.