Variable target value relaxed alternating projection method

• Authors: A. Cegielski;R. Dylewski
• Venue: Computational Optimization and Applications
• Year: 2010

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 驴. If Fix驴P A P B 驴驴 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 驴=inf驴 x驴A,y驴B 驴 x驴y 驴 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 +驴 k (P A P B x k 驴x k )) for 驴 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 驴. Instead of 驴 we apply its approximation which is updated in each iteration.