Relaxed Alternating Projection Methods

Let A and B be nonempty, convex and closed subsets of a Hilbert space H. In the practical considerations we need to find an element of the intersection A B or, more general, to solve the following problem: find a A and b B such that a - b = inf aA,bB a - b . One of the important methods generating sequences which converge weakly to a solution of above problems is the von Neumann alternating projection method xk+1 = PAPBxk. The method has found application in different areas of mathematics. These include probability and statistics, image reconstruction and intensity modulated radiation therapy, where the convex subsets are described by a large and sparse system of linear equations or inequalities. We deal with generalization of the von Neumann alternating projection method of the form xk+1 = PA(xk + k(xk)(PAPBxk - xk)), where Fix PAPB = . We give sufficient conditions for the weak convergence of the sequence (xk) to Fix PAPB in the general case and in the case A is a closed affine sub...

• Type: Journal
• Year: 2008
• Where: SIAMJO
• Authors: Andrzej Cegielski, Agnieszka Suchocka
• Keywords: Projection Method, SIAMJO 2008, Von Neumann, Weak Convergence