Extrapolated simultaneous block‐iterative cutter methods and applications

Abstract

Our goal of this paper is to solve the problem of common fixed point of a quite large family of demiclosed cutters on Hilbert spaces. We introduce two new extrapolated simultaneous block‐iterative methods. The first method converges weakly to a common fixed point of the family and the second one is a specific combination of the first and the steepest descent method and converges in norm. Its strong convergence is proved without additional conditions on the cutters such as the approximately shrinking property of each cutter and bounded regular assumption on their fixed point sets, assumed recently in the literature. Some particular cases of the last method, applications and computational experiments to a convex optimization over the intersection of level sets are given for illustration and comparison.