Nonstrictly Convex Minimization over the Bounded Fixed Point Set of a Nonexpansive Mapping
Nobuhiko Ogura, Isao Yamada · Numerical Functional Analysis and Optimization · 2003
In this paper, we consider, in a finite dimensional real Hilbert space , the variational inequality problem VIP: find , where is nonexpansive mapping with bounded and is paramonotone and Lipschitzian over . The nonstrictly convex minimization over the bounded fixed point set of a nonexpansive mapping is a typical example of such a variational inequality problem. We show that the hybrid steepest descent method, of which convergence properties were examined in some cases for example (Yamada, I. (2000 Yamada, I. 2000. Convex projection algorithm from POCS to Hybrid steepest descent method. The Journal of the IEICE, 83: 616–623. (in Japanese) [Google Scholar]). Convex projection algorithm from POCS to Hybrid steepest descent method. The Journal of the IEICE (in Japanese) 83:616–623; Yamada, I. (2001 Yamada, I. 2001. “The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings”. In Inherently Parallel Algorithm for Feasibility and Optimization Edited by: Butnariu, D., Censor, Y. and Reich, S. Elsevier. [Crossref] , [Google Scholar]). The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings. In: Butnariu, D., Censor, Y., Reich, S., eds. Inherently Parallel Algorithm for Feasibility and Optimization. Elsevier; Ogura, N., Yamada, I. (2002 Ogura, N. and Yamada, I. 2002. Non-strictly convex minimization over the fixed point set of an asymptotically shrinking nonexpansive mapping. Numer. Funct. Anal. Optim., 23: 113–137. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). Non-strictly convex minimization over the fixed point set of an asymptotically shrinking nonexpansive mapping. Numer. Funct. Anal. Optim. 23:113–137), is still applicable to the case where and T satisfy the above conditions.