Nonstrictly Convex Minimization over the Bounded Fixed Point Set of a Nonexpansive Mapping

• Published in: Numerical functional analysis and optimization Vol. 24; no. 1-2; pp. 129 - 135
• Main Authors: Ogura, Nobuhiko, Yamada, Isao
• Format: Journal Article
• Language: English
• Published: Taylor & Francis Group 06.01.2003
• Subjects: 2000 Mathematical Subject Classification: 47H05; Convex optimization; Fixed point theorem; Hybrid steepest descent method; Monotone operator; Nonexpansive mapping; Variational inequality problem
• ISSN: 0163-0563; 1532-2467
• DOI: 10.1081/NFA-120020250

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 ). Convex projection algorithm from POCS to Hybrid steepest descent method. The Journal of the IEICE (in Japanese) 83:616-623; Yamada, I. ( 2001 ). 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 ). 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.