NON-STRICTLY CONVEX MINIMIZATION OVER THE FIXED POINT SET OF AN ASYMPTOTICALLY SHRINKING NONEXPANSIVE MAPPING

• Published in: Numerical functional analysis and optimization Vol. 23; no. 1-2; pp. 113 - 137
• Main Authors: Ogura, Nobuhiko, Yamada, Isao
• Format: Journal Article
• Language: English
• Published: Taylor & Francis Group 05.01.2002
• Subjects: 2000 Mathematics Subject Classification; Convex optimization; Convex projection; Fixed point theorem; Generalized convex feasible set; Inverse problem; Monotone operator; Nonexpansive mapping; Steepest descent method; Variational inequality problem
• ISSN: 0163-0563; 1532-2467
• DOI: 10.1081/NFA-120003674

Suppose that T is a nonexpansive mapping on a real Hilbert space satisfying for some R > 0. Suppose also that a mapping is κ-Lipschitzian over and paramonotone over . Then it is shown that a variation of the hybrid steepest descent method (Yamada, Ogura, Yamashita and Sakaniwa (1998), Deutsch and Yamada (1998) and Yamada (1999-2001)): generates a sequence (u n ) satisfying , when is finite dimensional, where for all is the solution set of the variational inequality problem . This result relaxes the condition on and (λ n ) of the hybrid steepest descent method (Yamada (2001)), and makes the method applicable to the significantly wider class of convexly constrained inverse problems as well as the non-strictly convex minimization over the fixed point set of asymptotically shrinking nonexpansive mapping.