Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappings

• Published in: Numerical functional analysis and optimization Vol. 19; no. 1-2; pp. 33 - 56
• Main Authors: DEUTSCH, F, YAMADA, I
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Marcel Dakker Inc 01.01.1998; Taylor & Francis
• Subjects: 1991 Mathematics Subject Classification 47H09; 1991 Mathematics Subject Classification 47H10; 1991 Mathematics Subject Classification 49M45; 1991 Mathematics Subject Classification 65K05; 1991 Mathematics Subject Classification 65K10; 1991 Mathematics Subject Classification 90C25; 1991 Mathematics Subject Classification 90C30; best approximation; Calculus of variations and optimal control; convex feasibility problem; convex optimization; convex projection; Exact sciences and technology; fixed point theorem; generalized convex feasible set; Mathematical analysis; Mathematics; nonexpansive mapping; quadratic function; Sciences and techniques of general use; steepest descent method; Convex hull; Convex set; Fix point; Metric space; Function minimization; Hilbert space; Convex function; Algorithm
• ISSN: 0163-0563; 1532-2467
• DOI: 10.1080/01630569808816813

Let be nonexpansive mappings on a Hilbert space H, and let be a function which has a uniformly strongly positive and uniformly bounded second (Fréchet) derivative over the convex hull of T i (H) for some i. We first prove that Θ has a unique minimum over the intersection of the fixed point sets of all the T i 's at some point u * . Then a cyclic hybrid steepest descent algorithm is proposed and we prove that it converges to u * . This generalizes some recent results of Wittmann (1992), Combettes (1995), Bauschke (1996), and Yamada, Ogura, Yamashita, and Sakaniwa (1997). In particular, the minimization of Θ over the intersection of closed convex sets C i can be handled by taking T i to be the metric projection Pc i onto C i . We also propose a modification of our algorithm to handle the inconsistent case (i.e., when is empty as well.