Iterative Approaches to Convex Minimization Problems

Abstract

The aim of this paper is to generalize the results of Yamada et al. [Yamada, I., Ogura, N., Yamashita, Y., Sakaniwa, K. (1998). Quadratic approximation of fixed points of nonexpansive mappings in Hilbert spaces. Numer. Funct. Anal. Optimiz. 19(1):165–190], and to provide complementary results to those of Deutsch and Yamada [Deutsch, F., Yamada, I. (1998). Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappings. Numer. Funct. Anal. Optim. 19(1&2):33–56] in which they consider the minimization of some function θ over a closed convex set F, the nonempty intersection of N fixed point sets. We start by considering a quadratic function θ and providing a relaxation of conditions of Theorem 1 of Yamada et al. (1998) to obtain a sequence of fixed points of certain contraction maps, converging to the unique minimizer of θover F. We then extend Theorem 2 and obtain a complementary result to Theorem 3 of Yamada et al. (1998) by replacing the condition on the parameters by the more general condition lim _n→∞λ _n /λ _n+N  = 1. We next look at minimizing a more general function θ than a quadratic function which was proposed by Deutsch and Yamada (1998) and show that the sequence of fixed points of certain maps converge to the unique minimizer of θ over F. Finally, we prove a complementary result to that of Deutsch and Yamada (1998) by using the alternate condition on the parameters.

Key Words:

• Iterative approach
• Convex feasibility problem
• Convex minimization
• Nonexpansive mapping

2000 Mathematics Subject Classification:

• 47H09
• 65J15

Acknowledgment

The authors are grateful to the referees for their suggestions and comments which improved the presentation of this paper The third author was supported in part by the National Research Foundation of South Africa.