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

Abstract

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.