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

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 Ti(H) for some i. We first prove that Θ has a unique minimum over the intersection of the fixed point sets of all the Ti’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 Ci can be handled by taking Ti to be the metric projection Pci onto Ci. We also propose a modification of our algorithm to handle the inconsistent case (i.e., when is empty as well. Keywords: 1991 Mathematics Subject Classification 47H091991 Mathematics Subject Classification 47H101991 Mathematics Subject Classification 49M451991 Mathematics Subject Classification 65K051991 Mathematics Subject Classification 65K101991 Mathematics Subject Classification 90C251991 Mathematics Subject Classification 90C30nonexpansive mappingfixed point theoremconvex optimizationsteepest descent methodquadratic functionconvex projectionbest approximationconvex feasibility problemgeneralized convex feasible set