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.