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

Let T(i) (i = 1,2,...,N) be nonexpansive mappings on a Hilbert space H, and let Theta : H --> RU{infinity} be a function which has a uniformly strongly positive and uniformly bounded second (Frechet) derivative over the convex huh of T(i)(H) for some i. We first prove that Theta 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 Theta over the intersection boolean AND(1)(N)C(i) of closed convex sets C(i) can be handled by taking T(i) to be the metric projection P(Ci) onto C(i). We also propose a modification of our algorithm to handle the inconsistent case (i.e., when boolean AND(1)(N)C(i) is empty) as well.