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 ⊖: H → R∪{∞} be a function which has a uniformly strongly positive and uniformly bounded second (Frechet) derivative over the convex hull of T i (H) for some i. We first prove that ⊖ 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 ⊖ over the intersection ∩ l 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 ∩ l N C i is empty) as well.