Quadratic optimization of fixed points of nonexpansive mappings in hubert space

Finding an optimal point in the intersection of the fixed point sets of a family of nonexpansive mappings is a frequent problem in various areas of mathematical science and engineering. Let be nonexpansive mappings on a Hilbert space H, and let be a quadratic function defined by for all , where is a strongly positive bounded self-adjoint linear operator. Then, for each sequence of scalar parameters (λn) satisfying certain conditions, we propose an algorithm that generates a sequence converting strongly to a unique minimizer u* of Θ over the intersection of the fixed point sets of all the Ti's. This generalizes some results of Halpern (1967), Lions (1977), Wittmann (1992), and Bauschke (1996). In particular, the minimization of Θ over the intersection of closed convex sets Ci can be handled by taking Ti to the metric projection onto Ci without introducing any special inner products that depends on A. We also propose an algorithm that generates a sequence converging to a unique minimizer of Θ over , where K is a given closed convex set and for positive weights . This is applicable to the inconsistent case as well.