Quadratic optimization of fixed points of nonexpansive mappings in Hilbert 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 T(i) (i = 1, 2,..., N) be nonexpansive mappings on a Hilbert space H, and let Theta : H --> R be a quadratic function defined by Theta(x) := 1/2(Ax, x)-(b, x) for all x is an element of H, where A : H --> H is a strongly positive bounded self-adjoint linear operator. Then, for each sequence of scalar parameters (lambda(n)) satisfying certain conditions, we propose an algorithm that generates a sequence converging strongly to a unique minimizer u* of Theta over the intersection of the fixed point sets of all the T(i)'s. This generalizes some results of Halpern (1967), Lions (1977), Wittmann (1992), and Bauschke (1996). In particular, the minimization of Theta over the intersection (1) boolean AND(N) C(i) of closed convex sets C(i) can be handled by taking T(i) to the metric projection P(Ci) onto C(i) without introducing any special inner product that depends on A. We also propose an algorithm that generates a sequence converging to a unique minimizer of Theta over K(phi) := {u is an element of K / phi(u) = inf phi(K)} not equal 0, where K is a given closed convex set and phi(x) := Sigma(i=1)(N), w(i)d(x, C(i)) for positive weights w(i) (i = 1,...,N). This is applicable to the inconsistent case (1) boolean AND(N) C(i) = 0 as well.