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. Letbe nonexpansive mappings on a Hilbert space H, and letbe a quadratic function defined byfor all, whereis 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 intersectionof closed convex sets Cican be handled by taking Tito the metric projectiononto Ciwithout 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 andfor positive weights. This is applicable to the inconsistent caseas well.