An iterative approach to quadratic optimization

Assume that C-1,..., C-N are N closed convex subsets of a real Hilbert space H having a nonempty intersection C. Assume also that each C-i is the fixed point set of a nonexpansive mapping T-i of H. We devise an iterative algorithm which generates a sequence (x(n)) from an arbitrary initial x(0)is an element ofH. The sequence (x(n)) is shown to converge in norm to the unique solution of the quadratic minimization problem min(xis an element ofC)(1/2) (Ax, x) - (x,u), where A is a bounded linear strongly positive operator on H and u is a given point in H. Quadratic - quadratic minimization problems are also discussed.