An Iterative Approach to Quadratic Optimization

Assume that C₁, . . . , 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 x₀H. The sequence (x_n) is shown to converge in norm to the unique solution of the quadratic minimization problem min_xC(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.