THE HYBRID STEEPEST DESCENT METHOD FOR THE VARIATIONAL INEQUALITY PROBLEM OVER THE INTERSECTION OF FIXED POINT SETS OF NONEXPANSIVE MAPPINGS

This paper presents a simple algorithmic solution to the variational inequality prob-lem defined over the nonempty intersection of multiple fixed point sets of nonexpansive mappings in a real Hilbert space. The algorithmic solution is named the hybrid steepest descent method, because it is constructed by blending important ideas in the steepest de-scent method and in the fixed point theory, and generates a sequence converging strongly to the solution of the problem. The remarkable applicability of this method to the convexly constrained generalized pseudoinverse problem as well as to the convex feasibility problem is demonstrated by constructing nonexpansive mappings whose fixed point sets are the feasible sets of the problems