Schemes for finding minimum-norm solutions of variational inequalities

Consider the variational inequality (VI) of finding a point x* such that x* is an element of Fix(T) and <(I - S)x*, x - x*> >= 0, x is an element of Fix(T) (*) where T, S are nonexpansive self-mappings of a closed convex subset C of a Hilbert space, and Fix(T) is the set of fixed points of T. Assume that the solution set Omega of this VI is nonempty. This paper introduces two schemes, one implicit and one explicit, that can be used to find the minimum-norm solution of VI (*); namely, the unique solution x* to the quadratic minimization problem: x* = arg min(x is an element of Omega) parallel to x parallel to(2). (C) 2009 Elsevier Ltd. All rights reserved.