Convergence of hybrid steepest-descent methods for variational inequalities

Assume that F is a nonlinear operator on a real Hilbert space H which is η-strongly monotone and κ-Lipschitzian on a nonempty closed convex subset C of H. Assume also that C is the intersection of the fixed point sets of a finite number of nonexpansive mappings on H. We devise an iterative algorithm which generates a sequence (xn) from an arbitrary initial point x0 ∈ H. The sequence (xn) is shown to converge in)norm to the unique solution u* of the variational inequality 〈F(u*), υ u*〉 ≥0, for υ ∈ C. Applications to constrained pseudoinverse are included.