Convergence of hybrid steepest-descent methods for variational inequalities

Assume that F is a nonlinear operator on a real Hilbert space H which is eta-strongly monotone and kappa-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 (x(n)) from an arbitrary initial point x(0)is an element ofH. The sequence (x(n)) is shown to converge in norm to the unique solution u* of the variational inequality [F(u*), v - u*] greater than or equal to 0, for v is an element of C. Applications to constrained pseudoinverse are included.