Convergence of Hybrid Steepest-Descent Methods for Variational Inequalities

AbstractAssume 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 $$\left\langle {F(u*),\user1{v} - u*} \right\rangle \geqslant 0$$ Applications to constrained pseudoinverse are included.