Convergence Analysis of Modified Hybrid Steepest-Descent Methods with Variable Parameters 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 construct an iterative algorithm with variable parameters 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 $$\langle F(u^{\ast}), v - u^{\ast}\rangle \geq 0, \quad \forall v \in C.$$