Hybrid Steepest-Descent Methods for Solving Variational Inequalities Governed by Boundedly Lipschitzian and Strongly Monotone Operators

Hybrid Steepest-Descent Methods for Solving Variational Inequalities Governed by Boundedly Lipschitzian and Strongly Monotone Operators

Let H be a real Hilbert space and let F : H → H be a boundedly Lipschitzian and strongly monotone operator. We design three hybrid steepest descent algorithms for solving variational inequality VI C, F of finding a point x∗ ∈ C such that 〈Fx∗, x − x∗〉 ≥ 0, for all x ∈ C, where C is the set of fixed points of a strict pseudocontraction, or the set of common fixed points of finite strict pseudoco...

Iterative Algorithms for Variational Inequalities Governed by Boundedly Lipschitzian and Strongly Monotone Operators

Consider the variational inequality V I(C, F ) of finding a point x∗ ∈ C satisfying the property 〈Fx∗, x − x∗〉 ≥ 0 for all x ∈ C, where C is a nonempty closed convex subset of a real Hilbert space H and F : C → H is a nonlinear mapping. If F is boundedly Lipschitzian and strongly monotone, then we prove that V I(C, F ) has a unique solution and iterative algorithms can be devised to approximate...

Finite-Step Relaxed Hybrid Steepest-Descent Methods for Variational Inequalities

Mann-type Steepest-descent and Modified Hybrid Steepest-descent Methods for Variational Inequalities in Banach Spaces

1Department of Mathematics, Shanghai Normal University, Shanghai; and Scientific Computing Key Laboratory of Shanghai Universities, Shanghai, China 2Department of Mathematics & Statistics, College of Science, King Fahd University of Petroleum & Minerals, Dhahran, Saudi Arabia; and Department of Mathematics, Aligarh Muslim University, Aligarh, India 3Department of Applied Mathematics, National S...

Solving strongly monotone variational and quasi-variational inequalities

In this paper we develop a new and efficient method for variational inequality with Lipschitz continuous strongly monotone operator. Our analysis is based on a new strongly convex merit function. We apply a variant of the developed scheme for solving quasivariational inequality. As a result, we significantly improve the standard sufficient condition for existence and uniqueness of their solutio...