The Prediction-Correction and Relaxed Hybrid Steepest-Descent Method for Variational Inequalities

This paper proposes a new prediction-correction and relaxed hybrid steepest-descent method for a class of the variational inequality problem with a Lipschitzi and strongly monotone operator on a nonempty closed convex subset in real Hilbert space. In order to improve the computational efficiency, the proposed method integrates the Gauss-Seidel method into the prediction-correction method while taking advantage of some fundamental properties of the real Hilbert space. Finally, strong convergence of this method is proved under suitable assumptions.