Comparison of Convergence of the Modified and Relaxed Hybrid Steepest-Descent Methods for Variational Inequalities under Different Conditions

In order to reduce the difficulty and complexity on computing the projection from a real Hilbert space onto a nonempty closed convex subset, Yamada has provided the hybrid steepest-descent method for solving variational inequalities. Recently Xu has provided the modified and relaxed hybrid steepest-descent method for variational inequalities based on the minds of the Gauss-seidel method, and given out the convergence theorem under some suitable conditions(Condition 3.1). In this paper, we give out other different conditions(Condition 3.2) about the modified and relaxed hybrid steepest-descent method for variational inequalities, such the conditions can simplify proof and it is to be noted that the proof of strong convergence is different from the previous results. Furthermore we design a set of practical numerical experiments and numerical results demonstrated that the modified and relaxed hybrid steepest-descent method under the Condition 3.2 is more efficient than under the Condition 3.1.