Prox-Method with Rate of Convergence <i>O</i>(1/<i>t</i>) for Variational Inequalities with Lipschitz Continuous Monotone Operators and Smooth Convex-Concave Saddle Point Problems

Abstract

We propose a prox-type method with efficiency estimate $O(\epsilon^{-1})$ for approximating saddle points of convex-concave C$^{1,1}$ functions and solutions of variational inequalities with monotone Lipschitz continuous operators. Application examples include matrix games, eigenvalue minimization, and computing the Lovasz capacity number of a graph, and these are illustrated by numerical experiments with large-scale matrix games and Lovasz capacity problems.