A Primal-Dual Proximal Point Algorithm for Variational Inequality Problems

Abstract

The proximal point algorithm is an iterative method for finding a zero of a maximal monotone set-valued mapping, and has served as a fundamental method for solving Variational Inequality problems and, in particular, convex optimization problems. In this paper, we propose a new variant of the primal-dual proximal point algorithm for solving monotone Variational Inequality problems. The proposed method adopts a Gauss-Seidel-like procedure to solve a subproblem in each iteration, which updates primal variables and dual variables alternatively. We prove the convergence of the proposed method and show that our method is particularly effective for Variational Inequality problems with certain separable structure, such as traffic assignment problems.