The Proximal Method of Multipliers for a Class of Nonsmooth Convex Optimization

Abstract

This paper develops the proximal method of multipliers for a class of nonsmooth convex optimization. The method generates a sequence of minimization problems (subproblems). We show that the sequence of approximations to the solutions of the subproblems converges to a saddle point of the Lagrangian even if the original optimization problem may possess multiple solutions. The augmented Lagrangian due to Fortin appears in the subproblem. The remarkable property of the augmented Lagrangian over the standard Lagrangian is that it is always differentiable, and it is often semismoothly differentiable. This fact allows us to employ a nonsmooth Newton method for computing an approximation to the subproblem. The proximal term serves as the regularization of the objective function and guarantees the solvability of the Newton system without assuming strong convexity on the objective function. We exploit the theory of the nonsmooth Newton method to provide a rigorous proof for the global convergence of the proposed algorithm.