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

Abstract

We develop the theoretical foundation on the proximal method of multipliers for a class of nonsmooth convex optimization. The method generates a sequence of subproblems for the objective functions of the sum of the augmented Lagrangian due to Fortin and the proximal term. We show that the sequence of approximations to the subproblems converges to a saddle point of the standard Lagrangian even if the original optimization problem may possess multiple solutions. We employ a nonsmooth Newton method for computing an approximation to the subproblem. We exploit the theory of the nonsmooth analysis to provide a rigorous proof for the global convergence of the nonsmooth Newton algorithm.