  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.
