A general approach to convergence properties of some methods for nonsmooth convex optimization

Based on the notion of the ε-subgradient, we present a unified technique to establish convergence properties of several methods for nonsmooth convex minimization problems. Starting from the technical results, we obtain the global convergence of: (i) the variable metric proximal methods presented by Bonnans, Gilbert, Lemaréchal, and Sagastizábal, (ii) some algorithms proposed by Correa and Lemaréchal, and (iii) the proximal point algorithm given by Rockafellar. In particular, we prove that the Rockafellar-Todd phenomenon does not occur for each of the above mentioned methods. Moreover, we explore the convergence rate of {∥Xk∥} and {f(xk)} when {xk} is unbounded and {f(xk)} is bounded for the nonsmooth minimization methods (i), (ii), and (iii).