A GENERAL-APPROACH TO CONVERGENCE PROPERTIES OF SOME METHODS FOR NONSMOOTH CONVEX-OPTIMIZATION

Based on the notion of the E-subgradient, we present a unified tech ni que to establish convergence properties of several methods for nonsmoo th convex minimization problems. Starting from the technical results, we obtain the global convergence of: (i) the variable metric proximal methods presented by Bonnans, Gilbert, Lemarechal, and Sagastizabal, ( ii) some algorithms proposed by Correa and Lemarechal, and (iii) the p roximal point algorithm given by Rockafellar. In particular, we prove that the Rockafellar-Todd phenomenon does not occur for each of the ab ove mentioned methods. Moreover, we explore the convergence rate of {p arallel to x(k)parallel to}and {f(xk)} when {x(k)} is unbounded and {f (x(k))} is bounded for the nonsmooth minimization methods (i), (ii), a nd (iii).