Convergent Subgradient Methods for Nonsmooth Convex Minimization

In this paper, we develop new subgradient methods for solving nonsmooth convex optimization problems. These methods are the first ones, for which the whole sequence of test points is endowed with the worst-case performance guarantees. The new methods are derived from a relaxed estimating sequences condition, which allows reconstruction of the approximate primal-dual optimal solutions. Our methods are applicable as efficient real-time stabilization tools for potential systems with infinite horizon. As an example, we consider a model of privacy-respecting taxation, where the center has no information on the utility functions of the agents. Nevertheless, we show that by a proper taxation policy, the agents can be forced to apply in average the socially optimal strategies. Preliminary numerical experiments confirm a high efficiency of the new methods.