Excessive Gap Technique in Nonsmooth Convex Minimization

In this paper we introduce a new primal-dual technique for convergence analysis of gradient schemes for nonsmooth convex optimization. As an example of its application, we derive a primal-dual gradient method for a special class of structured nonsmooth optimization problems, which ensures a rate of convergence of order O( 1 k ), where k is the iteration count. Another example is a gradient scheme, which minimizes a nonsmooth strongly convex function with known structure with rate of convergence O( 1 k2 ). In both cases the efficiency of the methods is higher than the corresponding black-box lower complexity bounds by an order of magnitude.