In this paper we develop a new primal-dual subgradient method for nonsmooth convex optimization problems. This scheme is based on a self-concordant barrier for the basic feasible set. It is suitable for finding approximate solutions with certain relative accuracy. We discuss some applications of...

In many applications it is possible to justify a reasonable bound for possible variation of subgradients of objective function rather than for their uniform magnitude. In this paper we develop a new class of efficient primal-dual subgradient schemes for such problem classes.