Barrier Smoothing for Nonsmooth Convex Minimization

This paper proposes a smoothing technique for nonsmooth convex minimization using self-concordant barriers. To illustrate the main ideas, we compare our technique and the proximity smoothing approach (Nesterov2005) via the classical gradient method on both the theoretical and numerical aspects. While the barrier smoothing approach maintains the sublinear-convergence rate, it affords a new analytic step size, which significantly enhances the practical convergence of the gradient method as compared to proximity smoothing.