Recently Gilbert, Gonzaga and Karas [2001] constructed examples of ill-behaved central paths for convex programs. In this paper we show that under mild conditions the central path has sufficient smoothness to allow construction of a path-following interior point algorithm for non-differentiable convex programs. We show that starting from a point near the center of the first set an $\epsilon$-optimal solution can be obtained in a finite number of iterations converging linearly.
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View Analysis of a Path Following Method for Nonsmooth Convex Programs