In this paper we study a first-order primal-dual algorithm for convex optimization problems with known saddle-point structure. We prove convergence to a saddle-point with rate O(1/N ) in finite dimensions, which is optimal for the complete class of non-smooth problems we are considering in this paper. We further show accelerations of the proposed algorithm to yield optimal rates on easier problems. In particular we show that we can achieve O(1/N^2) convergence on problems, where the primal or the dual objective is uniformly convex, and we can show linear convergence, i.e. O(1/e^N ) on problems where both are uniformly convex. The wide ap- plicability of the proposed algorithm is demonstrated on several imaging problems such as image denoising, image deconvolution, image inpainting, motion estimation and image segmentation.
R.I. 685, CMAP, Ecole Polytechnique, 91128 Palaiseau Cedex, France, May 2010