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A First-Order Primal-Dual Algorithm for Convex Problems with Applications to Imaging

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Abstract

In this paper we study a first-order primal-dual algorithm for non-smooth convex optimization problems with known saddle-point structure. We prove convergence to a saddle-point with rate O(1/N) in finite dimensions for the complete class of problems. We further show accelerations of the proposed algorithm to yield improved rates on problems with some degree of smoothness. 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(ω N) for some ω∈(0,1), on smooth problems. The wide applicability of the proposed algorithm is demonstrated on several imaging problems such as image denoising, image deconvolution, image inpainting, motion estimation and multi-label image segmentation.

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Authors and Affiliations

  1. CMAP, Ecole Polytechnique, CNRS, 91128, Palaiseau, France

    Antonin Chambolle

  2. Institute for Computer Graphics and Vision, Graz University of Technology, 8010, Graz, Austria

    Thomas Pock

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  1. Antonin Chambolle
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  2. Thomas Pock
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Correspondence to Antonin Chambolle.

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Chambolle, A., Pock, T. A First-Order Primal-Dual Algorithm for Convex Problems with Applications to Imaging. J Math Imaging Vis 40, 120–145 (2011). https://doi.org/10.1007/s10851-010-0251-1

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