Mathematical Programming

, Volume 159, Issue 1–2, pp 253–287 | Cite as

On the ergodic convergence rates of a first-order primal–dual algorithm

  • Antonin Chambolle
  • Thomas Pock
Full Length Paper Series A


We revisit the proofs of convergence for a first order primal–dual algorithm for convex optimization which we have studied a few years ago. In particular, we prove rates of convergence for a more general version, with simpler proofs and more complete results. The new results can deal with explicit terms and nonlinear proximity operators in spaces with quite general norms.


Saddle-point problems First order algorithms Primal–dual algorithms Convergence rates Ergodic convergence 

Mathematics Subject Classification

49M29 65K10 65Y20 90C25 



This research is partially supported by the joint ANR/FWF Project Efficient Algorithms for Nonsmooth Optimization in Imaging (EANOI) FWF n. I1148/ANR-12-IS01-0003. A.C. would like to thank his colleague S. Gaiffas for stimulating discussions, as well as J. Fadili for very helpful discussions on nonlinear proximity operators. This work also benefited from the support of the “Gaspard Monge Program in Optimization and Operations Research” (PGMO), supported by EDF and the Fondation Mathématique Jacques Hadamard (FMJH). The authors also need to thank the referees for their careful reading of the manuscript and their numerous helpful comments.


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Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2015

Authors and Affiliations

  1. 1.CMAP, Ecole PolytechniqueCNRSPalaiseauFrance
  2. 2.Institute for Computer Graphics and VisionGraz University of TechnologyGrazAustria
  3. 3.Digital Safety and Security DepartmentAIT Austrian Institute of Technology GmbHViennaAustria

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