Set-Valued and Variational Analysis

, Volume 20, Issue 2, pp 307–330 | Cite as

Primal-Dual Splitting Algorithm for Solving Inclusions with Mixtures of Composite, Lipschitzian, and Parallel-Sum Type Monotone Operators

  • Patrick L. Combettes
  • Jean-Christophe Pesquet


We propose a primal-dual splitting algorithm for solving monotone inclusions involving a mixture of sums, linear compositions, and parallel sums of set-valued and Lipschitzian operators. An important feature of the algorithm is that the Lipschitzian operators present in the formulation can be processed individually via explicit steps, while the set-valued operators are processed individually via their resolvents. In addition, the algorithm is highly parallel in that most of its steps can be executed simultaneously. This work brings together and notably extends various types of structured monotone inclusion problems and their solution methods. The application to convex minimization problems is given special attention.


Maximal monotone operator Monotone inclusion Nonsmooth convex optimization Parallel sum Set-valued duality Splitting algorithm 

Mathematics Subject Classifications (2010)

47H05 49M29 49M27 90C25 49N15 


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Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  • Patrick L. Combettes
    • 1
  • Jean-Christophe Pesquet
    • 2
  1. 1.Laboratoire Jacques-Louis Lions, UMR CNRS 7598UPMC Université Paris 06ParisFrance
  2. 2.Laboratoire d’Informatique Gaspard Monge, UMR CNRS 8049Université Paris-EstMarne la Vallée Cedex 2France

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