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

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Abstract

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.

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Authors and Affiliations

  1. Laboratoire Jacques-Louis Lions, UMR CNRS 7598, UPMC Université Paris 06, 75005, Paris, France

    Patrick L. Combettes

  2. Laboratoire d’Informatique Gaspard Monge, UMR CNRS 8049, Université Paris-Est, 77454, Marne la Vallée Cedex 2, France

    Jean-Christophe Pesquet

Authors
  1. Patrick L. Combettes
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  2. Jean-Christophe Pesquet
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Correspondence to Patrick L. Combettes.

Additional information

This work was supported by the Agence Nationale de la Recherche under grants ANR-08-BLAN-0294-02 and ANR-09-EMER-004-03.

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Combettes, P.L., Pesquet, JC. Primal-Dual Splitting Algorithm for Solving Inclusions with Mixtures of Composite, Lipschitzian, and Parallel-Sum Type Monotone Operators. Set-Valued Anal 20, 307–330 (2012). https://doi.org/10.1007/s11228-011-0191-y

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