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Local Convergence Properties of Douglas–Rachford and Alternating Direction Method of Multipliers

Author

Listed:
  • Jingwei Liang

    (Normandie Univ)

  • Jalal Fadili

    (Normandie Univ)

  • Gabriel Peyré

    (DMA, ENS Paris)

Abstract

The Douglas–Rachford and alternating direction method of multipliers are two proximal splitting algorithms designed to minimize the sum of two proper lower semi-continuous convex functions whose proximity operators are easy to compute. The goal of this work is to understand the local linear convergence behaviour of Douglas–Rachford (resp. alternating direction method of multipliers) when the involved functions (resp. their Legendre–Fenchel conjugates) are moreover partly smooth. More precisely, when the two functions (resp. their conjugates) are partly smooth relative to their respective smooth submanifolds, we show that Douglas–Rachford (resp. alternating direction method of multipliers) (i) identifies these manifolds in finite time; (ii) enters a local linear convergence regime. When both functions are locally polyhedral, we show that the optimal convergence radius is given in terms of the cosine of the Friedrichs angle between the tangent spaces of the identified submanifolds. Under polyhedrality of both functions, we also provide conditions sufficient for finite convergence. The obtained results are illustrated by several concrete examples and supported by numerical experiments.

Suggested Citation

  • Jingwei Liang & Jalal Fadili & Gabriel Peyré, 2017. "Local Convergence Properties of Douglas–Rachford and Alternating Direction Method of Multipliers," Journal of Optimization Theory and Applications, Springer, vol. 172(3), pages 874-913, March.
    • Handle: RePEc:spr:joptap:v:172:y:2017:i:3:d:10.1007_s10957-017-1061-z
    DOI: 10.1007/s10957-017-1061-z
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    Cited by:

    1. Dirk A. Lorenz & Quoc Tran-Dinh, 2019. "Non-stationary Douglas–Rachford and alternating direction method of multipliers: adaptive step-sizes and convergence," Computational Optimization and Applications, Springer, vol. 74(1), pages 67-92, September.
    2. Walaa M. Moursi & Lieven Vandenberghe, 2019. "Douglas–Rachford Splitting for the Sum of a Lipschitz Continuous and a Strongly Monotone Operator," Journal of Optimization Theory and Applications, Springer, vol. 183(1), pages 179-198, October.
    3. Cesare Molinari & Jingwei Liang & Jalal Fadili, 2019. "Convergence Rates of Forward–Douglas–Rachford Splitting Method," Journal of Optimization Theory and Applications, Springer, vol. 182(2), pages 606-639, August.

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