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An Adaptive Alternating Direction Method of Multipliers

Author

Listed:
  • Sedi Bartz

    (University of Massachusetts Lowell)

  • Rubén Campoy

    (Universitat de València)

  • Hung M. Phan

    (University of Massachusetts Lowell)

Abstract

The alternating direction method of multipliers (ADMM) is a powerful splitting algorithm for linearly constrained convex optimization problems. In view of its popularity and applicability, a growing attention is drawn toward the ADMM in nonconvex settings. Recent studies of minimization problems for nonconvex functions include various combinations of assumptions on the objective function including, in particular, a Lipschitz gradient assumption. We consider the case where the objective is the sum of a strongly convex function and a weakly convex function. To this end, we present and study an adaptive version of the ADMM which incorporates generalized notions of convexity and penalty parameters adapted to the convexity constants of the functions. We prove convergence of the scheme under natural assumptions. To this end, we employ the recent adaptive Douglas–Rachford algorithm by revisiting the well-known duality relation between the classical ADMM and the Douglas–Rachford splitting algorithm, generalizing this connection to our setting. We illustrate our approach by relating and comparing to alternatives, and by numerical experiments on a signal denoising problem.

Suggested Citation

  • Sedi Bartz & Rubén Campoy & Hung M. Phan, 2022. "An Adaptive Alternating Direction Method of Multipliers," Journal of Optimization Theory and Applications, Springer, vol. 195(3), pages 1019-1055, December.
    • Handle: RePEc:spr:joptap:v:195:y:2022:i:3:d:10.1007_s10957-022-02098-9
    DOI: 10.1007/s10957-022-02098-9
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    References listed on IDEAS

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    1. Liang Chen & Defeng Sun & Kim-Chuan Toh, 2017. "A note on the convergence of ADMM for linearly constrained convex optimization problems," Computational Optimization and Applications, Springer, vol. 66(2), pages 327-343, March.
    2. Patrick L. Combettes & Jean-Christophe Pesquet, 2011. "Proximal Splitting Methods in Signal Processing," Springer Optimization and Its Applications, in: Heinz H. Bauschke & Regina S. Burachik & Patrick L. Combettes & Veit Elser & D. Russell Luke & Henry (ed.), Fixed-Point Algorithms for Inverse Problems in Science and Engineering, chapter 0, pages 185-212, Springer.
    3. Ernest K. Ryu & Yanli Liu & Wotao Yin, 2019. "Douglas–Rachford splitting and ADMM for pathological convex optimization," Computational Optimization and Applications, Springer, vol. 74(3), pages 747-778, December.
    4. Sedi Bartz & Rubén Campoy & Hung M. Phan, 2020. "Demiclosedness Principles for Generalized Nonexpansive Mappings," Journal of Optimization Theory and Applications, Springer, vol. 186(3), pages 759-778, September.
    5. Sedi Bartz & Minh N. Dao & Hung M. Phan, 2022. "Conical averagedness and convergence analysis of fixed point algorithms," Journal of Global Optimization, Springer, vol. 82(2), pages 351-373, February.
    Full references (including those not matched with items on IDEAS)

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