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A Simplified Form of Block-Iterative Operator Splitting and an Asynchronous Algorithm Resembling the Multi-Block Alternating Direction Method of Multipliers

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  • Jonathan Eckstein

    (Rutgers University)

Abstract

This paper develops what is essentially a simplified version of the block-iterative operator splitting method already proposed by the author and P. Combettes, but with more general initialization conditions. It then describes one way of implementing this algorithm asynchronously under a computational model inspired by modern high-performance computing environments, which consist of interconnected nodes each having multiple processor cores sharing a common local memory. The asynchronous implementation framework is then applied to derive an asynchronous algorithm which resembles the alternating direction method of multipliers with an arbitrary number of blocks of variables. Unlike earlier proposals for asynchronous variants of the alternating direction method of multipliers, the algorithm relies neither on probabilistic control nor on restrictive assumptions about the problem instance, instead making only standard convex-analytic regularity assumptions. It also allows the proximal parameters to range freely between arbitrary positive bounds, possibly varying with both iterations and subproblems.

Suggested Citation

  • Jonathan Eckstein, 2017. "A Simplified Form of Block-Iterative Operator Splitting and an Asynchronous Algorithm Resembling the Multi-Block Alternating Direction Method of Multipliers," Journal of Optimization Theory and Applications, Springer, vol. 173(1), pages 155-182, April.
    • Handle: RePEc:spr:joptap:v:173:y:2017:i:1:d:10.1007_s10957-017-1074-7
    DOI: 10.1007/s10957-017-1074-7
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    References listed on IDEAS

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    1. R. T. Rockafellar, 1976. "Augmented Lagrangians and Applications of the Proximal Point Algorithm in Convex Programming," Mathematics of Operations Research, INFORMS, vol. 1(2), pages 97-116, May.
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    Cited by:

    1. Mauricio Romero Sicre, 2020. "On the complexity of a hybrid proximal extragradient projective method for solving monotone inclusion problems," Computational Optimization and Applications, Springer, vol. 76(3), pages 991-1019, July.
    2. Yunda Dong, 2021. "Weak convergence of an extended splitting method for monotone inclusions," Journal of Global Optimization, Springer, vol. 79(1), pages 257-277, January.
    3. Weiyang Ding & Michael K. Ng & Wenxing Zhang, 2024. "A generalized alternating direction implicit method for consensus optimization: application to distributed sparse logistic regression," Journal of Global Optimization, Springer, vol. 90(3), pages 727-753, November.
    4. E. M. Bednarczuk & A. Jezierska & K. E. Rutkowski, 2018. "Proximal primal–dual best approximation algorithm with memory," Computational Optimization and Applications, Springer, vol. 71(3), pages 767-794, December.
    5. Majela Pentón Machado & Mauricio Romero Sicre, 2023. "A Projective Splitting Method for Monotone Inclusions: Iteration-Complexity and Application to Composite Optimization," Journal of Optimization Theory and Applications, Springer, vol. 198(2), pages 552-587, August.
    6. Luis M. Briceño-Arias & Giovanni Chierchia & Emilie Chouzenoux & Jean-Christophe Pesquet, 2019. "A random block-coordinate Douglas–Rachford splitting method with low computational complexity for binary logistic regression," Computational Optimization and Applications, Springer, vol. 72(3), pages 707-726, April.
    7. Gilles Bareilles & Yassine Laguel & Dmitry Grishchenko & Franck Iutzeler & Jérôme Malick, 2020. "Randomized Progressive Hedging methods for multi-stage stochastic programming," Annals of Operations Research, Springer, vol. 295(2), pages 535-560, December.
    8. Howard Heaton & Yair Censor, 2019. "Asynchronous sequential inertial iterations for common fixed points problems with an application to linear systems," Journal of Global Optimization, Springer, vol. 74(1), pages 95-119, May.
    9. Dong, Yunda, 2024. "Extended splitting methods for systems of three-operator monotone inclusions with continuous operators," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 223(C), pages 86-107.
    10. Luis M. Briceño-Arias & Fernando Roldán, 2022. "Four-Operator Splitting via a Forward–Backward–Half-Forward Algorithm with Line Search," Journal of Optimization Theory and Applications, Springer, vol. 195(1), pages 205-225, October.
    11. Patrick R. Johnstone & Jonathan Eckstein, 2021. "Single-forward-step projective splitting: exploiting cocoercivity," Computational Optimization and Applications, Springer, vol. 78(1), pages 125-166, January.

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