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Complexity of the relaxed Peaceman–Rachford splitting method for the sum of two maximal strongly monotone operators

Author

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  • Renato D. C. Monteiro

    (Georgia Institute of Technology)

  • Chee-Khian Sim

    (University of Portsmouth)

Abstract

This paper considers the relaxed Peaceman–Rachford (PR) splitting method for finding an approximate solution of a monotone inclusion whose underlying operator consists of the sum of two maximal strongly monotone operators. Using general results obtained in the setting of a non-Euclidean hybrid proximal extragradient framework, we extend a previous convergence result on the iterates generated by the relaxed PR splitting method, as well as establish new pointwise and ergodic convergence rate results for the method whenever an associated relaxation parameter is within a certain interval. An example is also discussed to demonstrate that the iterates may not converge when the relaxation parameter is outside this interval.

Suggested Citation

  • Renato D. C. Monteiro & Chee-Khian Sim, 2018. "Complexity of the relaxed Peaceman–Rachford splitting method for the sum of two maximal strongly monotone operators," Computational Optimization and Applications, Springer, vol. 70(3), pages 763-790, July.
    • Handle: RePEc:spr:coopap:v:70:y:2018:i:3:d:10.1007_s10589-018-9996-z
    DOI: 10.1007/s10589-018-9996-z
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    References listed on IDEAS

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    1. M. V. Solodov & B. F. Svaiter, 2000. "An Inexact Hybrid Generalized Proximal Point Algorithm and Some New Results on the Theory of Bregman Functions," Mathematics of Operations Research, INFORMS, vol. 25(2), pages 214-230, May.
    2. Damek Davis & Wotao Yin, 2017. "Faster Convergence Rates of Relaxed Peaceman-Rachford and ADMM Under Regularity Assumptions," Mathematics of Operations Research, INFORMS, vol. 42(3), pages 783-805, August.
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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. Chee-Khian Sim, 2023. "Convergence Rates for the Relaxed Peaceman-Rachford Splitting Method on a Monotone Inclusion Problem," Journal of Optimization Theory and Applications, Springer, vol. 196(1), pages 298-323, January.
    3. 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.

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