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A Projective Splitting Method for Monotone Inclusions: Iteration-Complexity and Application to Composite Optimization

Author

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  • Majela Pentón Machado

    (Universidade Federal da Bahia)

  • Mauricio Romero Sicre

    (Universidade Federal da Bahia)

Abstract

We propose an inexact projective splitting method to solve the problem of finding a zero of a sum of maximal monotone operators. We perform convergence and complexity analyses of the method by viewing it as a special instance of an inexact proximal point method proposed by Solodov and Svaiter in 2001, for which pointwise and ergodic complexity results have been studied recently by Sicre. Also, for this latter method, we establish convergence rates and complexity bounds for strongly monotone inclusions, from where we obtain linear convergence for our projective splitting method under strong monotonicity and cocoercivity assumptions. We apply the proposed projective splitting scheme to composite convex optimization problems and establish pointwise and ergodic function value convergence rates, extending a recent work of Johnstone and Eckstein.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:198:y:2023:i:2:d:10.1007_s10957-023-02214-3
    DOI: 10.1007/s10957-023-02214-3
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    References listed on IDEAS

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    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. 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.
    3. 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.
    4. 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.
    5. Majela Pentón Machado, 2019. "Projective method of multipliers for linearly constrained convex minimization," Computational Optimization and Applications, Springer, vol. 73(1), pages 237-273, May.
    6. Majela Pentón Machado, 2018. "On the Complexity of the Projective Splitting and Spingarn’s Methods for the Sum of Two Maximal Monotone Operators," Journal of Optimization Theory and Applications, Springer, vol. 178(1), pages 153-190, July.
    7. L. C. Ceng & B. S. Mordukhovich & J. C. Yao, 2010. "Hybrid Approximate Proximal Method with Auxiliary Variational Inequality for Vector Optimization," Journal of Optimization Theory and Applications, Springer, vol. 146(2), pages 267-303, August.
    8. M. Marques Alves & Jonathan Eckstein & Marina Geremia & Jefferson G. Melo, 2020. "Relative-error inertial-relaxed inexact versions of Douglas-Rachford and ADMM splitting algorithms," Computational Optimization and Applications, Springer, vol. 75(2), pages 389-422, March.
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