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Backward-Forward-Reflected-Backward Splitting for Three Operator Monotone Inclusions

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  • Rieger, Janosch
  • Tam, Matthew K.

Abstract

In this work, we propose and analyse two splitting algorithms for finding a zero of the sum of three monotone operators, one of which is assumed to be Lipschitz continuous. Each iteration of these algorithms require one forward evaluation of the Lipschitz continuous operator and one resolvent evaluation of each of the other two operators. By specialising to two operator inclusions, we recover the forward-reflected-backward and the reflected-forward-backward splitting methods as particular cases. The inspiration for the proposed algorithms arises from interpretations of the aforementioned reflected splitting algorithms as discretisations of the continuous-time proximal point algorithm.

Suggested Citation

  • Rieger, Janosch & Tam, Matthew K., 2020. "Backward-Forward-Reflected-Backward Splitting for Three Operator Monotone Inclusions," Applied Mathematics and Computation, Elsevier, vol. 381(C).
    • Handle: RePEc:eee:apmaco:v:381:y:2020:i:c:s0096300320302174
    DOI: 10.1016/j.amc.2020.125248
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    References listed on IDEAS

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    1. Taylor, A. & Hendrickx, J. & Glineur, F., 2015. "Smooth Strongly Convex Interpolation and Exact Worst-case Performance of First-order Methods," LIDAM Discussion Papers CORE 2015013, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. Ernest K. Ryu & Bằng Công Vũ, 2020. "Finding the Forward-Douglas–Rachford-Forward Method," Journal of Optimization Theory and Applications, Springer, vol. 184(3), pages 858-876, March.
    3. Francisco J. Aragón Artacho & Rubén Campoy, 2019. "Computing the Resolvent of the Sum of Maximally Monotone Operators with the Averaged Alternating Modified Reflections Algorithm," Journal of Optimization Theory and Applications, Springer, vol. 181(3), pages 709-726, June.
    4. B. Abbas & H. Attouch & Benar F. Svaiter, 2014. "Newton-Like Dynamics and Forward-Backward Methods for Structured Monotone Inclusions in Hilbert Spaces," Journal of Optimization Theory and Applications, Springer, vol. 161(2), pages 331-360, May.
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    Cited by:

    1. Yonghong Yao & Abubakar Adamu & Yekini Shehu, 2024. "Forward–Reflected–Backward Splitting Algorithms with Momentum: Weak, Linear and Strong Convergence Results," Journal of Optimization Theory and Applications, Springer, vol. 201(3), pages 1364-1397, June.
    2. Francisco J. Aragón Artacho & Rubén Campoy & Matthew K. Tam, 2021. "Strengthened splitting methods for computing resolvents," Computational Optimization and Applications, Springer, vol. 80(2), pages 549-585, November.
    3. Rubén Campoy, 2022. "A product space reformulation with reduced dimension for splitting algorithms," Computational Optimization and Applications, Springer, vol. 83(1), pages 319-348, September.
    4. 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.

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