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Four-Operator Splitting via a Forward–Backward–Half-Forward Algorithm with Line Search

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  • Luis M. Briceño-Arias

    (Universidad Técnica Federico Santa María)

  • Fernando Roldán

    (Universidad Técnica Federico Santa María)

Abstract

In this article, we provide a splitting method for solving monotone inclusions in a real Hilbert space involving four operators: a maximally monotone, a monotone-Lipschitzian, a cocoercive, and a monotone-continuous operator. The proposed method takes advantage of the intrinsic properties of each operator, generalizing the forward–backward–half-forward splitting and the Tseng’s algorithm with line search. At each iteration, our algorithm defines the step size by using a line search in which the monotone-Lipschitzian and the cocoercive operators need only one activation. We also derive a method for solving nonlinearly constrained composite convex optimization problems in real Hilbert spaces. Finally, we implement our algorithm in a nonlinearly constrained least-square problem and we compare its performance with available methods in the literature.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:195:y:2022:i:1:d:10.1007_s10957-022-02074-3
    DOI: 10.1007/s10957-022-02074-3
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    References listed on IDEAS

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    1. 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.
    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. Boţ, Radu Ioan & Csetnek, Ernö Robert & Hendrich, Christopher, 2015. "Inertial Douglas–Rachford splitting for monotone inclusion problems," Applied Mathematics and Computation, Elsevier, vol. 256(C), pages 472-487.
    5. Rieger, Janosch & Tam, Matthew K., 2020. "Backward-Forward-Reflected-Backward Splitting for Three Operator Monotone Inclusions," Applied Mathematics and Computation, Elsevier, vol. 381(C).
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