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Extended splitting methods for systems of three-operator monotone inclusions with continuous operators

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  • Dong, Yunda

Abstract

In this article, we propose two new splitting methods for solving systems of three-operator monotone inclusions in real Hilbert spaces, where the first operator is continuous monotone, the second is maximal monotone and the third is maximal monotone and is linearly composed. These methods primarily involve evaluating the first operator and computing resolvents with respect to the other two operators. For one method corresponding to Lipschitz continuous operator, we give back-tracking techniques to determine step lengths. Moreover, we propose a dual-first version of this method. For the other method, which corresponds to a uniformly continuous operator, we develop innovative back-tracking techniques, incorporating additional conditions to determine step lengths. The weak convergence of either method is proven using characteristic operator techniques. Notably, either method fully decouples the third operator from its linear composition operator. Numerical results demonstrate the effectiveness of our proposed splitting methods, together with their special cases and variants, in solving test problems.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:matcom:v:223:y:2024:i:c:p:86-107
    DOI: 10.1016/j.matcom.2024.03.024
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    References listed on IDEAS

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    1. Dong, Yunda, 2023. "A new splitting method for systems of monotone inclusions in Hilbert spaces," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 203(C), pages 518-537.
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    4. Laurent Condat, 2013. "A Primal–Dual Splitting Method for Convex Optimization Involving Lipschitzian, Proximable and Linear Composite Terms," Journal of Optimization Theory and Applications, Springer, vol. 158(2), pages 460-479, August.
    5. 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.
    6. 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.
    7. Puya Latafat & Panagiotis Patrinos, 2017. "Asymmetric forward–backward–adjoint splitting for solving monotone inclusions involving three operators," Computational Optimization and Applications, Springer, vol. 68(1), pages 57-93, September.
    8. Jonathan Eckstein & Michael C. Ferris, 1998. "Operator-Splitting Methods for Monotone Affine Variational Inequalities, with a Parallel Application to Optimal Control," INFORMS Journal on Computing, INFORMS, vol. 10(2), pages 218-235, May.
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