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Forward-Primal-Dual-Half-Forward Algorithm for Splitting Four Operators

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  • Fernando Roldán

    (Universidad de Concepción)

Abstract

In this article, we propose a splitting algorithm to find zeros of the sum of four maximally monotone operators in real Hilbert spaces. In particular, we consider a Lipschitz operator, a cocoercive operator, and a linear composite term. In the case where the Lipschitz operator is absent, our method reduces to the Condat-Vũ algorithm. On the other hand, when the linear composite term is absent, the algorithm reduces to the Forward-Backward-Half-Forward algorithm (FBHF). Additionally, in each case, the set of step-sizes that guarantee the weak convergence of those methods are recovered. Therefore, our algorithm can be seen as a combination of Condat-Vũ and FBHF. Moreover, we propose extensions and applications of our method in multivariate monotone inclusions and saddle point problems. Finally, we present a numerical experiment on image deblurring problems.

Suggested Citation

  • Fernando Roldán, 2025. "Forward-Primal-Dual-Half-Forward Algorithm for Splitting Four Operators," Journal of Optimization Theory and Applications, Springer, vol. 204(1), pages 1-26, January.
    • Handle: RePEc:spr:joptap:v:204:y:2025:i:1:d:10.1007_s10957-024-02560-w
    DOI: 10.1007/s10957-024-02560-w
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    References listed on IDEAS

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    1. 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.
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