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Primal-dual splittings as fixed point iterations in the range of linear operators

Author

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  • Luis Briceño-Arias

    (Universidad Técnica Federico Santa María)

  • Fernando Roldán

    (Universidad Técnica Federico Santa María)

Abstract

In this paper we study the convergence of the relaxed primal-dual algorithm with critical preconditioners for solving composite monotone inclusions in real Hilbert spaces. We prove that this algorithm define Krasnosel’skiĭ-Mann (KM) iterations in the range of a particular monotone self-adjoint linear operator with non-trivial kernel. Our convergence result generalizes (Condat in J Optim Theory Appl 158: 460–479, 2013, Theorem 3.3) and follows from that of KM iterations defined in the range of linear operators, which is a real Hilbert subspace under suitable conditions. The Douglas–Rachford splitting (DRS) with a non-standard metric is written as a particular instance of the primal-dual algorithm with critical preconditioners and we recover classical results from this new perspective. We implement the algorithm in total variation reconstruction, verifying the advantages of using critical preconditioners and relaxation steps.

Suggested Citation

  • Luis Briceño-Arias & Fernando Roldán, 2023. "Primal-dual splittings as fixed point iterations in the range of linear operators," Journal of Global Optimization, Springer, vol. 85(4), pages 847-866, April.
    • Handle: RePEc:spr:jglopt:v:85:y:2023:i:4:d:10.1007_s10898-022-01237-w
    DOI: 10.1007/s10898-022-01237-w
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    References listed on IDEAS

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    1. Luis Briceño-Arias & Sergio López Rivera, 2019. "A Projected Primal–Dual Method for Solving Constrained Monotone Inclusions," Journal of Optimization Theory and Applications, Springer, vol. 180(3), pages 907-924, March.
    2. Luis Briceño & Roberto Cominetti & Cristián Cortés & Francisco Martínez, 2008. "An Integrated Behavioral Model of Land Use and Transport System: A Hyper-network Equilibrium Approach," Networks and Spatial Economics, Springer, vol. 8(2), pages 201-224, September.
    3. 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.
    4. Luis Briceño-Arias & Julio Deride & Cristian Vega, 2022. "Random Activations in Primal-Dual Splittings for Monotone Inclusions with a Priori Information," Journal of Optimization Theory and Applications, Springer, vol. 192(1), pages 56-81, January.
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