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Relaxed and Inertial Nonlinear Forward–Backward with Momentum

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  • Fernando Roldán

    (Universidad de Concepción)

  • Cristian Vega

    (Universidad de Tarapacá)

Abstract

In this article, we study inertial algorithms for numerically solving monotone inclusions involving the sum of a maximally monotone and a cocoercive operator. In particular, we analyze the convergence of inertial and relaxed versions of the nonlinear forward–backward with momentum (NFBM). We propose an inertial version of NFBM including a relaxation step and a second version considering a double-inertial step with additional momentum. By applying NFBM to specific monotone inclusions, we derive inertial and relaxed versions of algorithms such as forward–backward, forward-half-reflected-backward (FHRB), Chambolle–Pock, Condat–Vũ, among others, thereby recovering and extending previous results from the literature for solving monotone inclusions involving maximally monotone, cocoercive, monotone and Lipschitz, and linear bounded operators. We also present numerical experiments on image restoration, comparing the proposed inertial and relaxed algorithms. In particular, we compare the inertial and relaxed FHRB with its non-inertial and momentum versions. Additionally, we compare the numerical convergence for larger step-sizes versus relaxation parameters and introduce a restart strategy that incorporates larger step-sizes and inertial steps to further enhance numerical convergence.

Suggested Citation

  • Fernando Roldán & Cristian Vega, 2025. "Relaxed and Inertial Nonlinear Forward–Backward with Momentum," Journal of Optimization Theory and Applications, Springer, vol. 206(2), pages 1-30, August.
    • Handle: RePEc:spr:joptap:v:206:y:2025:i:2:d:10.1007_s10957-025-02694-5
    DOI: 10.1007/s10957-025-02694-5
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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. M. Marques Alves & Jonathan Eckstein & Marina Geremia & Jefferson G. Melo, 2020. "Relative-error inertial-relaxed inexact versions of Douglas-Rachford and ADMM splitting algorithms," Computational Optimization and Applications, Springer, vol. 75(2), pages 389-422, March.
    3. 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.
    4. Daniel Cortild & Juan Peypouquet, 2025. "Krasnoselskii–Mann Iterations: Inertia, Perturbations and Approximation," Journal of Optimization Theory and Applications, Springer, vol. 204(2), pages 1-30, February.
    5. 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.
    6. 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.
    7. 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.
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