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Krasnoselskii–Mann Iterations: Inertia, Perturbations and Approximation

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  • Daniel Cortild

    (University of Groningen)

  • Juan Peypouquet

    (University of Groningen)

Abstract

This paper is concerned with the study of a family of fixed point iterations combining relaxation with different inertial (acceleration) principles. We provide a systematic, unified and insightful analysis of the hypotheses that ensure their weak, strong and linear convergence, either matching or improving previous results obtained by analysing particular cases separately. We also show that these methods are robust with respect to different kinds of perturbations–which may come from computational errors, intentional deviations, as well as regularisation or approximation schemes–under surprisingly weak assumptions. Although we mostly focus on theoretical aspects, numerical illustrations in image inpainting and electricity production markets reveal possible trends in the behaviour of these types of methods.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:204:y:2025:i:2:d:10.1007_s10957-024-02600-5
    DOI: 10.1007/s10957-024-02600-5
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    References listed on IDEAS

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    1. Juan Peypouquet, 2012. "Coupling the Gradient Method with a General Exterior Penalization Scheme for Convex Minimization," Journal of Optimization Theory and Applications, Springer, vol. 153(1), pages 123-138, April.
    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. Nahla Noun & Juan Peypouquet, 2013. "Forward–Backward Penalty Scheme for Constrained Convex Minimization Without Inf-Compactness," Journal of Optimization Theory and Applications, Springer, vol. 158(3), pages 787-795, September.
    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.
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    Cited by:

    1. 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.

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