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A Forward–Backward Algorithm With Different Inertial Terms for Structured Non-Convex Minimization Problems

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  • Szilárd Csaba László

    (Technical University of Cluj-Napoca)

Abstract

We investigate an inertial forward–backward algorithm in connection with the minimization of the sum of a non-smooth and possibly non-convex and a non-convex differentiable function. The algorithm is formulated in the spirit of the famous FISTA method; however, the setting is non-convex and we allow different inertial terms. Moreover, the inertial parameters in our algorithm can take negative values too. We also treat the case when the non-smooth function is convex, and we show that in this case a better step size can be allowed. Further, we show that our numerical schemes can successfully be used in DC-programming. We prove some abstract convergence results which applied to our numerical schemes allow us to show that the generated sequences converge to a critical point of the objective function, provided a regularization of the objective function satisfies the Kurdyka–Łojasiewicz property. Further, we obtain a general result that applied to our numerical schemes ensures convergence rates for the generated sequences and for the objective function values formulated in terms of the KL exponent of a regularization of the objective function. Finally, we apply our results to image restoration.

Suggested Citation

  • Szilárd Csaba László, 2023. "A Forward–Backward Algorithm With Different Inertial Terms for Structured Non-Convex Minimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 198(1), pages 387-427, July.
    • Handle: RePEc:spr:joptap:v:198:y:2023:i:1:d:10.1007_s10957-023-02204-5
    DOI: 10.1007/s10957-023-02204-5
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    References listed on IDEAS

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    1. Hédy Attouch & Jérôme Bolte & Patrick Redont & Antoine Soubeyran, 2010. "Proximal Alternating Minimization and Projection Methods for Nonconvex Problems: An Approach Based on the Kurdyka-Łojasiewicz Inequality," Mathematics of Operations Research, INFORMS, vol. 35(2), pages 438-457, May.
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    3. Pierre Frankel & Guillaume Garrigos & Juan Peypouquet, 2015. "Splitting Methods with Variable Metric for Kurdyka–Łojasiewicz Functions and General Convergence Rates," Journal of Optimization Theory and Applications, Springer, vol. 165(3), pages 874-900, June.
    4. Zhongming Wu & Min Li, 2019. "General inertial proximal gradient method for a class of nonconvex nonsmooth optimization problems," Computational Optimization and Applications, Springer, vol. 73(1), pages 129-158, May.
    5. Emilie Chouzenoux & Jean-Christophe Pesquet & Audrey Repetti, 2014. "Variable Metric Forward–Backward Algorithm for Minimizing the Sum of a Differentiable Function and a Convex Function," Journal of Optimization Theory and Applications, Springer, vol. 162(1), pages 107-132, July.
    6. Peter Ochs, 2018. "Local Convergence of the Heavy-Ball Method and iPiano for Non-convex Optimization," Journal of Optimization Theory and Applications, Springer, vol. 177(1), pages 153-180, April.
    7. A. Chambolle & Ch. Dossal, 2015. "On the Convergence of the Iterates of the “Fast Iterative Shrinkage/Thresholding Algorithm”," Journal of Optimization Theory and Applications, Springer, vol. 166(3), pages 968-982, September.
    8. 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.
    9. Zhongming Wu & Chongshou Li & Min Li & Andrew Lim, 2021. "Inertial proximal gradient methods with Bregman regularization for a class of nonconvex optimization problems," Journal of Global Optimization, Springer, vol. 79(3), pages 617-644, March.
    10. Yaohua Hu & Chong Li & Kaiwen Meng & Xiaoqi Yang, 2021. "Linear convergence of inexact descent method and inexact proximal gradient algorithms for lower-order regularization problems," Journal of Global Optimization, Springer, vol. 79(4), pages 853-883, April.
    11. Patrick R. Johnstone & Pierre Moulin, 2017. "Local and global convergence of a general inertial proximal splitting scheme for minimizing composite functions," Computational Optimization and Applications, Springer, vol. 67(2), pages 259-292, June.
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