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Iterative regularization methods with new stepsize rules for solving variational inclusions

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Abstract

The paper concerns with three iterative regularization methods for solving a variational inclusion problem of the sum of two operators, the one is maximally monotone and the another is monotone and Lipschitz continuous, in a Hilbert space. We first describe how to incorporate regularization terms in the methods of forward-backward types, and then establish the strong convergence of the resulting methods. With several new stepsize rules considered, the methods can work with or without knowing previously the Lipschitz constant of cost operator. Unlike known hybrid methods, the strong convergence of the proposed methods comes from the regularization technique. Several applications to signal recovery problems and optimal control problems together with numerical experiments are also presented in this paper. Our numerical results have illustrated the fast convergence and computational effectiveness of the new methods over known hybrid methods.

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Acknowledgements

The authors sincerely thank the Editor and anonymous reviewers for their constructive comments which helped to improve the quality and presentation of this paper. The research of first and third authors is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant Number 101.01-2020.06. The research of last author is supported by the Namur Institute for Complex Systems, naXys, University of Namur, Belgium.

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Authors and Affiliations

  1. TIMAS - Thang Long University, Hanoi, Vietnam

    Dang Van Hieu

  2. Department of Mathematics, University of Science, Vietnam National University, Hanoi, Vietnam

    Pham Ky Anh

  3. Institute of Mathematics, VAST, Hanoi, 18 Hoang Quoc Viet, Hanoi, Vietnam

    Le Dung Muu

  4. Department of Mathematics, naXys, Namur Institute for Complex Systems, University of Namur, Namur, Belgium

    Jean Jacques Strodiot

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  1. Dang Van Hieu
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  2. Pham Ky Anh
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  3. Le Dung Muu
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  4. Jean Jacques Strodiot
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Correspondence to Dang Van Hieu.

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Van Hieu, D., Anh, P.K., Muu, L.D. et al. Iterative regularization methods with new stepsize rules for solving variational inclusions. J. Appl. Math. Comput. 68, 571–599 (2022). https://doi.org/10.1007/s12190-021-01534-9

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  • DOI: https://doi.org/10.1007/s12190-021-01534-9

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