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Iterative methods for solving monotone variational inclusions without prior knowledge of the Lipschitz constant of the single-valued operator

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Abstract

In this work, we investigate a contraction-type method for solving monotone variational inclusion problems in real Hilbert spaces. We obtain strong convergence theorems for two algorithms with a self-adaptive step size for solving monotone variational inclusions. The advantage of our algorithms is that we do not require a cocoercivity assumption nor do we need to know the Lipschitz-type constant of the single-valued operator. Moreover, a convergence rate is derived in the case where one of the operators is maximally and strongly monotone, and the other is monotone and Lipschitz continuous. The performance of our proposed methods is illustrated by numerical experiments regarding signal recovery. Our results improve and extend some known results, and our experiments show that our proposed algorithms are efficient and outperform other algorithms which are available in the literature.

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Data availability

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

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Acknowledgements

The authors are thankful to an anonymous reviewer for comments and remarks which substantially improved the quality of the paper. We would also like to express our gratitude to Professor Patrick Combettes, Editor, for giving us the opportunity to revise and resubmit this manuscript.

Funding

P. Cholamjiak was supported by the Thailand Science Research and Innovation Fund and the University of Phayao (FF67).

Author information

Authors and Affiliations

  1. Faculty of Mathematical Economics, National Economics University, Hanoi City, Vietnam

    Duong Viet Thong

  2. Department of Mathematics, The Technion – Israel Institute of Technology, 32000, Haifa, Israel

    Simeon Reich

  3. School of Science, University of Phayao, 56000, Phayao, Thailand

    Prasit Cholamjiak

  4. Falculty of Maths, FPT University, HCM Saigon Hi-Tech Park, Ho Chi Minh City, Vietnam

    Le Dinh Long

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  1. Duong Viet Thong
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  2. Simeon Reich
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  4. Le Dinh Long
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Contributions

DVT, SR, and LDL wrote the main manuscript text, and PC prepared Figs. 1, 2, 3, 4, 5, 6, 7 and 8 and Tables 1 and 2. All authors reviewed the manuscript carefully. All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

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Correspondence to Simeon Reich.

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Thong, D.V., Reich, S., Cholamjiak, P. et al. Iterative methods for solving monotone variational inclusions without prior knowledge of the Lipschitz constant of the single-valued operator. Numer Algor 97, 1267–1300 (2024). https://doi.org/10.1007/s11075-024-01749-4

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