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An Explicit Extragradient Algorithm for Solving Variational Inequalities

Author

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  • Dang Van Hieu

    (Ton Duc Thang University)

  • Jean Jacques Strodiot

    (University of Namur)

  • Le Dung Muu

    (TIMAS, Thang Long University)

Abstract

In this paper, we introduce an explicit iterative algorithm for solving a (pseudo) monotone variational inequality under Lipschitz condition in a Hilbert space. The algorithm is constructed around some projections incorporated by inertial terms. It uses variable stepsizes which are generated at each iteration by some simple computations. Furthermore, it can be easily implemented without the prior knowledge of the Lipschitz constant of the operator. Theorems of weak convergence are established under mild conditions, and some numerical results are reported for the purpose of comparison with other algorithms. The obtained results in this paper extend some related works in the literature.

Suggested Citation

  • Dang Van Hieu & Jean Jacques Strodiot & Le Dung Muu, 2020. "An Explicit Extragradient Algorithm for Solving Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 185(2), pages 476-503, May.
    • Handle: RePEc:spr:joptap:v:185:y:2020:i:2:d:10.1007_s10957-020-01661-6
    DOI: 10.1007/s10957-020-01661-6
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    References listed on IDEAS

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    1. Q. L. Dong & J. Z. Huang & X. H. Li & Y. J. Cho & Th. M. Rassias, 2019. "MiKM: multi-step inertial Krasnosel’skiǐ–Mann algorithm and its applications," Journal of Global Optimization, Springer, vol. 73(4), pages 801-824, April.
    2. Dang Hieu & Duong Viet Thong, 2018. "New extragradient-like algorithms for strongly pseudomonotone variational inequalities," Journal of Global Optimization, Springer, vol. 70(2), pages 385-399, February.
    3. L. C. Ceng & M. Teboulle & J. C. Yao, 2010. "Weak Convergence of an Iterative Method for Pseudomonotone Variational Inequalities and Fixed-Point Problems," Journal of Optimization Theory and Applications, Springer, vol. 146(1), pages 19-31, July.
    4. Q. L. Dong & Y. J. Cho & L. L. Zhong & Th. M. Rassias, 2018. "Inertial projection and contraction algorithms for variational inequalities," Journal of Global Optimization, Springer, vol. 70(3), pages 687-704, March.
    5. Y. Censor & A. Gibali & S. Reich, 2011. "The Subgradient Extragradient Method for Solving Variational Inequalities in Hilbert Space," Journal of Optimization Theory and Applications, Springer, vol. 148(2), pages 318-335, February.
    6. Martin Seydenschwanz, 2015. "Convergence results for the discrete regularization of linear-quadratic control problems with bang–bang solutions," Computational Optimization and Applications, Springer, vol. 61(3), pages 731-760, July.
    7. Pham Khanh & Phan Vuong, 2014. "Modified projection method for strongly pseudomonotone variational inequalities," Journal of Global Optimization, Springer, vol. 58(2), pages 341-350, February.
    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. Q. L. Dong & Y. C. Tang & Y. J. Cho & Th. M. Rassias, 2018. "“Optimal” choice of the step length of the projection and contraction methods for solving the split feasibility problem," Journal of Global Optimization, Springer, vol. 71(2), pages 341-360, June.
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    Cited by:

    1. Zhong-bao Wang & Xue Chen & Jiang Yi & Zhang-you Chen, 2022. "Inertial projection and contraction algorithms with larger step sizes for solving quasimonotone variational inequalities," Journal of Global Optimization, Springer, vol. 82(3), pages 499-522, March.
    2. Pham Ky Anh & Trinh Ngoc Hai, 2021. "Dynamical system for solving bilevel variational inequalities," Journal of Global Optimization, Springer, vol. 80(4), pages 945-963, August.

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