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Inertial Iterative Schemes with Variable Step Sizes for Variational Inequality Problem Involving Pseudomonotone Operator

Author

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  • Jamilu Abubakar

    (Department of Mathematics, King Mongkut’s University of Technology Thonburi, Bangkok 10140, Thailand
    Department of Mathematics, Usmanu Danfodiyo University, Sokoto 840001, Nigeria
    Current address: Department of Mathematics, King Mongkut’s University of Technology Thonburi, 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand.)

  • Poom Kumam

    (Department of Mathematics, King Mongkut’s University of Technology Thonburi, Bangkok 10140, Thailand
    Center of Excellence in Theoretical and Computational Science (TaCS-CoE), Science Laboratory Building, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand
    Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan)

  • Habib ur Rehman

    (Department of Mathematics, King Mongkut’s University of Technology Thonburi, Bangkok 10140, Thailand)

  • Abdulkarim Hassan Ibrahim

    (Department of Mathematics, King Mongkut’s University of Technology Thonburi, Bangkok 10140, Thailand)

Abstract

Two inertial subgradient extragradient algorithms for solving variational inequality problems involving pseudomonotone operator are proposed in this article. The iterative schemes use self-adaptive step sizes which do not require the prior knowledge of the Lipschitz constant of the underlying operator. Furthermore, under mild assumptions, we show the weak and strong convergence of the sequences generated by the proposed algorithms. The strong convergence in the second algorithm follows from the use of viscosity method. Numerical experiments both in finite- and infinite-dimensional spaces are reported to illustrate the inertial effect and the computational performance of the proposed algorithms in comparison with the existing state of the art algorithms.

Suggested Citation

  • Jamilu Abubakar & Poom Kumam & Habib ur Rehman & Abdulkarim Hassan Ibrahim, 2020. "Inertial Iterative Schemes with Variable Step Sizes for Variational Inequality Problem Involving Pseudomonotone Operator," Mathematics, MDPI, vol. 8(4), pages 1-25, April.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:4:p:609-:d:346181
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    References listed on IDEAS

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    1. Rapeepan Kraikaew & Satit Saejung, 2014. "Strong Convergence of the Halpern Subgradient Extragradient Method for Solving Variational Inequalities in Hilbert Spaces," Journal of Optimization Theory and Applications, Springer, vol. 163(2), pages 399-412, November.
    2. 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.
    3. 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.
    4. 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.
    5. 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.
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    Cited by:

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