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On Bilevel Monotone Inclusion and Variational Inequality Problems

Author

Listed:
  • Austine Efut Ofem

    (School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban 4041, South Africa)

  • Jacob Ashiwere Abuchu

    (School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban 4041, South Africa
    Department of Mathematics, University of Calabar, Calabar 540271, Nigeria)

  • Hossam A. Nabwey

    (Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam Bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia
    Department of Basic Engineering, Faculty of Engineering, Menoufia University, Shibin el Kom 32511, Egypt)

  • Godwin Chidi Ugwunnadi

    (Department of Mathematics, University of Eswatini, Kwaluseni M201, Eswatini
    Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, P.O. Box 94, Pretoria 0204, South Africa)

  • Ojen Kumar Narain

    (School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban 4041, South Africa)

Abstract

In this article, the problem of solving a strongly monotone variational inequality problem over the solution set of a monotone inclusion problem in the setting of real Hilbert spaces is considered. To solve this problem, two methods, which are improvements and modifications of the Tseng splitting method, and projection and contraction methods, are presented. These methods are equipped with inertial terms to improve their speed of convergence. The strong convergence results of the suggested methods are proved under some standard assumptions on the control parameters. Also, strong convergence results are achieved without prior knowledge of the operator norm. Finally, the main results of this research are applied to solve bilevel variational inequality problems, convex minimization problems, and image recovery problems. Some numerical experiments to show the efficiency of our methods are conducted.

Suggested Citation

  • Austine Efut Ofem & Jacob Ashiwere Abuchu & Hossam A. Nabwey & Godwin Chidi Ugwunnadi & Ojen Kumar Narain, 2023. "On Bilevel Monotone Inclusion and Variational Inequality Problems," Mathematics, MDPI, vol. 11(22), pages 1-28, November.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:22:p:4643-:d:1279998
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    References listed on IDEAS

    as
    1. Jacob Ashiwere Abuchu & Austine Efut Ofem & Godwin Chidi Ugwunnadi & Ojen Kumar Narain & Azhar Hussain & Xian-Ming Gu, 2023. "Hybrid Alternated Inertial Projection and Contraction Algorithm for Solving Bilevel Variational Inequality Problems," Journal of Mathematics, Hindawi, vol. 2023, pages 1-23, March.
    2. Duong Viet Thong & Pham Ky Anh & Vu Tien Dung & Do Thi My Linh, 2023. "A Novel Method for Finding Minimum-norm Solutions to Pseudomonotone Variational Inequalities," Networks and Spatial Economics, Springer, vol. 23(1), pages 39-64, March.
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