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Levitin–Polyak Well-Posedness by Perturbations for the Split Hemivariational Inequality Problem on Hadamard Manifolds

Author

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  • Vo Minh Tam

    (Dong Thap University)

  • Nguyen Hung

    (Posts and Telecommunications Institute of Technology)

  • Zhenhai Liu

    (Guangxi Minzu University)

  • Jen Chih Yao

    (China Medical University)

Abstract

The purpose of this paper is to establish some new results on the Levitin–Polyak well-posedness to a class of split hemivariational inequality problems on Hadamard manifolds. We first consider a new class of split hemivariational inequality problems (for short, SHIP) on Hadamard manifolds and introduce the regularized gap functions for these problems. Then, we study the notion of Levitin–Polyak well-posedness by perturbations to SHIP and show the equivalence between the Levitin–Polyak well-posedness by perturbations and the existence of solutions for SHIP under suitable conditions. Furthermore, based on the regularized gap functions for the perturbed SHIP, we establish the criterion for the Levitin–Polyak well-posedness by perturbations for SHIP via the split optimization problems on Hadamard manifolds. Our main results presented in paper are new even in the special case of hemivariational inequality problems.

Suggested Citation

  • Vo Minh Tam & Nguyen Hung & Zhenhai Liu & Jen Chih Yao, 2022. "Levitin–Polyak Well-Posedness by Perturbations for the Split Hemivariational Inequality Problem on Hadamard Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 195(2), pages 684-706, November.
    • Handle: RePEc:spr:joptap:v:195:y:2022:i:2:d:10.1007_s10957-022-02111-1
    DOI: 10.1007/s10957-022-02111-1
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    References listed on IDEAS

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    1. Fang, Ya-Ping & Huang, Nan-Jing & Yao, Jen-Chih, 2010. "Well-posedness by perturbations of mixed variational inequalities in Banach spaces," European Journal of Operational Research, Elsevier, vol. 201(3), pages 682-692, March.
    2. Xiao-bo Li & Nan-jing Huang & Qamrul Hasan Ansari & Jen-Chih Yao, 2019. "Convergence Rate of Descent Method with New Inexact Line-Search on Riemannian Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 180(3), pages 830-854, March.
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