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Strong convergence results for quasimonotone variational inequalities

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Abstract

A survey of the existing literature reveals that results on quasimonotone variational inequality problems are scanty in the literature. Moreover, the few existing results are either obtained in finite dimensional Hilbert spaces or the authors were only able to obtain weak convergence results in infinite dimensional Hilbert spaces. In this paper, we study the quasimonotone variational inequality problem and variational inequality problem without monotonicity. We introduce two new inertial iterative schemes with self-adaptive step sizes for approximating a solution of the variational inequality problem. Our proposed methods combine the inertial Tseng extragradient method with viscosity approximation method. We prove some strong convergence results for the proposed algorithms without the knowledge of the Lipschitz constant of the cost operator in infinite dimensional Hilbert spaces. Finally, we provide some numerical experiments to demonstrate the efficiency of our proposed methods in comparison with some recently announced results in the literature in this direction.

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Acknowledgements

The authors sincerely thank the reviewers for their careful reading, constructive comments and fruitful suggestions that improved the manuscript. The research of the first author is wholly supported by the University of KwaZulu-Natal, Durban, South Africa Postdoctoral Fellowship. He is grateful for the funding and financial support. The second author is supported by the National Research Foundation (NRF) of South Africa Incentive Funding for Rated Researchers (Grant Number 119903). Opinions expressed and conclusions arrived are those of the authors and are not necessarily to be attributed to the NRF.

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Authors and Affiliations

  1. School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban, South Africa

    Timilehin O. Alakoya & Oluwatosin T. Mewomo

  2. College of Mathematics and Computer Science, Zhejiang Normal University, Jinhua, 321004, People’s Republic of China

    Yekini Shehu

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  1. Timilehin O. Alakoya
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  2. Oluwatosin T. Mewomo
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  3. Yekini Shehu
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Correspondence to Yekini Shehu.

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Appendix

Appendix

Algorithm 2.1 in Ye and He (2015).

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Alakoya, T.O., Mewomo, O.T. & Shehu, Y. Strong convergence results for quasimonotone variational inequalities. Math Meth Oper Res 95, 249–279 (2022). https://doi.org/10.1007/s00186-022-00780-2

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