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Convergence of Relaxed Inertial Subgradient Extragradient Methods for Quasimonotone Variational Inequality Problems

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Abstract

In this paper, we present two new relaxed inertial subgradient extragradient methods for solving variational inequality problems in a real Hilbert space. We establish the convergence of the sequence generated by these methods when the cost operator is quasimonotone and Lipschitz continuous, and when it is Lipschitz continuous without any form of monotonicity. The methods combine both the inertial and relaxation techniques in order to achieve high convergence speed, and the techniques used are quite different from the ones in most papers for solving variational inequality problems. Furthermore, we present some experimental results to illustrate the profits gained from the relaxed inertial steps.

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Acknowledgements

The authors are grateful to the anonymous referees and the handling Editor for their insightful comments which have improved the earlier version of the manuscript greatly. The first author acknowledges with thanks the scholarship and financial support from the University of KwaZulu-Natal (UKZN) Doctoral Scholarship. The research of the second author is wholly supported by the National Research Foundation (NRF) South Africa (S& F-DSI/NRF Free Standing Postdoctoral Fellowship; Grant Number: 120784). The second author also acknowledges the financial support from DSI/NRF, South Africa Center of Excellence in Mathematical and Statistical Sciences (CoE-MaSS) Postdoctoral Fellowship. The fourth 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 CoE-MaSS and NRF. This paper is dedicated to the loving memory of late Professor Charles Ejike Chidume (1947–2021).

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Authors and Affiliations

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

    G. N. Ogwo, C. Izuchukwu & O. T. Mewomo

  2. DSI-NRF Center of Excellence in Mathematical and Statistical Sciences (CoE-MaSS), Johannesburg, South Africa

    C. Izuchukwu

  3. Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, People’s Republic of China

    Y. Shehu

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  1. G. N. Ogwo
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Correspondence to Y. Shehu.

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Ogwo, G.N., Izuchukwu, C., Shehu, Y. et al. Convergence of Relaxed Inertial Subgradient Extragradient Methods for Quasimonotone Variational Inequality Problems. J Sci Comput 90, 10 (2022). https://doi.org/10.1007/s10915-021-01670-1

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