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Explicit extragradient-like method with adaptive stepsizes for pseudomonotone variational inequalities

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Abstract

The purpose of this paper is to introduce a new modified subgradient extragradient method for finding an element in the set of solutions of the variational inequality problem for a pseudomonotone and Lipschitz continuous mapping in real Hilbert spaces. It is well known that for the existing subgradient extragradient methods, the step size requires the line-search process or the knowledge of the Lipschitz constant of the mapping, which restrict the applications of the method. To overcome this barrier, in this work we present a modified subgradient extragradient method with adaptive stepsizes and do not require extra projection or value of the mapping. The advantages of the proposed method only use one projection to compute and the strong convergence proved without the prior knowledge of the Lipschitz constant of the inequality variational mapping. Numerical experiments illustrate the performances of our new algorithm and provide a comparison with related algorithms.

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Acknowledgements

The authors would like to thank two anonymous reviewers for their comments on the manuscript which helped us very much in improving and presenting the original version of this paper.

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Authors and Affiliations

  1. Institute of Research and Development, Duy Tan University, Da Nang, 550000, Vietnam

    Duong Viet Thong

  2. School of Mathematics and Statistics, Xidian University, Xi’an, People’s Republic of China

    Jun Yang

  3. Department of Mathematics Education and the RINS, Gyeongsang National University, Jinju, 52828, Korea

    Yeol Je Cho

  4. School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, 611731, Sichuan, People’s Republic of China

    Yeol Je Cho

  5. Department of Mathematics, National Technical University of Athens, Zografou Campus, 15780, Athens, Greece

    Themistocles M. Rassias

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  1. Duong Viet Thong
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  2. Jun Yang
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  4. Themistocles M. Rassias
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Correspondence to Duong Viet Thong.

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Thong, D.V., Yang, J., Cho, Y.J. et al. Explicit extragradient-like method with adaptive stepsizes for pseudomonotone variational inequalities. Optim Lett 15, 2181–2199 (2021). https://doi.org/10.1007/s11590-020-01678-w

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  • DOI: https://doi.org/10.1007/s11590-020-01678-w

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