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Two modified extragradient algorithms for solving variational inequalities

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  • Trinh Ngoc Hai

    (Hanoi University of Science and Technology)

Abstract

In this paper, we discuss two modified extragradient methods for variational inequalities. The first one can be applied when the Lipschitz constant of the involving operator is unknown. In contrast to the work by Hieu and Thong (J Glob Optim 70:385–399, 2018) and by Khanh (Numer Funct Anal Optim 37:1131–1143, 2016), the new algorithm does not require its step-sizes tending to zero. This feature helps to speed up our method. The second algorithm solves variational inequalities with non-Lipschitz continuous operators. Under the pseudomonotonicity assumption, the proposed algorithm converges to a solution of the problem. In contrast to other solution methods for this class of problems, the new algorithm does not require the step sizes being square summable. Some numerical experiments show that the new algorithms are more effective than the existing ones.

Suggested Citation

  • Trinh Ngoc Hai, 2020. "Two modified extragradient algorithms for solving variational inequalities," Journal of Global Optimization, Springer, vol. 78(1), pages 91-106, September.
    • Handle: RePEc:spr:jglopt:v:78:y:2020:i:1:d:10.1007_s10898-020-00895-y
    DOI: 10.1007/s10898-020-00895-y
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    References listed on IDEAS

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    1. M.A. Noor, 2003. "Extragradient Methods for Pseudomonotone Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 117(3), pages 475-488, June.
    2. Pham Khanh & Phan Vuong, 2014. "Modified projection method for strongly pseudomonotone variational inequalities," Journal of Global Optimization, Springer, vol. 58(2), pages 341-350, February.
    3. Dang Hieu & Duong Viet Thong, 2018. "New extragradient-like algorithms for strongly pseudomonotone variational inequalities," Journal of Global Optimization, Springer, vol. 70(2), pages 385-399, February.
    4. P. Anh & T. Hai & P. Tuan, 2016. "On ergodic algorithms for equilibrium problems," Journal of Global Optimization, Springer, vol. 64(1), pages 179-195, January.
    5. J. Bello Cruz & A. Iusem, 2010. "Convergence of direct methods for paramonotone variational inequalities," Computational Optimization and Applications, Springer, vol. 46(2), pages 247-263, June.
    6. J. Y. Bello Cruz & R. Díaz Millán, 2014. "A Direct Splitting Method for Nonsmooth Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 161(3), pages 728-737, June.
    7. Y. J. Wang & N. H. Xiu & J. Z. Zhang, 2003. "Modified Extragradient Method for Variational Inequalities and Verification of Solution Existence," Journal of Optimization Theory and Applications, Springer, vol. 119(1), pages 167-183, October.
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