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Modified Extragradient Method for Variational Inequalities and Verification of Solution Existence

Author

Listed:
  • Y. J. Wang

    (Nanjing Normal University
    Qufu Normal University)

  • N. H. Xiu

    (Northern Jiaotong University)

  • J. Z. Zhang

    (City University of Hong Kong)

Abstract

In this paper, we propose a modified extragradient method for solving variational inequalities (VI) which has the following nice features: (i) The generated sequence possesses an expansion property with respect to the starting point; (ii) the existence of the solution to a VI problem can be verified through the behavior of the generated sequence from the fact that the iterative sequence diverges to infinity if and only if the solution set is empty. Global convergence of the method is guaranteed under mild conditions. Our preliminary computational experience is also reported.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:119:y:2003:i:1:d:10.1023_b:jota.0000005047.30026.b8
    DOI: 10.1023/B:JOTA.0000005047.30026.b8
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    References listed on IDEAS

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    1. I. V. Konnov, 1997. "A Class of Combined Iterative Methods for Solving Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 94(3), pages 677-693, September.
    2. M. Seetharama Gowda & Jong-Shi Pang, 1992. "On Solution Stability of the Linear Complementarity Problem," Mathematics of Operations Research, INFORMS, vol. 17(1), pages 77-83, February.
    3. Y. J. Wang & N. H. Xiu & C. Y. Wang, 2001. "Unified Framework of Extragradient-Type Methods for Pseudomonotone Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 111(3), pages 641-656, December.
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    Cited by:

    1. E. M. Bednarczuk & A. Jezierska & K. E. Rutkowski, 2018. "Proximal primal–dual best approximation algorithm with memory," Computational Optimization and Applications, Springer, vol. 71(3), pages 767-794, December.
    2. Trinh Ngoc Hai, 2020. "Two modified extragradient algorithms for solving variational inequalities," Journal of Global Optimization, Springer, vol. 78(1), pages 91-106, September.
    3. X. Wang & S. Li & X. Kou & Q. Zhang, 2015. "A new alternating direction method for linearly constrained nonconvex optimization problems," Journal of Global Optimization, Springer, vol. 62(4), pages 695-709, August.
    4. Guo-ji Tang & Nan-jing Huang, 2012. "Korpelevich’s method for variational inequality problems on Hadamard manifolds," Journal of Global Optimization, Springer, vol. 54(3), pages 493-509, November.
    5. P. Anh & H. Le Thi, 2013. "An Armijo-type method for pseudomonotone equilibrium problems and its applications," Journal of Global Optimization, Springer, vol. 57(3), pages 803-820, November.
    6. Chuanwei Wang & Yiju Wang & Chuanliang Xu, 2007. "A projection method for a system of nonlinear monotone equations with convex constraints," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 66(1), pages 33-46, August.
    7. R. Díaz Millán & O. P. Ferreira & J. Ugon, 2024. "Extragradient method with feasible inexact projection to variational inequality problem," Computational Optimization and Applications, Springer, vol. 89(2), pages 459-484, November.

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