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On the convergence properties of non-Euclidean extragradient methods for variational inequalities with generalized monotone operators

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  • Cong Dang
  • Guanghui Lan

Abstract

In this paper, we study a class of generalized monotone variational inequality (GMVI) problems whose operators are not necessarily monotone (e.g., pseudo-monotone). We present non-Euclidean extragradient (N-EG) methods for computing approximate strong solutions of these problems, and demonstrate how their iteration complexities depend on the global Lipschitz or Hölder continuity properties for their operators and the smoothness properties for the distance generating function used in the N-EG algorithms. We also introduce a variant of this algorithm by incorporating a simple line-search procedure to deal with problems with more general continuous operators. Numerical studies are conducted to illustrate the significant advantages of the developed algorithms over the existing ones for solving large-scale GMVI problems. Copyright Springer Science+Business Media New York 2015

Suggested Citation

  • Cong Dang & Guanghui Lan, 2015. "On the convergence properties of non-Euclidean extragradient methods for variational inequalities with generalized monotone operators," Computational Optimization and Applications, Springer, vol. 60(2), pages 277-310, March.
    • Handle: RePEc:spr:coopap:v:60:y:2015:i:2:p:277-310
    DOI: 10.1007/s10589-014-9673-9
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    Cited by:

    1. Duong Viet Thong & Aviv Gibali & Mathias Staudigl & Phan Tu Vuong, 2021. "Computing Dynamic User Equilibrium on Large-Scale Networks Without Knowing Global Parameters," Networks and Spatial Economics, Springer, vol. 21(3), pages 735-768, September.
    2. Zhen-Ping Yang & Gui-Hua Lin, 2021. "Variance-Based Single-Call Proximal Extragradient Algorithms for Stochastic Mixed Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 190(2), pages 393-427, August.
    3. Xiao-Juan Zhang & Xue-Wu Du & Zhen-Ping Yang & Gui-Hua Lin, 2019. "An Infeasible Stochastic Approximation and Projection Algorithm for Stochastic Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 183(3), pages 1053-1076, December.
    4. Aswin Kannan & Uday V. Shanbhag, 2019. "Optimal stochastic extragradient schemes for pseudomonotone stochastic variational inequality problems and their variants," Computational Optimization and Applications, Springer, vol. 74(3), pages 779-820, December.
    5. Boţ, R.I. & Csetnek, E.R. & Vuong, P.T., 2020. "The forward–backward–forward method from continuous and discrete perspective for pseudo-monotone variational inequalities in Hilbert spaces," European Journal of Operational Research, Elsevier, vol. 287(1), pages 49-60.
    6. Fedor Stonyakin & Alexander Gasnikov & Pavel Dvurechensky & Alexander Titov & Mohammad Alkousa, 2022. "Generalized Mirror Prox Algorithm for Monotone Variational Inequalities: Universality and Inexact Oracle," Journal of Optimization Theory and Applications, Springer, vol. 194(3), pages 988-1013, September.
    7. Ahmet Alacaoglu & Yura Malitsky & Volkan Cevher, 2021. "Forward-reflected-backward method with variance reduction," Computational Optimization and Applications, Springer, vol. 80(2), pages 321-346, November.

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