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Variance-Based Single-Call Proximal Extragradient Algorithms for Stochastic Mixed Variational Inequalities

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  • Zhen-Ping Yang

    (Jiaying University)

  • Gui-Hua Lin

    (Shanghai University)

Abstract

In the study of stochastic variational inequalities, the extragradient algorithms attract much attention. However, such schemes require two evaluations of the expected mapping at each iteration in general. In this paper, we present several variance-based single-call proximal extragradient algorithms for solving a class of stochastic mixed variational inequalities by aiming at alleviating the cost of an extragradient step. One salient feature of the proposed algorithms is that they require only one evaluation of the expected mapping at each iteration, and hence, the computation load may be significantly reduced. We show that the proposed algorithms can achieve sublinear ergodic convergence rate in terms of the restricted merit function. Furthermore, under the strongly Minty variational inequality condition, we derive some results related to convergence rate of the distance between iterates and solutions, the iteration and oracle complexities for the proposed algorithms when the sample size increases at a geometric or polynomial rate. Numerical experiments indicate that the proposed algorithms are quite competitive with some existing algorithms.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:190:y:2021:i:2:d:10.1007_s10957-021-01882-3
    DOI: 10.1007/s10957-021-01882-3
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    References listed on IDEAS

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    1. Zhen-Ping Yang & Yuliang Wang & Gui-Hua Lin, 2020. "Variance-Based Modified Backward-Forward Algorithm with Line Search for Stochastic Variational Inequality Problems and Its Applications," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 37(03), pages 1-33, April.
    2. 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.
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    5. 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.
    6. 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.
    7. Jinlong Lei & Uday V. Shanbhag & Jong-Shi Pang & Suvrajeet Sen, 2020. "On Synchronous, Asynchronous, and Randomized Best-Response Schemes for Stochastic Nash Games," Mathematics of Operations Research, INFORMS, vol. 45(1), pages 157-190, February.
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    10. Jie Jiang & Xiaojun Chen & Zhiping Chen, 2020. "Quantitative analysis for a class of two-stage stochastic linear variational inequality problems," Computational Optimization and Applications, Springer, vol. 76(2), pages 431-460, June.
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