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Variable sample-size optimistic mirror descent algorithm for stochastic mixed variational inequalities

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Abstract

In this paper, we propose a variable sample-size optimistic mirror descent algorithm under the Bregman distance for a class of stochastic mixed variational inequalities. Different from those conventional variable sample-size extragradient algorithms to evaluate the expected mapping twice at each iteration, our algorithm requires only one evaluation of the expected mapping and hence can significantly reduce the computation load. In the monotone case, the proposed algorithm can achieve \({\mathcal {O}}(1/t)\) ergodic convergence rate in terms of the expected restricted gap function and, under the strongly generalized monotonicity condition, the proposed algorithm has a locally linear convergence rate of the Bregman distance between iterations and solutions when the sample size increases geometrically. Furthermore, we derive some results on stochastic local stability under the generalized monotonicity condition. Numerical experiments indicate that the proposed algorithm compares favorably with some existing methods.

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Data availability

The datasets generated during the current study are available from the corresponding author on reasonable request.

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Authors and Affiliations

  1. School of Mathematics, Jiaying University, Meizhou, 514015, China

    Zhen-Ping Yang

  2. College of Mathematics and Statistics, Chongqing Jiaotong University, Chongqing, 400074, China

    Yong Zhao

  3. School of Management, Shanghai University, Shanghai, 200444, China

    Gui-Hua Lin

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  1. Zhen-Ping Yang
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This work was supported in part by NSFC (Nos. 12101262, 12071280, 12001072, 12271067), the Guangdong Basic and Applied Basic Research Foundation (No. 2022A1515010263), the Chongqing Natural Science Foundation (cstc2019jcyj-zdxmX0016, CSTB2022NSCQ-MSX1318), the Key Laboratory for Optimization and Control of Ministry of Education, Chongqing Normal University (No. CSSXKFKTZ202001), and the Group Building Scientific Innovation Project for universities in Chongqing (CXQT21021).

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Yang, ZP., Zhao, Y. & Lin, GH. Variable sample-size optimistic mirror descent algorithm for stochastic mixed variational inequalities. J Glob Optim 89, 143–170 (2024). https://doi.org/10.1007/s10898-023-01346-0

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