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An Infeasible Stochastic Approximation and Projection Algorithm for Stochastic Variational Inequalities

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Abstract

In this paper, we consider a stochastic variational inequality, in which the mapping involved is an expectation of a given random function. Inspired by the work of He (Appl Math Optim 35:69–76, 1997) and the extragradient method proposed by Iusem et al. (SIAM J Optim 29:175–206, 2019), we propose an infeasible projection algorithm with line search scheme, which can be viewed as a modification of the above-mentioned method of Iusem et al. In particular, in the correction step, we replace the projection by computing search direction and stepsize, that is, we need only one projection at each iteration, while the method of Iusem et al. requires two projections at each iteration. Moreover, we use dynamic sampled scheme with line search to cope with the absence of Lipschitz constant and choose the stepsize to be bounded away from zero and the direction to be a descent direction. In the process of stochastic approximation, we iteratively reduce the variance of a stochastic error. Under appropriate assumptions, we derive some properties related to convergence, convergence rate, and oracle complexity. In particular, compared with the method of Iusem et al., our method uses less projections and has the same iteration complexity, which, however, has a higher oracle complexity for a given tolerance in a finite dimensional space. Finally, we report some numerical experiments to show its efficiency.

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Acknowledgements

This work was supported in part by NSFC (Nos. 11671250, 11431004, 71831008) and Humanity and Social Science Foundation of Ministry of Education of China (No. 15YJA630034). The authors are grateful to an anonymous referee for his/her helpful comments and suggestions, which have led to much improvement of the paper.

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Authors and Affiliations

  1. College of Mobile Telecommunications, Chongqing University of Posts and Telecommunications, Chongqing, 400065, China

    Xiao-Juan Zhang

  2. College of Mathematical Sciences, Chongqing Normal University, Chongqing, 401131, China

    Xue-Wu Du

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

    Zhen-Ping Yang & Gui-Hua Lin

Authors
  1. Xiao-Juan Zhang
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  2. Xue-Wu Du
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  3. Zhen-Ping Yang
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  4. Gui-Hua Lin
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Correspondence to Gui-Hua Lin.

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Alfredo N. Iusem.

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Zhang, XJ., Du, XW., Yang, ZP. et al. An Infeasible Stochastic Approximation and Projection Algorithm for Stochastic Variational Inequalities. J Optim Theory Appl 183, 1053–1076 (2019). https://doi.org/10.1007/s10957-019-01578-9

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  • DOI: https://doi.org/10.1007/s10957-019-01578-9

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