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Improved variance reduction extragradient method with line search for stochastic variational inequalities

Author

Listed:
  • Ting Li

    (Nanjing Normal University)

  • Xingju Cai

    (Nanjing Normal University)

  • Yongzhong Song

    (Nanjing Normal University)

  • Yumin Ma

    (Nanjing Normal University)

Abstract

In this paper, we investigate the numerical methods for solving stochastic variational inequalities. Using line search scheme, we propose an improved variance based stochastic extragradient method with different step sizes in the prediction and correction steps. The range of correction step size which can guarantee the convergence is also given. For the initial line search step size of each iteration, an adaptive method is adopted. Rather than the same scale for each reduction, a proportional reduction related to the problem is used to meet the line search criteria. Under the assumptions of Lipschitz continuous, pseudo-monotone operator and independent identically distributed sampling, the iterative complexity and the oracle complexity are obtained. When estimating the upper bound of the second order moment of the martingale difference sequence, we give a more convenient and comprehensible proof instead of using the Burkholder-Davis-Gundy inequality. The proposed algorithm is applied to fractional programming problems and the $$l_2$$ l 2 regularized logistic regression problem. The numerical results demonstrate its superiority.

Suggested Citation

  • Ting Li & Xingju Cai & Yongzhong Song & Yumin Ma, 2023. "Improved variance reduction extragradient method with line search for stochastic variational inequalities," Journal of Global Optimization, Springer, vol. 87(2), pages 423-446, November.
    • Handle: RePEc:spr:jglopt:v:87:y:2023:i:2:d:10.1007_s10898-022-01135-1
    DOI: 10.1007/s10898-022-01135-1
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    References listed on IDEAS

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    1. Alfredo N. Iusem & Alejandro Jofré & Philip Thompson, 2019. "Incremental Constraint Projection Methods for Monotone Stochastic Variational Inequalities," Mathematics of Operations Research, INFORMS, vol. 44(1), pages 236-263, February.
    2. Xingju Cai & Guoyong Gu & Bingsheng He, 2014. "On the O(1/t) convergence rate of the projection and contraction methods for variational inequalities with Lipschitz continuous monotone operators," Computational Optimization and Applications, Springer, vol. 57(2), pages 339-363, March.
    3. Huifu Xu, 2010. "Sample Average Approximation Methods For A Class Of Stochastic Variational Inequality Problems," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 27(01), pages 103-119.
    4. B. S. He & L. Z. Liao, 2002. "Improvements of Some Projection Methods for Monotone Nonlinear Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 112(1), pages 111-128, January.
    5. Xingju Cai & Deren Han & Lingling Xu, 2013. "An improved first-order primal-dual algorithm with a new correction step," Journal of Global Optimization, Springer, vol. 57(4), pages 1419-1428, December.
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