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Regularizations for Stochastic Linear Variational Inequalities

Author

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  • Yanfang Zhang

    (Chinese Academy of Sciences)

  • Xiaojun Chen

    (The Hong Kong Polytechnic University)

Abstract

This paper applies the Moreau–Yosida regularization to a convex expected residual minimization (ERM) formulation for a class of stochastic linear variational inequalities. To have the convexity of the corresponding sample average approximation (SAA) problem, we adopt the Tikhonov regularization. We show that any cluster point of minimizers of the Tikhonov regularization for the SAA problem is a minimizer of the ERM formulation with probability one as the sample size goes to infinity and the Tikhonov regularization parameter goes to zero. Moreover, we prove that the minimizer is the least $$l_2$$ l 2 -norm solution of the ERM formulation. We also prove the semismoothness of the gradient of the Moreau–Yosida and Tikhonov regularizations for the SAA problem.

Suggested Citation

  • Yanfang Zhang & Xiaojun Chen, 2014. "Regularizations for Stochastic Linear Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 163(2), pages 460-481, November.
    • Handle: RePEc:spr:joptap:v:163:y:2014:i:2:d:10.1007_s10957-013-0514-2
    DOI: 10.1007/s10957-013-0514-2
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    References listed on IDEAS

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    Cited by:

    1. Yong Zhao & Jin Zhang & Xinmin Yang & Gui-Hua Lin, 2017. "Expected Residual Minimization Formulation for a Class of Stochastic Vector Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 175(2), pages 545-566, November.

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