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 \(l_2\)-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.
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The author’s work was supported in part by the NSF foundation (11001011) of China and China Postdoctoral Science Foundation (2013M541065).
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Communicated by Masao Fukushima.
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Zhang, Y., Chen, X. Regularizations for Stochastic Linear Variational Inequalities. J Optim Theory Appl 163, 460–481 (2014). https://doi.org/10.1007/s10957-013-0514-2
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DOI: https://doi.org/10.1007/s10957-013-0514-2
Keywords
- Stochastic variational inequality
- Epi-convergence
- Semismooth
- Expected residual minimization
- Sample average approximations