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Regularized Sample Average Approximation Approach for Two-Stage Stochastic Variational Inequalities

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Abstract

Sample average approximation (SAA) approach for two-stage stochastic variational inequalities (SVIs) with continuous probability distributions, where the second-stage problems have multiple solutions, may not promise convergence assertions as the sample size tends to infinity. In this paper, a regularized SAA approach is proposed to numerically solve a class of two-stage SVIs with continuous probability distributions, where the second-stage problems are monotone and allowed to have multiple solutions. We first give some structural properties. After that, the convergence analysis of the regularized SAA approach for two-stage SVIs is investigated as the regularization parameter tends to zero and the sample size tends to infinity. Finally, we employ the progressive hedging algorithm to report some numerical results.

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Acknowledgements

Jie Jiang was supported by China Postdoctoral Science Foundation (Grant No. 2020M673117) and CAS AMSS-PolyU Joint Laboratory of Applied Mathematics and Fundamental Research Funds for the Central Universities (Grant Nos. 2020CDJQY-A039 and 2021CDJQY-009). Shengjie Li was supported by the National Natural Science Foundation of China (Grant No. 11971078). The authors would like to thank editors for organizing such a professional and efficient reviewing process. They also appreciate two anonymous referees for their insightful and important comments, which improve the presentation and quality of the paper.

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  1. College of Mathematics and Statistics, Chongqing University, Chongqing, China

    Jie Jiang & Shengjie Li

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  1. Jie Jiang
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  2. Shengjie Li
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Correspondence to Shengjie Li.

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Communicated by Akhtar A. Khan.

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Jiang, J., Li, S. Regularized Sample Average Approximation Approach for Two-Stage Stochastic Variational Inequalities. J Optim Theory Appl 190, 650–671 (2021). https://doi.org/10.1007/s10957-021-01905-z

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