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Expected Residual Minimization Formulation for a Class of Stochastic Vector Variational Inequalities

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Abstract

This paper considers a class of vector variational inequalities. First, we present an equivalent formulation, which is a scalar variational inequality, for the deterministic vector variational inequality. Then we concentrate on the stochastic circumstance. By noting that the stochastic vector variational inequality may not have a solution feasible for all realizations of the random variable in general, for tractability, we employ the expected residual minimization approach, which aims at minimizing the expected residual of the so-called regularized gap function. We investigate the properties of the expected residual minimization problem, and furthermore, we propose a sample average approximation method for solving the expected residual minimization problem. Comprehensive convergence analysis for the approximation approach is established as well.

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Acknowledgments

This work was supported in part by the NSFC Grant (No. 11431004), the Hongkong Baptist University FRG1/15-16/027 and RC-NACAN-ZHANG J, the Humanity and Social Science Foundation of Ministry of Education of China (No. 15YJA630034) and the Innovation Program of Shanghai Municipal Education Commission (No. 14ZS086). The authors are grateful to Professor F. Giannessi, Professor G.Y. Chen, and two anonymous referees for their valuable comments and constructive suggestions, which help to improve the original manuscript.

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Authors and Affiliations

  1. Department of Mathematics, Sichuan University, Chengdu, 610065, China

    Yong Zhao

  2. Department of Mathematics, Hong Kong Baptist University, Hongkong, China

    Jin Zhang

  3. Department of Mathematics, Chongqing Normal University, Chongqing, 401131, China

    Xinmin Yang

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

    Gui-Hua Lin

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  1. Yong Zhao
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  2. Jin Zhang
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  3. Xinmin Yang
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  4. Gui-Hua Lin
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Correspondence to Yong Zhao.

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Communicated by Guang-ya Chen.

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Zhao, Y., Zhang, J., Yang, X. et al. Expected Residual Minimization Formulation for a Class of Stochastic Vector Variational Inequalities. J Optim Theory Appl 175, 545–566 (2017). https://doi.org/10.1007/s10957-016-0939-5

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  • DOI: https://doi.org/10.1007/s10957-016-0939-5

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