STOCHASTIC VARIATIONAL INEQUALITY APPROACHES TO THE STOCHASTIC GENERALIZED NASH EQUILIBRIUM WITH SHARED CONSTRAINTS

doi:10.4208/jcm.2109-m2020-0099

Journal of Computational Mathematics ›› 2023, Vol. 41 ›› Issue (3): 415-436.DOI: 10.4208/jcm.2109-m2020-0099

STOCHASTIC VARIATIONAL INEQUALITY APPROACHES TO THE STOCHASTIC GENERALIZED NASH EQUILIBRIUM WITH SHARED CONSTRAINTS

Yanfang Zhang

College of Science, Minzu University of China, Beijing 100081, China

Received:2020-04-16 Revised:2021-06-09 Published:2023-03-14
Contact: Yanfang Zhang, Email:zhangyanfang@muc.edu.cn
Supported by:
The author's work was supported by the National Natural Science Foundation of China (No. 11601541, No. 12171027), State Key Laboratory of Scientific and Engineering Computing, Chinese Academy of Sciences, and the Youth Foundation of Minzu University of China (No. 2021QNPY98).

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Yanfang Zhang. STOCHASTIC VARIATIONAL INEQUALITY APPROACHES TO THE STOCHASTIC GENERALIZED NASH EQUILIBRIUM WITH SHARED CONSTRAINTS[J]. Journal of Computational Mathematics, 2023, 41(3): 415-436.

In this paper, we consider the generalized Nash equilibrium with shared constraints in the stochastic environment, and we call it the stochastic generalized Nash equilibrium. The stochastic variational inequalities are employed to solve this kind of problems, and the expected residual minimization model and the conditional value-at-risk formulations defined by the residual function for the stochastic variational inequalities are discussed. We show the risk for different kinds of solutions for the stochastic generalized Nash equilibrium by the conditional value-at-risk formulations. The properties of the stochastic quadratic generalized Nash equilibrium are shown. The smoothing approximations for the expected residual minimization formulation and the conditional value-at-risk formulation are employed. Moreover, we establish the gradient consistency for the measurable smoothing functions and the integrable functions under some suitable conditions, and we also analyze the properties of the formulations. Numerical results for the applications arising from the electricity market model illustrate that the solutions for the stochastic generalized Nash equilibrium given by the ERM model have good properties, such as robustness, low risk and so on.

Key words: Stochastic generalized Nash equilibrium, Stochastic variational inequalities, Expected residual minimization model, CVaR formulation, Gradient consistency

CLC Number:

90C15
90C33

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STOCHASTIC VARIATIONAL INEQUALITY APPROACHES TO THE STOCHASTIC GENERALIZED NASH EQUILIBRIUM WITH SHARED CONSTRAINTS

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