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Yanfang Zhang   

  1. 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,
  • 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).


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.

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[1] K. Arrow and G. Debreu, Existence of an equilibrium for a competitive economy, Econometrica, 22:3(1954), 265-290.
[2] J.V. Burke, X. Chen, and H. Sun, Subdifferentiation and smoothing of nonsmooth integral functionals, Math. Program., 181(2019), 229-264.
[3] J.V. Burke, T. Hoheisel, and C. Kanzow, Gradient consistency for integral-convolution smoothing fucntions, Set-Valued Var. Anal., 21(2013), 359-376.
[4] X. Chen, Methods for nonsmooth, nonconvex minization, Math. Program., 134:1(2012), 71-99.
[5] X. Chen, T.K. Pang and R.J.B. Wets, Two-stage stochastic variational inequalities:an ERMsolution procedure, Math. Program., 165:1(2017), 71-111.
[6] X. Chen, H. Sun and H. Xu, Discrete approximation of two-stage stochastic and distributionally robust linear complementarity problems, Math. Program., 177(2019), 255-289.
[7] X. Chen and Z. Wang, Differential variational inequality approach to dynamic games with shared constraints, Math. Program., 146:1-2(2014), 379-408.
[8] X. Chen, R.J.B. Wets, and Y. Zhang, Stochastic variational inequalities:residual minimization smoothing/sample average approximations, SIAM J. Optim., 22:2(2012), 649-673.
[9] X. Chen, A. Shapiro and H. Sun, Convergence analysis of smaple average approximation of twostage stochastic generalized equations, SIAM. J. Optim., 29:1(2019), 135-161.
[10] F.H. Clarke, Optimization and nonsmooth analysis, John Wiley, New York, 1983.
[11] A. Conejo, F. Nogales, and J. Arroyo, Risk-constrained self-scheduling of a thermal power producer, IEEE Trans. Power Syst., 19:3(2004), 1569-1574.
[12] F. Facchinei, A. Fischer, and V. Piccialli, On generalized Nash games and variational inequalites, Oper. Res. Lett., 35:2(2007), 159-164.
[13] P.T. Harker, Generalized Nash games and quasi-variational inequalities, Eur. J. Oper. Res., 54:1(1991), 81-94.
[14] B.F. Hobbs, Linear complementarity models of Nash-cournot competition in Bilateral and POOLCO power markets, IEEE Trans. Power Syst., 16:2(2001), 194-202.
[15] B.F. Hobbs and J.S. Pang, Spatial oligopolistic equilibria with arbitrage, shared resources, and price function conjectures, Math. Program., 101:1(2004), 57-94.
[16] T. Ichiishi, Game theory for economic analysis, Academic press, New York, 1983.
[17] B. Jadamba and F. Raciti, Variational inequality approach to stochastic nash equilibrium problems with an application to cournot oligopoly, J. Optim. Theory Appl., 165:3(2015), 1050-1070.
[18] H. Jiang, U. Shanbhag, and S. Meyn, Distributed computation of equilibria in misspecifie convex stochastic nash games, IEEE Trans. Autom. Control, 63:2(2015), 360-371.
[19] H. Jiang and H. Xu, Stochastic approximation approximation approaches to the stochastic variational inequality problems, IEEE Trans. Autom. Control, 53:6(2008), 1462-1475.
[20] A. Kannan, U. Shanbhag, and H. Kim, Addressing supply-side risk in uncertain power markets:stochastic nash models, scalable algorithms and error analysis, Optim. Methods Softw., 28:5(2013), 1095-1138.
[21] J. Koshal, A. Nedić, and U.V. Shanbhag, Regularized iterative stochastic approximation methods for stochastic variational inequality problems, IEEE Trans. Autom. Control, 58:3(2013), 594-609.
[22] O. Mangasarian and T.H. Shiau, Lipschitz continuity of solutions of linear inequalities, programs and complementarity problems, SIAM J. Control Optim., 25:3(1987), 583-595.
[23] L. Mckenzie, On the existence of a general equilibrium for a competitive market, Econometrica, 27:1(1959), 55-71.
[24] K. Nabetani, P. Tseng, and M. Fukushima, Parametrized variational inequality approaches to generalized Nash equilibrium problems with shared constraints, Comput. Optim. Appl., 48:3(2011), 423-452.
[25] J. Nash, Equilibrium points in n-person games, Proc. Natl. Acad. Sci., 36:1(1950), 48-49.
[26] J.S. Pang and M. Fukushima, Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games, Comput. Manag. Sci., 2:1(2005), 21-56.
[27] U. Ravat and U.V. Shanbhag, On the charaterization of solution sets of smooth and nonsmooth convex stochastic Nash games, SIAM J. Optim., 21:3(2011), 1168-1199.
[28] R.T. Rockafellar and J. Sun, Solving monotone stochastic variational inequalities and complementarity problems by progressive hedging, Math. Program., 174(2019), 453-471.
[29] R.T. Rockafellar and S. Uryasev, Optimization of conditional value-at-risk, J. Risk, 2(2000), 493-517.
[30] R.T. Rockafellar and R.J.B. Wets, Variational Analysis, Springer, New York, 1998.
[31] R.T. Rockafellar and R.J.B. Wets, Stochastic variational inequalities:single-stage to multistage, Math. Program., 165:1(2017), 331-360.
[32] J. Rosen, Existence and uniqueness of equilibrium points for concave n-person games, Econometrica, 33:3(1965), 520-534.
[33] V. Singh, O. Jouini, and A. Lisser, Existence of nash equilibrium for chance-constrained games, Oper. Res. Lett., 44:5(2016), 640-644.
[34] V.V. Singh and A. Lisser, Variational inequality formulation for the games with random payoffs, J. Glob. Optim., 72:4(2018), 743-760.
[35] H. Sun and X. Chen, Two-stage stochastic variational inequalities:Theory, Algorithms and Applications, J. Oper. Res. Soc. China, 9(2019), 1-32.
[36] H. Xu and D. Zhang, Stochastic nash equilibrium problems:sample average approximation and applications, Comput. Optim. Appl., 55:3(2013), 597-645.
[1] Jie Jiang, Yun Shi, Xiaozhou Wang, Xiaojun Chen. REGULARIZED TWO-STAGE STOCHASTIC VARIATIONAL INEQUALITIES FOR COURNOT-NASH EQUILIBRIUM UNDER UNCERTAINTY [J]. Journal of Computational Mathematics, 2019, 37(6): 813-842.
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