Abstract
In this note, we consider a class of variational inequalities on probabilistic Lebesgue spaces, where the constraints are satisfied on average, and provide an approximation procedure for the solutions. As an application, we investigate the Nash–Cournot oligopoly problem with uncertain data and compare the solutions obtained when the constraints are satisfied on average with the ones obtained when the constraints are satisfied almost surely.
Similar content being viewed by others
References
Chen, X., Fukushima, M.: Expected residual minimization method for stochastic linear complementarity problems. Math. Oper. Res. 30, 1022–1038 (2005)
Chen, X., Zhang, C., Fukushima, M.: Robust solution of monotone stochastic linear complementarity problems. Math. Program. B 117, 51–80 (2009)
Patriksson, M.: On the applicability and solution of bilevel optimization models in transportation science: a study on the existence, stability and computation of optimal solutions to stochastic mathematical programs with equilibrium constraints. Trans. Res. B 42, 843–860 (2008)
Patriksson, M., Wynter, L.: Stochastic mathematical programs with equilibrium constraints. Oper. Res. Lett. 25, 159–167 (1999)
Ravat, U., Shanbhag, U.V.: On the characterization of solution sets of smooth and nonsmooth stochastic Nash games. In: Proceedings of the American Control Conference (ACC), Baltimore (2010)
Ravat, U., Shanbhag, U.V.: On the characterization of solution sets of smooth and nonsmooth convex stochastic Nash games. SIAM J. Optim. 21, 1168–1199 (2011)
Ravat, U., Shanbhag, U.V.: On the existence of solutions to stochastic variational inequality and complementarity problems. arXiv:1306.0586v1 [math.OC] (2013)
Lu, S., Budhiraja, A.: Confidence regions for stochastic variational inequalities. Math. Oper. Res. 38, 545–568 (2013)
Gürkan, G., Özge, A.Y., Robinson, S.M.: Sample-path solution of stochastic variational inequalities. Math. Program. 84, 313–333 (1999)
Shapiro, A., Xu, H.: Stochastic mathematical programs with equilibrium constraints, modelling and sample average approximation. Optimization 57, 395–418 (2008)
Daniele, P., Giuffré, S.: Random variational inequalities and the random traffic equilibrium problem. J. Optim. Theory Appl. 167, 363–381 (2015)
Gwinner, J., Raciti, F.: On a class of random variational inequalities on random sets. Numer. Funct. Anal. Optim. 27, 619–636 (2006)
Gwinner, J., Raciti, F.: On monotone variational inequalities with random data. J. Math. Inequal. 3, 443–453 (2009)
Gwinner, J., Raciti, F.: Some equilibrium problems under uncertainty and random variational inequalities. Ann. Oper. Res. 200, 299–319 (2012)
Patriche, M.: Equilibrium of Bayesian fuzzy economies and quasi-variational inequalities with random fuzzy mappings. J. Inequal. Appl. 2013, 374 (18 pages) (2013). doi:10.1186/1029-242X-2013-374
Jadamba, B., Khan, A.A., Raciti, F.: Regularization of stochastic variational inequalities and a comparison of an \(L_p\) and a sample-path approach. Nonlinear Anal. 94, 65–83 (2014)
Gwinner, J., Raciti, F.: Random equilibrium problems on networks. Math. Comput. Model. 43, 880–891 (2006)
Jadamba, B., Raciti, F.: On the modelling of some environmental games with uncertain data. J. Optim. Theory Appl. 167, 959–968 (2015)
Jadamba, B., Raciti, F.: Variational inequality approach to stochastic Nash equilibrium problems with an application to Cournot oligopoly. J. Optim. Theory Appl. 165, 1050–1070 (2015)
Jadamba, B., Raciti, F.: A stochastic model of oligopolistic market equilibrium problems. In: Optimization in Science and Engineering, pp. 263–271. Springer, New York (2014)
Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and their Applications. Academic Press, New York (1980)
Jacod, J., Protter, P.: Probability Essentials, 2nd edn. Springer, Berlin (2002)
Mikosch, T.: Elementary Stochastic Calculus with Finance in View. World Scientific, Singapore (1998)
Mosco, U.: Convergence of convex sets and of solutions of variational inequalities. Adv. Math. 3, 510–585 (1969)
Gwinner, J.: A class of random variational inequalities and simple random unilateral boundary value problems - existence, discretization, finite elements approximations. Stoch. Anal. Appl. 18, 967–993 (2000)
Barbagallo, A., Mauro, P.: Evolutionary variational formulation for oligopolistic market equilibrium problems with production excesses. J. Optim. Theory Appl. 155, 288–314 (2012)
Gabay, D., Moulin, H.: On the uniqueness and stability of Nash equilibria in non cooperative games. In: Applied Stochastic Control in Econometrics and Management Sciences, pp. 271–293. North Holland, Amsterdam (1980)
Marcotte, P.: Advantages and drawback of variational inequalities formulations. In: Variational Inequalities and Network Equilibrium Problems, pp. 179–194. Plenum, New York (1995)
Murphy, F.H., Sherali, H., Soyster, A.L.: A mathematical programming approach for determining oligopolistic market equilibrium. Math. Program. 24, 92–106 (1982)
Bonenti, F., Oggioni, G., Allevi, E., Marangoni, G.: Evaluating the EU ETS impacts on profits, investments and prices of the Italian electricity market. Energy Policy 59, 242–256 (2013)
Faraci, F., Raciti, F.: On generalized Nash equilibrium in infinite dimension: the Lagrange multipliers approach. Optimization 64, 321–338 (2015)
Nagurney, A.: Network Economics: A Variational Inequality Approach. Kluwer Academic Publishers, Dordrecht (1993)
Raciti, F., Falsaperla, P.: Improved non-iterative algorithm for the calculation of the equilibrium in the traffic network problem. J. Optim. Theory. Appl. 133, 401–411 (2007)
Falsaperla, P., Raciti, F.: Global approximation of Lipschitz solutions of time-dependent variational inequalities with error estimates. Numer. Funct. Anal. Optim. 35, 1018–1042 (2014)
Dentcheva, D., Ruszczyński, A.: Optimization with stochastic dominance constraints. SIAM J. Optim. 14, 548–566 (2003)
Acknowledgments
The work of one of the authors, F.R., has been partially supported by GNAMPA-INdAM.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Fabian Flores-Bazàn.
Rights and permissions
About this article
Cite this article
Faraci, F., Jadamba, B. & Raciti, F. On Stochastic Variational Inequalities with Mean Value Constraints. J Optim Theory Appl 171, 675–693 (2016). https://doi.org/10.1007/s10957-016-0888-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10957-016-0888-z
Keywords
- Stochastic variational inequalities
- Nash equilibrium
- Cournot oligopoly
- Mosco convergence
- Probabilistic constraints