Journal of Global Optimization

, Volume 53, Issue 4, pp 587–614 | Cite as

Nonsmooth optimization reformulations of player convex generalized Nash equilibrium problems



Using a regularized Nikaido-Isoda function, we present a (nonsmooth) constrained optimization reformulation of the player convex generalized Nash equilibrium problem (GNEP). Further we give an unconstrained reformulation of a large subclass of player convex GNEPs which, in particular, includes the jointly convex GNEPs. Both approaches characterize all solutions of a GNEP as minima of optimization problems. The smoothness properties of these optimization problems are discussed in detail, and it is shown that the corresponding objective functions are continuous and piecewise continuously differentiable under mild assumptions. Some numerical results based on the unconstrained optimization reformulation being applied to player convex GNEPs are also included.


Generalized Nash equilibrium problem Jointly convex Player convex Optimization reformulation Continuity PC1mapping Constant rank constraint qualification 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Burke J.V., Lewis A.S., Overton M.L.: A robust gradient sampling algorithm for nonsmooth, nonconvex optimization. SIAM J. Optim. 15, 751–779 (2005)CrossRefGoogle Scholar
  2. 2.
    Chaney R.W.: Piecewise C k functions in nonsmooth analysis. Nonlinear Anal. Theory Methods Appl. 15, 649–660 (1990)CrossRefGoogle Scholar
  3. 3.
    Chinchuluun A., Pardalos P.M., Migdalas A., Pitsoulis L.: Pareto Optimality, Game Theory and Equilibria. Optimization and Its Applications, vol. 17. Springer, Berlin (2008)CrossRefGoogle Scholar
  4. 4.
    Clarke F.H.: Optimization and Nonsmooth Analysis. SIAM, Philadelphia (1990)CrossRefGoogle Scholar
  5. 5.
    Dreves, A., Kanzow, C.: Nonsmooth optimization reformulations characterizing all solutions of jointly convex generalized Nash equilibrium problems. Comput. Optim. Appl. (to appear)Google Scholar
  6. 6.
    Facchinei, F., Fischer, A., Piccialli, V.: Generalized Nash equilibrium problems and Newton methods. Math. Program. 117, 163–194 doi:10.1007/s10589-009-9314-x
  7. 7.
    Facchinei F., Kanzow C.: Generalized Nash equilibrium problems. Ann. Oper. Res. 175, 177–211 (2010)CrossRefGoogle Scholar
  8. 8.
    Facchinei F., Kanzow C.: Penalty methods for the solution of generalized Nash equilibrium problems. SIAM J. Optim. 20, 2228–2253 (2010)CrossRefGoogle Scholar
  9. 9.
    Facchinei F., Pang J.-S.: Finite-Dimensional Variational Inequalities and Complementarity Problems I. Springer, New York (2003)Google Scholar
  10. 10.
    Facchinei F., Pang J.-S.: Exact penalty functions for generalized Nash problems. In: Di Pillo, G., Roma, M. (eds) Large Scale Nonlinear Optimization, pp. 115–126. Springer, Berlin (2006)CrossRefGoogle Scholar
  11. 11.
    Facchinei F., Pang J.-S.: Nash equilibria: the variational approach. In: Palomar, D.P., Eldar, Y.C. (eds) Convex Optimization in Signal Processing and Communications, pp. 443–493. Cambridge University Press, Cambridge (2010)Google Scholar
  12. 12.
    Facchinei, F., Sagratella, S.: On the computation of all solutions of jointly convex generalized Nash equilibrium problems. Optim. Lett. (to appear)Google Scholar
  13. 13.
    Forgó, F., Szép, J., Szidarovszky, F.: Introduction to the Theory of Games. Concepts, Methods, Applications. Kluwer Academic Publishers, Dordrecht, Netherlands (1999). doi:10.1007/s11590-010-0218-6
  14. 14.
    Fukushima, M.: Restricted generalized Nash equilibria and controlled penalty algorithm. Comput. Manag. Sci. (to appear)Google Scholar
  15. 15.
    Fudenberg, D., Tirole, J.: Game Theory. MIT Press, Cambridge, MA (1991). doi:10.1007/s10287-009-0097-4
  16. 16.
    Gürkan G., Pang J.S.: Approximations of Nash equilibria. Math. Program. 117, 223–253 (2009)CrossRefGoogle Scholar
  17. 17.
    Hogan W.W.: Point-to-set maps in mathematical programming. SIAM Rev. 15, 591–603 (1973)CrossRefGoogle Scholar
  18. 18.
    Janin R.: Directional derivative of the marginal function in nonlinear programming. Math. Program. Study 21, 110–126 (1984)CrossRefGoogle Scholar
  19. 19.
    Kesselman A., Leonardi S., Bonifaci V.: Game-theoretic analysis of internet switching with selfish users. Lect. Notes Comput. Sci. 3828, 236–245 (2005). doi:10.1007/s10107-010-0386-2 CrossRefGoogle Scholar
  20. 20.
    Myerson R.B.: Game Theory. Analysis of Conflict. Harvard University Press, Cambridge, MA (1991)Google Scholar
  21. 21.
    Nabetani K., Tseng P., Fukushima M.: Parametrized variational inequality approaches to generalized Nash equilibrium problems with shared constraints. Comput. Optim. Appl. 48, 423–452 (2011)CrossRefGoogle Scholar
  22. 22.
    Nagurney, A.: Oligopolistic market equilibrium. In: Floudas, C.A., Pardalos, P.M. (eds.) Encyclopedia of Optimization, 2nd edn, pp. 2691–2694 (2009)Google Scholar
  23. 23.
    Nisan, N., Roughgarden, T., Tardos, E., Vazirani, V.V. (eds): Algorithmic Game Theory. Cambridge University Press, New York (2007)Google Scholar
  24. 24.
    Pang, J.-S.: Computing generalized Nash equilibria. Technical Report, Department of Mathematical Sciences, The Johns Hopkins University, Baltimore, MD (October 2002)Google Scholar
  25. 25.
    Pang J.-S., Fukushima M.: Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games. Comput. Manag. Sci. 2, 21–56 (2005)CrossRefGoogle Scholar
  26. 26.
    Pang J.-S., Ralph D.: Piecewise smoothness, local invertibility, and parametric analysis of normal maps. Math. Oper. Res. 21, 401–426 (1996)CrossRefGoogle Scholar
  27. 27.
    Peters H.: Game Theory. A Multi-Leveled Approach. Springer, Berlin, Germany (2008)Google Scholar
  28. 28.
    Ralph D., Dempe S.: Directional derivatives of the solution of a parametric nonlinear program. Math. Program. 70, 159–172 (1995)Google Scholar
  29. 29.
    Rockafellar R.T., Wets R.J.-B.: Variational Analysis. A Series of Comprehensive Studies in Mathematics, vol. 317. Springer, Berlin, Heidelberg (1998)Google Scholar
  30. 30.
    van Damme E.: Stability and Perfection of Nash Euilibria, 2nd edn. Springer, Berlin (1996)Google Scholar
  31. 31.
    von Heusinger A., Kanzow C.: Optimization reformulations of the generalized Nash equilibrium problem using Nikaido-Isoda-type functions. Comput. Optim. Appl. 43, 353–377 (2009)CrossRefGoogle Scholar
  32. 32.
    von Heusinger, A., Kanzow, C., Fukushima, M.: Newton’s method for computing a normalized equilibrium in the generalized Nash game through fixed point formulation. Math. Program. (to appear)Google Scholar

Copyright information

© Springer Science+Business Media, LLC. 2011

Authors and Affiliations

  1. 1.Institute of MathematicsUniversity of WürzburgWürzburgGermany
  2. 2.Karlsruhe Institute of TechnologyInstitute of Operations ResearchKarlsruheGermany

Personalised recommendations