Journal of Optimization Theory and Applications

, Volume 170, Issue 2, pp 687–709 | Cite as

The Cone Condition and Nonsmoothness in Linear Generalized Nash Games

  • Oliver Stein
  • Nathan Sudermann-Merx


We consider linear generalized Nash games and introduce the so-called cone condition, which characterizes the smoothness of a gap function that arises from a reformulation of the generalized Nash equilibrium problem as a piecewise linear optimization problem based on the Nikaido–Isoda function. Other regularity conditions such as the linear independence constraint qualification or the strict Mangasarian–Fromovitz condition are only sufficient for smoothness, but have the advantage that they can be verified more easily than the cone condition. Therefore, we present special cases, where these conditions are not only sufficient, but also necessary for smoothness of the gap function. Our main tool in the analysis is a global extension of the gap function that allows us to overcome the common difficulty that its domain may not cover the whole space.


Generalized Nash equilibrium problem Nikaido–Isoda function Piecewise linear function Constraint qualification Genericity Parametric optimization 

Mathematics Subject Classification

91A06 91A10 90C31 



We thank the anonymous referees for their precise and substantial remarks, which helped to significantly improve the paper. Furthermore, we would like to thank Christian Kanzow and Axel Dreves for fruitful discussions on the subject of this paper. This research was partially supported by the DFG (Deutsche Forschungsgemeinschaft) under grant STE 772/13-1.


  1. 1.
    Nash, J.: Non-cooperative games. Ann. Math. 54, 286–295 (1951)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Arrow, K.J., Debreu, G.: Existence of an equilibrium for a competitive economy. Econometrica 22, 265–290 (1954)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Debreu, G.: A social equilibrium existence theorem. Proc. Natl. Acad. Sci. 38, 886–893 (1952)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Facchinei, F., Kanzow, C.: Generalized Nash equilibrium problems. Ann. Oper. Res. 175, 177–211 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Fischer, A., Herrich, M., Schönefeld, K.: Generalized Nash equilibrium problems—recent advances and challenges. Pesquisa Oper. 34, 521–558 (2014)CrossRefGoogle Scholar
  6. 6.
    Schiro, D.A., Pang, J.-S., Shanbhag, U.V.: On the solution of affine generalized Nash equilibrium problems with shared constraints by Lemke’s method. Math. Program. 142, 1–46 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Dreves, A.: Finding all solutions of affine generalized Nash equilibrium problems with one-dimensional strategy sets. Math. Methods Oper. Res. 80, 139–159 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dreves, A., Facchinei, F., Kanzow, C., Sagratella, S.: On the solution of the KKT conditions of generalized Nash equilibrium problems. SIAM J. Optim. 21, 1082–1108 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Von Neumann, J., Morgenstern, O.: Theory of Games and Economic Behavior. Princeton University Press, Princeton (1944)zbMATHGoogle Scholar
  10. 10.
    Von Heusinger, A., Kanzow, C.: Optimization reformulations of the generalized Nash equilibrium problem using Nikaido-Isoda-type functions. Comput. Optimiz. Appl. 43, 353–377 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Auslender, A.: Optimisation: Méthodes Numériques. Masson, Paris (1976)zbMATHGoogle Scholar
  12. 12.
    Giannessi, F.: Separation of sets and gap functions for quasi-variational inequalities. In: Giannessi, F., Maugeri, A. (eds.) Variational Inequality and Network Equilibrium Problems, pp. 101–121. Plenum Press, New York (1995)CrossRefGoogle Scholar
  13. 13.
    Harms, N., Kanzow, C., Stein, O.: On differentiability properties of player convex generalized Nash equilibrium problems. Optimization 64, 365–388 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Stein, O., Sudermann-Merx, N.: On smoothness properties of optimal value functions at the boundary of their domain under complete convexity. Math. Methods Oper. Res. 79, 327–352 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Nikaido, H., Isoda, K.: Note on noncooperative convex games. Pac. J. Math. 5, 807–815 (1955)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Dreves, A., Kanzow, C., Stein, O.: Nonsmooth optimization reformulations of player convex generalized Nash equilibrium problems. J. Global Optim. 53, 587–614 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Danskin, J.M.: The Theory of Max-Min and its Applications to Weapons Allocation Problems. Springer, New York (1967)CrossRefzbMATHGoogle Scholar
  18. 18.
    Kyparisis, J.: On uniqueness of Kuhn-Tucker multipliers in nonlinear programming. Math. Program. 32, 242–246 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Wachsmuth, G.: On LICQ and the uniqueness of Lagrange multipliers. Operat. Res. Lett. 41, 78–80 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)CrossRefzbMATHGoogle Scholar
  21. 21.
    Cooper, W.W., Seiford, L.M., Tone, K.: Data Envelopment Analysis. Springer, New York (2007)zbMATHGoogle Scholar
  22. 22.
    Harms, N., Hoheisel, T., Kanzow, C.: On a smooth dual gap function for a class of player convex generalized Nash equilibrium problems. J. Optim. Theory Appl., online first, (2014) doi: 10.1007/s10957-014-0631-6
  23. 23.
    Jongen, HTh, Jonker, P., Twilt, F.: Nonlinear Optimization in Finite Dimensions. Kluwer, Dordrecht (2000)zbMATHGoogle Scholar
  24. 24.
    Stein, O.: Bi-Level Strategies in Semi-Infinite Programming. Kluwer, Boston (2003)CrossRefzbMATHGoogle Scholar
  25. 25.
    Ralph, D., Stein, O.: The C-Index: a new stability concept for quadratic programs with complementarity constraints. Math. Oper. Res. 36, 503–526 (2011)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Clarke, F.H.: Optimization and Nonsmooth Analysis. SIAM, Philadelphia (1990)CrossRefzbMATHGoogle Scholar
  27. 27.
    Van Hang, N.T., Yen, N.D.: On the problem of minimizing a difference of polyhedral convex functions under linear constraints. J. Optim. Theory Appl., online first, (2015). doi: 10.1007/s10957-015-0769-x
  28. 28.
    Izmailov, A.F., Solodov, M.V.: Critical Lagrange multipliers: what we currently know about them, how they spoil our lives, and what we can do about it. TOP 23, 1–26 (2015)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Karlsruhe Institute of TechnologyKarlsruheGermany

Personalised recommendations