Computational Optimization and Applications

, Volume 43, Issue 3, pp 353–377 | Cite as

Optimization reformulations of the generalized Nash equilibrium problem using Nikaido-Isoda-type functions

  • Anna von HeusingerEmail author
  • Christian Kanzow


We consider the generalized Nash equilibrium problem which, in contrast to the standard Nash equilibrium problem, allows joint constraints of all players involved in the game. Using a regularized Nikaido-Isoda-function, we then present three optimization problems related to the generalized Nash equilibrium problem. The first optimization problem is a complete reformulation of the generalized Nash game in the sense that the global minima are precisely the solutions of the game. However, this reformulation is nonsmooth. We then modify this approach and obtain a smooth constrained optimization problem whose global minima correspond to so-called normalized Nash equilibria. The third approach uses the difference of two regularized Nikaido-Isoda-functions in order to get a smooth unconstrained optimization problem whose global minima are, once again, precisely the normalized Nash equilibria. Conditions for stationary points to be global minima of the two smooth optimization problems are also given. Some numerical results illustrate the behaviour of our approaches.


Generalized Nash equilibria Normalized Nash equilibria Joint constraints Regularized Nikaido-Isoda-function Constrained optimization reformulation Unconstrained optimization reformulation 


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  1. 1.
    Adida, E., Perakis, G.: Dynamic pricing and inventory control: uncertainty and competition. Part B: an algorithm for the normalized Nash equilibrium. Operations Research Center, Sloan School of Management, MIT, Technical Report (December 2005) Google Scholar
  2. 2.
    Adida, E., Perakis, G.: Dynamic pricing and inventory control: uncertainty and competition. Part A: existence of Nash equilibrium. Operations Research Center, Sloan School of Management, MIT, Technical Report (January 2006) Google Scholar
  3. 3.
    Başar, T.: Relaxation techniques and asynchronous algorithms for on-line computation of non-cooperative equilibria. J. Econ. Dyn. Control 11, 531–549 (1987) zbMATHCrossRefGoogle Scholar
  4. 4.
    Barzilai, J., Borwein, J.M.: Two point step size gradient method. IMA J. Numer. Anal. 8, 141–148 (1988) zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bensoussan, A.: Points de Nash dans le cas de fonctionelles quadratiques et jeux differentiels lineaires a N personnes. SIAM J. Control 12, 460–499 (1974) zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Contreras, J., Klusch, M., Krawczyk, J.B.: Numerical solutions to Nash-Cournot equilibria in coupled constraint electricity markets. IEEE Trans. Power Syst. 19, 195–206 (2004) CrossRefGoogle Scholar
  7. 7.
    Dai, Y.H., Liao, L.Z.: R-linear convergence of the Barzilai and Borwein gradient method. IMA J. Numer. Anal. 22, 1–10 (2002) zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Facchinei, F., Pang, J.-S.: Finite-Dimensional Variational Inequalities and Complementarity Problems, vol. I. Springer, New York (2003) Google Scholar
  9. 9.
    Facchinei, F., Pang, J.-S.: Finite-Dimensional Variational Inequalities and Complementarity Problems, vol. II. Springer, New York (2003) Google Scholar
  10. 10.
    Facchinei, F., Fischer, A., Piccialli, V.: On generalized Nash games and variational inequalities. Oper. Res. Lett. 35, 159–164 (2007) zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Facchinei, F., Fischer, A., Piccialli, V.: Generalized Nash equilibrium problems and Newton methods. Math. Program. Ser. B (2007). doi: 10.1007/s10107-007-0160-2 Google Scholar
  12. 12.
    Flåm, S.D., Antipin, A.S.: Equilibrium programming using proximal-like algorithms. Math. Program. 78, 29–41 (1997) CrossRefGoogle Scholar
  13. 13.
    Flåm, S.D., Ruszczyński, A.: Noncooperative convex games: computing equilibria by partial regularization. IIASA, Laxenburg, Austria, Working Paper 94-42 (May 1994) Google Scholar
  14. 14.
    Fletcher, R.: On the Barzilai-Borwein method. Department of Mathematics, University of Dundee, Dundee, United Kingdom, Numerical Analysis Report NA/207 (2001) Google Scholar
  15. 15.
    Fukushima, M.: Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems. Math. Program. 53, 99–110 (1992) zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Grippo, L., Sciandrone, M.: Nonmonotone globalization techniques for the Barzilai-Borwein gradient method. Comput. Optim. Appl. 23, 143–169 (2002) zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Gürkan, G., Pang, J.-S.: Approximations of Nash equilibria. Math. Program. Ser. B (2007). doi: 10.1007/s10107-007-0156-y zbMATHGoogle Scholar
  18. 18.
    Harker, P.T.: Generalized Nash games and quasivariational inequalities. Eur. J. Oper. Res. 54, 81–94 (1991) zbMATHCrossRefGoogle Scholar
  19. 19.
    Hobbs, B.F.: Linear complementarity models of Nash-Cournot competition in bilateral and POOLCO power markets. IEEE Trans. Power Syst. 16, 194–202 (2001) CrossRefGoogle Scholar
  20. 20.
    Hogan, W.W.: Point-to-set maps in mathematical programming. SIAM Rev. 15, 591–603 (1973) zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Kanzow, C., Fukushima, M.: Theoretical and numerical investigation of the D-gap function for box constrained variational inequalities. Math. Program. 83, 55–87 (1998) MathSciNetGoogle Scholar
  22. 22.
    Kesselman, A., Leonardi, S., Bonifaci, V.: Game-theoretic analysis of Internet switching with selfish users. In: Lecture Notes in Computer Science, vol. 3828, pp. 236–245 (2005) Google Scholar
  23. 23.
    Krawczyk, J.B.: Coupled constraint Nash equilibria in environmental games. Resour. Energy Econ. 27, 157–181 (2005) CrossRefGoogle Scholar
  24. 24.
    Krawczyk, J.B., Uryasev, S.: Relaxation algorithms to find Nash equilibria with economic applications. Environ. Model. Assess. 5, 63–73 (2000) CrossRefGoogle Scholar
  25. 25.
    Li, S., Başar, T.: Distributed algorithms for the computation of noncooperative equilibria. Automatica 23, 523–533 (1987) zbMATHCrossRefGoogle Scholar
  26. 26.
    Mastroeni, G.: Gap functions for equilibrium problems. J. Global Optim. 27, 411–426 (2003) zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Nikaido, H., Isoda, K.: Note on noncooperative convex games. Pac. J. Math. 5, 807–815 (1955) MathSciNetGoogle Scholar
  28. 28.
    Pang, J.-S., Fukushima, M.: Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games. Comput. Manag. Sci. 2, 21–56 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Pang, J.-S., Qi, L.: Nonsmooth equations: motivation and algorithms. SIAM J. Optim. 3, 443–465 (1993) zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Peng, J.-M.: Equivalence of variational inequality problems to unconstrained optimization. Math. Program. 78, 347–356 (1997) Google Scholar
  31. 31.
    Qi, L.: Convergence analysis of some algorithms for solving nonsmooth equations. Math. Oper. Res. 18, 227–244 (1993) zbMATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Qi, L., Sun, J.: A nonsmooth version of Newton’s method. Math. Program. 58, 353–367 (1993) CrossRefMathSciNetGoogle Scholar
  33. 33.
    Raydan, M.: On the Barzilai and Borwein choice of the steplength for the gradient method. IMA J. Numer. Anal. 13, 321–326 (1993) zbMATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Raydan, M.: The Barzilai and Borwein gradient method for the large scale unconstrained minimization problem. SIAM J. Optim. 7, 26–33 (1997) zbMATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Rosen, J.B.: Existence and uniqueness of equilibrium points for concave N-person games. Econometrica 33, 520–534 (1965) zbMATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Uryasev, S., Rubinstein, R.Y.: On relaxation algorithms in computation of noncooperative equilibria. IEEE Trans. Autom. Control 39, 1263–1267 (1994) zbMATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    Yamashita, N., Taji, K., Fukushima, M.: Unconstrained optimization reformulations of variational inequality problems. J. Optim. Theory Appl. 92, 439–456 (1997) zbMATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    Zhang, L., Han, J.: Unconstrained optimization reformulation of equilibrium problems. Institute of Applied Mathematics, Chinese Academy of Sciences, Beijing, China, Technical Report (2006) Google Scholar

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© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Institute of MathematicsUniversity of WürzburgWürzburgGermany

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