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Some projection-like methods for the generalized Nash equilibria

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Abstract

A generalized Nash game is an m-person noncooperative game in which each player’s strategy depends on the rivals’ strategies. Based on a quasi-variational inequality formulation for the generalized Nash game, we present two projection-like methods for solving the generalized Nash equilibria in this paper. It is shown that under certain assumptions, these methods are globally convergent. Preliminary computational experience is also reported.

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References

  1. Arrow, K.J., Debreu, G.: Existence of an equilibrium for a competitive economy. Econometrica 22, 265–290 (1954)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bensoussan, A.: Points de Nash dans le cas de fontionnelles quadratiques et jeux differentiels linéaires a N persons. SIAM J. Control 12, 460–499 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  3. Cash, J.: Solution set in a special case of generalized Nash equilibrium games. Kybernetika 37, 21–37 (2001)

    MathSciNet  Google Scholar 

  4. Cournot, A.A.: Researches into the Mathematical Principles of the Theory of Wealth. Macmillan, New York (1897)

    Google Scholar 

  5. Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequality and Complementarity Problems. Springer, New York (2003)

    Google Scholar 

  6. Gafni, E.M., Bertsekas, D.P.: Two-metric projection problems and descent methods for asymmetric variational inequality problems. Math. Program. 53, 99–110 (1984)

    Google Scholar 

  7. Harker, P.T.: Generalized Nash games and quasi-variational inequalities. Eur. J. Oper. Res. 54, 81–94 (1991)

    Article  MATH  Google Scholar 

  8. Iusem, A.N.: An iterative algorithm for the variational inequality problem. Math. Appl. Comput. 13, 103–114 (1994)

    MATH  MathSciNet  Google Scholar 

  9. Khobotov, E.N.: Modification of the extragradient method for solving variational inequalities and certain optimization problem. U.S.S.R. Comput. Math. Math. Phys. 17, 120–127 (1987)

    Article  Google Scholar 

  10. Kocvara, M., Outrata, J.V.: On a class of quasi-variational inequalities. Optim. Methods Software 5, 275–295 (1995)

    Article  Google Scholar 

  11. Korpelevich, G.M.: The extragradient method for finding saddle points and other problems. Matecon 12, 747–756 (1976)

    MATH  Google Scholar 

  12. Marcotte, P.: Application of Khobotov’s algorithm to variational inequalities and network equilibrium problems. Inf. Syst. Oper. Res. 29, 258–270 (1991)

    MATH  Google Scholar 

  13. McKenzie, L.W.: On the existence of general equilibrium for a competitive market. Econometrica 27, 54–71 (1959)

    Article  MATH  MathSciNet  Google Scholar 

  14. Nash, J.F.: Equilibrium points in n-person games. Proc. Nat. Acad. Sci. U.S.A. 36, 48–49 (1950)

    Article  MATH  MathSciNet  Google Scholar 

  15. Nash, J.F.: Non-cooperative games. Ann. Math. 54, 286–295 (1951)

    Article  MathSciNet  Google Scholar 

  16. Outrata, J., Zowe, J.: A numerical approach to optimization problems with variational inequality constraints. Math. Program. 68, 105–130 (1995)

    MathSciNet  Google Scholar 

  17. Pang, J.S.: Computing generalized Nash equilibria. Department of Mathematical sciences, The Johns Hopkins University (2002). Manuscript

  18. Pang, J.S., Fukushima, M.: Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games. Comput. Manage. Sci. 2, 21–56 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  19. Robinson, S.M.: Shadow prices for measures of effectiveness. I. Linear Model. Oper. Res. 41, 518–535 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  20. Robinson, S.M.: Shadow prices for measures of effectiveness. II. General Model. Oper. Res. 41, 536–548 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  21. Sun, D.: An iterative method for solving variational inequality problems and complementarity problems. Numer. Math. J. Chin. Univ. 16, 145–153 (1994)

    MATH  Google Scholar 

  22. Toint, Ph.L.: Global convergence of a class of trust region methods for nonconvex minimization in Hilbert space. IMA J. Numer. Anal. 8, 231–252 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  23. von Heusinger, A., Kanzow, C.: Optimization reformulations of the generalized Nash equilibrium problem using Nikaido-Isoda-type functions. Comput. Optim. Appl. (2007). doi: 10.1007/s10589-007-9145-6

    Google Scholar 

  24. von Neumann, J., Morgenstern, O.: Theory of Games and Economic Behavior. Princeton University Press, Princeton (1944)

    MATH  Google Scholar 

  25. Wang, Y.J., Xiu, N.H., Wang, C.Y.: A new version of extragradient method for variational inequality problems. Comput. Math. Appl. 42, 969–979 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  26. Wei, J.Y., Smeers, Y.: Spatial oligopolistic electricity models with Cournot generators and regulated transmission prices. Oper. Res. 47, 102–112 (1999)

    Article  MATH  Google Scholar 

  27. Xiu, N.H., Zhang, J.Z.: Some recent advances in projection-type methods for variational inequalities. J. Comput. Appl. Math. 152, 559–585 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  28. Xiu, N.H., Wang, Y.J., Zhang, X.S.: Modified fixed-point equations and related iterative methods for variational inequalities. Comput. Math. Appl. 47, 913–920 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  29. Zarantonello, E.H.: Projections on convex sets in Hilbert space and spectral theory. In: Zarantonello, E.H. (ed.) Contributions to Nonlinear Functional Analysis. Academic Press, New York (1971)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, City University of Hong Kong, Hong Kong, Hong Kong

    Jianzhong Zhang

  2. Institute of Operations Research, Qufu Normal University, Qufu, China

    Biao Qu

  3. Department of Applied Mathematics, Beijing Jiaotong University, Beijing, China

    Biao Qu & Naihua Xiu

Authors
  1. Jianzhong Zhang
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  2. Biao Qu
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  3. Naihua Xiu
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Corresponding author

Correspondence to Jianzhong Zhang.

Additional information

This research was supported by Hong Kong University Grant Council under the CERG Project CityU and CUHK 103105, and the National Natural Science Foundation of China (No. 70471002, 10571106, 10701047).

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Zhang, J., Qu, B. & Xiu, N. Some projection-like methods for the generalized Nash equilibria. Comput Optim Appl 45, 89–109 (2010). https://doi.org/10.1007/s10589-008-9173-x

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  • DOI: https://doi.org/10.1007/s10589-008-9173-x

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