Newton's method for computing a normalized equilibrium in the generalized Nash game through fixed point formulation

Mathematical Programming

Volumn 132, Issue 1-2, 2012, Pages 99-123

  Von Heusinger, Anna ^a   Kanzow, Christian ^a   Fukushima, Masao ^b  

^a UNIVERSITY OF WÜRZBURG   (Germany)

^b KYOTO UNIVERSITY   (Japan)

Author keywords

Computable generalized Jacobian; Constant rank constraint qualification; Fixed point characterization; Generalized Nash equilibrium problem; Local superlinear quadratic convergence; Nonsmooth Newton method; Normalized equilibrium

Indexed keywords

CONSTANT RANK; FIXED-POINT CHARACTERIZATION; GENERALIZED JACOBIAN; GENERALIZED NASH EQUILIBRIUM; LOCAL SUPERLINEAR/QUADRATIC CONVERGENCE; NONSMOOTH NEWTON METHOD;

FUNCTIONS; GAME THEORY; NUMERICAL METHODS;

NEWTON-RAPHSON METHOD;

EID: 84858439352     PISSN: 00255610     EISSN: 14364646     Source Type: Journal    
DOI: 10.1007/s10107-010-0386-2     Document Type: Article

References (33)

1. E. Allgower K. Georg 1980 Simplicial and continuation methods for approximating fixed points and solutions to systems of equations SIAM Rev. 22 28 85 554709 0432.65027 10.1137/1022003
2. K.J. Arrow G. Debreu 1954 Existence of an equilibrium for a competitive economy Econometrica 22 265 290 77069 0055.38007 10.2307/1907353
3. Aubin J.-P.: Mathematical Methods of Game and Economic Theory. North-Holland, Amsterdam (1979)
4. S.C. Billups 2002 A homotopy-based algorithm for mixed complementarity problems SIAM J. Optim. 12 583 605 1884907 1056.90132 10.1137/S1052623498337431
5. Clarke, F.H.: Optimization and Nonsmooth Analysis. John Wiley, New York (1983) reprinted by SIAM, Philadelphia (1990)
6. J. Contreras M. Klusch J.B. Krawczyk 2004 Numerical solutions to Nash-Cournot equilibria in coupled constraint electricity markets IEEE Trans. Power Syst. 19 195 206 10.1109/TPWRS.2003.820692
7. G. Debreu 1952 A social equilibrium existence theorem Proc. Natl. Acad. Sci. 38 10 886 893 50251 0047.38804 10.1073/pnas.38.10.886
8. S. Dempe D. Pallaschke 1997 Quasidifferentiability of optimal solutions in parametric nonlinear optimization Optimization 40 1 24 1442029 0872.90091 10.1080/02331939708844295
9. S. Dempe S. Vogel 2001 The generalized Jacobian of the optimal solution in parametric optimization Optimization 50 387 405 1892911 1054.90074 10.1080/02331930108844570 (Pubitemid 33817601)
10. F. Facchinei A. Fischer V. Piccialli 2009 Generalized Nash equilibrium problems and Newton methods Math. Program. 117 163 194 2421304 1166.90015 10.1007/s10107-007-0160-2
11. F. Facchinei C. Kanzow 2007 Generalized Nash equilibrium problems 4OR Q. J. Oper. Res. 5 173 210 2353415 1211.91162 10.1007/s10288-007-0054-4
12. Fukushima, M.: Restricted generalized Nash equilibria and controlled penalty algorithm. Comput. Manag. Sci. (to appear)
13. S. Govindan R. Wilson 2003 A global Newton method to compute Nash equilibria J. Econ. Theory 110 65 86 1984254 1042.91001 10.1016/S0022-0531(03) 00005-X (Pubitemid 36802525)
14. P.J.-J. Herings R. Peeters 2010 Homotopy methods to compute equilibria in game theory Econ. Theory 42 119 156 2551731 1185.91028 10.1007/s00199-009-0441- 5
15. 1 optimization reformulations of the generalized Nash equilibrium problem Optim. Methods Softw. 23 953 973 2464695 1154.91356 10.1080/10556780802319370
16. A. von Heusinger C. Kanzow 2009 Optimization reformulations of the generalized Nash equilibrium problem using Nikaido-Isoda-type functions Comput. Optim. Appl. 43 353 377 2511919 1170.90495 10.1007/s10589-007-9145-6
17. A. von Heusinger C. Kanzow 2009 Relaxation methods for generalized Nash equilibrium problems with inexact line search J. Optim. Theory Appl. 143 159 183 2545947 1182.91024 10.1007/s10957-009-9553-0
18. W.W. Hogan 1973 Point-to-set maps in mathematical programming SIAM Rev. 15 591 603 345641 0256.90042 10.1137/1015073
19. R. Janin 1984 Directional derivative of the marginal function in nonlinear programming Math. Program. Study 21 110 126 751246 0549.90082 10.1007/BFb0121214 (Pubitemid 14621990)
20. A. Kesselman S. Leonardi V. Bonifaci 2005 Game-theoretic analysis of internet switching with selfish users Lect. Notes Comput. Sci. 3828 236 245 10.1007/11600930-23 (Pubitemid 43849261)
21. 1 equations J. Oper. Res. Soc. Jpn. 29 352 372 875584 0611.65032
22. J.B. Krawczyk 2005 Coupled constraint Nash equilibria in environmental games Resour. Energy Econ. 27 157 181 10.1016/j.reseneeco.2004.08.001 (Pubitemid 40621220)
23. J.B. Krawczyk S. Uryasev 2000 Relaxation algorithms to find Nash equilibria with economic applications Environ. Model. Assess. 5 63 73 10.1023/A:1019097208499
24. Nabetani, K.; Tseng, P.; Fukushima, M.: Parametrized variational inequality approaches to generalized Nash equilibrium problems with shared constraints. Comput. Optim. Appl. (to appear)
25. J. Nash 1951 Non-cooperative games Ann. Math. 54 286 295 43432 0045.08202 10.2307/1969529
26. H. Nikaido K. Isoda 1955 Note on noncooperative convex games Pac. J. Math. 5 807 815 73910
27. Outrata J.; Kocvara M.; Zowe J.: Nonsmooth Approach to Optimization Problems with Equilibrium Constraints. Kluwer, Dordrecht (1998)
28. J. Outrata J. Zowe 1995 A numerical approach to optimization problems with variational inequality constraints Math. Program. 68 105 130 1312107 0835.90093
29. J.-S. Pang D. Ralph 1996 Piecewise smoothness, local invertibility, and parametric analysis of normal maps Math. Oper. Res. 21 401 426 1397221 0857.90122 10.1287/moor.21.2.401 (Pubitemid 126600327)
30. L. Qi 1993 Convergence analysis of some algorithms for solving nonsmooth equations Math. Oper. Res. 18 227 244 1250115 0776.65037 10.1287/moor.18.1.227
31. J.B. Rosen 1965 Existence and uniqueness of equilibrium points for concave N-person games Econometrica 33 520 534 194210 0142.17603 10.2307/1911749
32. Sun D.; Fukushima M.; Qi L.: A computable generalized Hessian of the D-gap function and Newton-type methods for variational inequality problems. In: Ferris, M.C.; Pang, J.-S. (eds) Complementarity and Variational Problems: State of the Art, pp. 452-472. SIAM, Philadelphia (1997)
33. S. Uryasev R.Y. Rubinstein 1994 On relaxation algorithms in computation of noncooperative equilibria IEEE Trans. Autom. Control 39 1263 1267 1283705 0811.90117 10.1109/9.293193