Penalty methods for the solution of generalized Nash equilibrium problems

SIAM Journal on Optimization

Volumn 20, Issue 5, 2010, Pages 2228-2253

  Facchinei, Francisco ^a   Kanzow, Christian ^b  

^a SAPIENZA UNIVERSITY OF ROME   (Italy)

^b UNIVERSITY OF WÜRZBURG   (Germany)

Author keywords

Exact penalty function; Generalized Nash equilibrium problem; Global convergence; Jointly convex problem; Nash equilibrium problem

Indexed keywords

CONVEX PROBLEMS; EXACT PENALTIES; GENERALIZED NASH EQUILIBRIUM; GLOBAL CONVERGENCE; NASH EQUILIBRIUM;

NUMERICAL ANALYSIS;

CONVERGENCE OF NUMERICAL METHODS;

EID: 77956081926     PISSN: 10526234     EISSN: None     Source Type: Journal    
DOI: 10.1137/090749499     Document Type: Article

References (40)

1. K. J. Arrow and G. Debreu, Existence of an equilibrium for a competitive economy, Econometrica, 22 (1954), pp. 265-290.
2. A. Bensoussan, Points de Nash dans le cas de fonctionelles quadratiques et jeux differentiels lineaires a N personnes, SIAM J. Control, 12 (1974), pp. 460-499.
3. D. P. Bertsekas, Nonlinear Programming, 2nd ed., Athena Scientific, Belmont, MA, 1999.
4. J. V. Burke, A robust trust region method for constrained nonlinear programming problems, SIAM J. Optim., 2 (1992), pp. 325-347.
5. F. H. Clarke, Optimization and Nonsmooth Analysis, John Wiley, New York, 1983.
6. J. Contreras, M. Klusch, and J. B. Krawczyk, Numerical solutions to Nash-Cournot equilibria in coupled constraint electricity markets, IEEE Trans. Power Syst., 19 (2004), pp. 195-206.
7. G. Debreu, A social equilibrium existence theorem, Proc. Nat. Acad. Sci. U.S.A., 38 (1952), pp. 886-893.
8. G. Di Pillo and F. Facchinei, Exact penalty functions for nondifferentiable programming problems, in Nonsmooth Optimization and Related Topics, F. H. Clarke, V. F. Dem'yanov, and F. Giannessi, eds., Plenum Press, New York, 1989, pp. 89-107.
9. G. Di Pillo and F. Facchinei, Exact barrier methods for Lipschitz programs, Appl. Math. Optim., 32 (1995), pp. 1-31.
10. F. Facchinei, Refinements of necessary conditions for optimality in nonlinear programming, J. Optim. Theory Appl., 73 (1992), pp. 65-74.
11. F. Facchinei, A. Fischer, and V. Piccialli, On generalized Nash games and VIs, Oper. Res. Lett., 35 (2007), pp. 159-164.
12. F. Facchinei, A. Fischer, and V. Piccialli, Generalized Nash equilibrium problems and Newton methods, Math. Program., 117 (2009), pp. 163-194.
13. F. Facchinei and C. Kanzow, Generalized Nash equilibrium problems, 4OR, 5 (2007), pp. 173-210.
14. F. Facchinei and C. Kanzow, Penalty Methods for the Solution of Generalized Nash Equilibrium Problems (with Complete Test Problems), Technical report, Institute of Mathematics, University of Würzburg, Würzburg, Germany, 2009.
15. F. Facchinei and L. Lampariello, Partial Penalization for the Solution of Generalized Nash Equilibrium Problems, DIS Technical report, Sapienza University of Rome, Roma, Italy, 2010, J. Global Optim., submitted.
16. F. Facchinei and J.-S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Vol.I, Springer, New York, 2003.
17. F. Facchinei and J.-S. Pang, Exact penalty functions for generalized Nash problems, in Large Scale Nonlinear Optimization, G. Di Pillo and M. Roma, eds., Springer, New York, 2006, pp. 115-126.
18. F. Facchinei, V. Piccialli, and M. Sciandrone, Decomposition algorithms for generalized potential games, Comput. Optim. Appl., to appear.
19. S. D. Flåm and A. S. Antipin, Equilibrium programming using proximal-like algorithms, Math. Programming, 78 (1997), pp. 29-41.
20. M. Fukushima, Restricted generalized Nash equilibria and controlled penalty algorithm, Comput. Manag. Sci., to appear.
21. P. T. Harker, Generalized Nash games and quasi-variational inequalities, European J. Oper. Res., 54 (1991), pp. 81-94.
22. A. von Heusinger and C. Kanzow, Optimization reformulations of the generalized Nash equilibrium problem using Nikaido-Isoda-type functions, Comput. Optim. Appl., 43 (2009), pp. 353-377.
23. 1 optimization reformulations of the generalized Nash equilibrium problem, Optim. Methods Softw., 23 (2008), pp. 953-973.
24. A. von Heusinger and C. Kanzow, Relaxation methods for generalized Nash equilibrium problems with inexact line search, J. Optim. Theory Appl., 143 (2009), pp. 159-183.
25. A. von Heusinger, C. Kanzow, and M. Fukushima, Newton's method for computing a normalized Nash equilibrium in the generalized Nash game through fixed point formulation, Math. Program., to appear.
26. J.-B. Hiriart-Urruty and C. Lemaréchal, Convex Analysis and Minimization Algorithms I, Springer, Berlin, 1993.
27. A. D. Ioffe and J. V. Outrata, On metric and calmness qualification conditions in subdifferential calculus, Set-Valued Anal., 16 (2008), pp. 199-227.
28. A. Kesselman, S. Leonardi, and V. Bonifaci, Game-theoretic analysis of internet switching with selfish users, in Internet and Network Economics, Lecture Notes in Comput. Sci. 3828, Springer, Berlin, 2005, pp. 236-245.
29. J. B. Krawczyk and S. Uryasev, Relaxation algorithms to find Nash equilibria with economic applications, Environ. Model. Assess., 5 (2000), pp. 63-73.
30. K. Kubota and M. Fukushima, Gap function approach to the generalized Nash equilibrium problem, J. Optim. Theory Appl., 144 (2010), pp. 511-531.
31. K. Nabetani, Variational Inequality Approaches to Generalized Nash Equilibrium Problems, Master thesis, Department of Applied Mathematics and Physics, Kyoto University, Kyoto, Japan, 2008.
32. K. Nabetani, P. Tseng, and M. Fukushima, Parametrized variational inequality approaches to generalized Nash equilibrium problems with shared constraints, Comput. Optim. Appl., to appear.
33. J. V. Outrata, M. Kočvara, and J. Zowe, Nonsmooth Approach to Optimization Problems with Equilibrium Constraints, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1998.
34. J.-S. Pang and M. Fukushima, Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games, Comput. Manag. Sci., 2 (2005), pp. 21-56.
35. J.-S. Pang and M. Fukushima, Erratum: Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games, Comput. Manag. Sci., 6 (2009), pp. 373-375.
36. J.-S. Pang, G. Scutari, F. Facchinei, and C. Wang, Distributed power allocation with rate constraints in Gaussian parallel interference channels, IEEE Trans. Inform. Theory, 54 (2008), pp. 3471-3489.
37. S. M. Robinson, Shadow prices for measures of effectiveness, I: Linear model, Oper. Res., 41 (1993), pp. 518-535.
38. S. M. Robinson, Shadow prices for measures of effectiveness, II: General model, Oper. Res., 41 (1993), pp. 536-548.
39. J. B. Rosen, Existence and uniqueness of equilibrium points for concave N-person games, Econometrica, 33 (1965), pp. 520-533.
40. S. Uryas'ev and Y. Rubinstein, On relaxation algorithms in computation of noncooperative equilibria, IEEE Trans. Automat. Control, 39 (1994), pp. 1263-1267.