Summary
We propose the use exact penalty functions for the solution of generalized Nash equilibrium problems (GNEPs). We show that by this approach it is possible to reduce the solution of a GNEP to that of a usual Nash problem. This paves the way to the development of numerical methods for the solution of GNEPs. We also introduce the notion of generalized stationary point of a GNEP and argue that convergence to generalized stationary points is an appropriate aim for solution algorithms.
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References
A. Bensoussan. Points de Nash dans le cas de fonctionnelles quadratiques et jeux differentiels lineaires a N personnes. SIAM Journal on Control 12 (1974) 460–499.
T. Başar. Relaxation techniques and asynchronous algorithms for on-line computation of non-cooperative equilibria. Journal of Economic Dynamics and Control 11 (1987) 531–549.
S. Berridge and J.B. Krawczyc. Relaxation algorithms in finding Nash equilibria. Economic working papers archives (1997) URL: http://econwpa.wustl.edu/eprints/comp/papers/9707/9707002.abs
J. Contreras, M.K. Klusch and J.B. Krawczyc. Numerical solution to Nash-Cournot equilibria in coupled constraints electricity markets. IEEE Transactions on Power Systems 19 (2004) 195–206.
V.F. Demyanov, G. Di Pillo, and F. Facchinei. Exact penalization via Dini and Hadamard constrained derivatives. Optimization Methods and Software 9 (1998) 19–36.
G. Di Pillo and F. Facchinei. Exact penalty functions for nondifferentiable programming problems. In F.H. Clarke, V.F. Demyanov, and F. Giannessi, editors, Nonsmooth Optimization and Related Topics, 89–107, Plenum Press (New York 1989)
G. Di Pillo and F. Facchinei. Regularity conditions and exact penalty functions in Lipschitz programming problems. In F. Giannessi, editor, Nonsmooth Optimization Methods and Applications, 107–120, Gordon and Breach (New York 1992)
G. Di Pillo and F. Facchinei. Exact barrier functions methods for Lipschitz Programs. Applied Mathematics and Optimization 32 (1995) 1–31.
F. Facchinei and J.-S. Pang. Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer Verlag (New York 2003).
M. Fukushima and J.-S. Pang. Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games. Computational Management Science, to appear.
P.T. Harker. Generalized Nash games and quasivariational inequalities. European Journal of Operations research 54 (1991) 81–94.
J.-B. Hiriart-Urruty, and C. Lemaréchal. Convex Analysis and Minimization Algorithms, Springer Verlag (New York 1993).
J.B. Krawczyc and S. Uryasev. Relaxation algorithms to find Nash equilibria with economic applications. Environmental Modelling and Assessment 5 (2000) 63–73.
S. Li and T. Başar. Distributed algorithms for the computation of noncooperative equilibria. Automatica 23 (1987) 523–533.
H. Nikaido and K. Isoda. Note on noncooperative convex games. Pacific Journal of Mathematics 5 (1955) 807–815.
G.P. Papavassilopoulos. Iterative techniques for the Nash solution in quadratic games with unknown parameters. SIAM Journal on Control and Optimization 24 (1986) 821–834.
S.M. Robinson. Shadow prices for measures of effectiveness. II. Linear Model. Operations Research 41 (1993) 518–535.
S.M. Robinson. Shadow prices for measures of effectiveness. II. General Model. Operations Research 41 (1993) 536–548.
S. Uryasev and R. Y. Rubinstein. On relaxation algorithms in computation of noncooperative equilibria. IEEE Transactions on Automatic Control 39 (1994) 1263–1267.
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Facchinei, F., Pang, JS. (2006). Exact penalty functions for generalized Nash problems. In: Di Pillo, G., Roma, M. (eds) Large-Scale Nonlinear Optimization. Nonconvex Optimization and Its Applications, vol 83. Springer, Boston, MA. https://doi.org/10.1007/0-387-30065-1_8
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DOI: https://doi.org/10.1007/0-387-30065-1_8
Publisher Name: Springer, Boston, MA
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