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

Content Provider	Springer Nature Link
Author	Heusinger, Anna Fukushima, Masao Kanzow, Christian
Copyright Year	2010
Abstract	We consider the generalized Nash equilibrium problem (GNEP), where not only the players’ cost functions but also their strategy spaces depend on the rivals’ decision variables. Existence results for GNEPs are typically shown by using a fixed point argument for a certain set-valued function. Here we use a regularization of this set-valued function in order to obtain a single-valued function that is easier to deal with from a numerical point of view. We show that the fixed points of the latter function constitute an important subclass of the generalized equilibria called normalized equilibria. This fixed point formulation is then used to develop a nonsmooth Newton method for computing a normalized equilibrium. The method uses a so-called computable generalized Jacobian that is much easier to compute than Clarke generalized Jacobian or B-subdifferential. We establish local superlinear/quadratic convergence of the method under the constant rank constraint qualification, which is weaker than the frequently used linear independence constraint qualification, and a suitable second-order condition. Some numerical results are presented to illustrate the performance of the method.
Ending Page	123
Page Count	25
Starting Page	99
File Format	PDF
ISSN	00255610
e-ISSN	14364646
Journal	Mathematical Programming
Issue Number	1-2
Volume Number	132
Language	English
Publisher	Springer-Verlag
Publisher Date	2010-05-29
Publisher Place	Berlin, Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Local superlinear/quadratic convergence Noncooperative games Theoretical, Mathematical and Computational Physics Nonsmooth Newton method Computable generalized Jacobian Generalized Nash equilibrium problem Mathematical Methods in Physics Constant rank constraint qualification Mathematics of Computing Calculus of Variations and Optimal Control; Optimization Numerical Analysis Newton-type methods Fixed point characterization Nonlinear programming Combinatorics Normalized equilibrium
Content Type	Text
Resource Type	Article
Subject	Mathematics Software

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