Journal
MATHEMATICAL PROGRAMMING
Volume 132, Issue 1-2, Pages 99-123Publisher
SPRINGER HEIDELBERG
DOI: 10.1007/s10107-010-0386-2
Keywords
Generalized Nash equilibrium problem; Normalized equilibrium; Fixed point characterization; Nonsmooth Newton method; Computable generalized Jacobian; Constant rank constraint qualification; Local superlinear/quadratic convergence
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Funding
- international doctorate program Identification, Optimization, and Control with Applications in Modern Technologies
- Elite-Network of Bavaria
- Japan Society for the Promotion of Science
- Grants-in-Aid for Scientific Research [22500256] Funding Source: KAKEN
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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.
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