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

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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.

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Authors and Affiliations

  1. Institute of Mathematics, University of Würzburg, Am Hubland, 97074, Würzburg, Germany

    Anna von Heusinger & Christian Kanzow

  2. Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Kyoto, 606-8501, Japan

    Masao Fukushima

Authors
  1. Anna von Heusinger
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  2. Christian Kanzow
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  3. Masao Fukushima
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Correspondence to Anna von Heusinger.

Additional information

This research was supported by a grant from the international doctorate program “Identification, Optimization, and Control with Applications in Modern Technologies” within the Elite-Network of Bavaria. This research was also supported in part by a Grant-in-Aid for Scientific Research from Japan Society for the Promotion of Science.

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von Heusinger, A., Kanzow, C. & Fukushima, M. Newton’s method for computing a normalized equilibrium in the generalized Nash game through fixed point formulation. Math. Program. 132, 99–123 (2012). https://doi.org/10.1007/s10107-010-0386-2

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  • DOI: https://doi.org/10.1007/s10107-010-0386-2

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