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Inexact Newton Method for Solving Generalized Nash Equilibrium Problems

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Abstract

In this article, we present an inexact Newton method to solve generalized Nash equilibrium problems (GNEPs). Two types of GNEPs are studied: player convex and jointly convex. We reformulate the GNEP into an unconstrained optimization problem using a complementarity function and solve it by the proposed method. It is found that the proposed numerical scheme has the global convergence property for both types of GNEPs. The strong BD-regularity assumption for the reformulated system of GNEP plays a crucial role in global convergence. In fact, the strong BD-regularity assumption and a suitable choice of a forcing sequence expedite the inexact Newton method to Q-quadratic convergence. The efficiency of the proposed numerical scheme is shown for a collection of problems, including the realistic internet switching problem, where selfish users generate traffic. A comparison of the proposed method with the existing semi-smooth Newton method II for GNEP is provided, which indicates that the proposed scheme is more efficient.

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Data Availability

The authors declare that the manuscript has no associated data. For the numerical part of the paper and MATLAB codes for the proposed algorithms, readers are requested to contact the first author.

Notes

  1. Throughout the paper, in \(\mathbb {R}^n\), we use the notations \(\Vert \cdot \Vert \) and \(\langle \cdot , \cdot \rangle \) to represent the usual Euclidean norm and inner product, respectively.

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Acknowledgements

The authors are truly thankful to the editor and reviewers for their constructive comments that substantially improved the quality of the paper. Debdas Ghosh acknowledges the financial support of the research grants MATRICS (MTR/2021/000696) and Core Research Grant (CRG/2022/001347) by the Science and Engineering Research Board, India.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Indian Institute of Technology (Banaras Hindu University), Varanasi, Uttar Pradesh, 221005, India

    Abhishek Singh & Debdas Ghosh

  2. Department of Mathematics, Aligarh Muslim University, Aligarh, 202002, India

    Qamrul Hasan Ansari

  3. College of Sciences, Chongqing University of Technology, Chongqing, 400054, China

    Qamrul Hasan Ansari

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  1. Abhishek Singh
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Singh, A., Ghosh, D. & Ansari, Q.H. Inexact Newton Method for Solving Generalized Nash Equilibrium Problems. J Optim Theory Appl (2024). https://doi.org/10.1007/s10957-024-02411-8

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