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A nonmonotone trust-region method for generalized Nash equilibrium and related problems with strong convergence properties

Author

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  • Leonardo Galli

    (University of Florence)

  • Christian Kanzow

    (University of Würzburg)

  • Marco Sciandrone

    (University of Florence)

Abstract

The generalized Nash equilibrium problem (GNEP) is often difficult to solve by Newton-type methods since the problem tends to have locally nonunique solutions. Here we take an existing trust-region method which is known to be locally fast convergent under a relatively mild error bound condition, and modify this method by a nonmonotone strategy in order to obtain a more reliable and efficient solver. The nonmonotone trust-region method inherits the nice local convergence properties of its monotone counterpart and is also shown to have the same global convergence properties. Numerical results indicate that the nonmonotone trust-region method is significantly better than the monotone version, and is at least competitive to an existing software applied to the same reformulation used within our trust-region framework. Additional tests on quasi-variational inequalities (QVI) are also presented to validate efficiency of the proposed extension.

Suggested Citation

  • Leonardo Galli & Christian Kanzow & Marco Sciandrone, 2018. "A nonmonotone trust-region method for generalized Nash equilibrium and related problems with strong convergence properties," Computational Optimization and Applications, Springer, vol. 69(3), pages 629-652, April.
    • Handle: RePEc:spr:coopap:v:69:y:2018:i:3:d:10.1007_s10589-017-9960-3
    DOI: 10.1007/s10589-017-9960-3
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    References listed on IDEAS

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    1. X. J. Tong & L. Qi, 2004. "On the Convergence of a Trust-Region Method for Solving Constrained Nonlinear Equations with Degenerate Solutions," Journal of Optimization Theory and Applications, Springer, vol. 123(1), pages 187-211, October.
    2. L. Qi & X. J. Tong & D. H. Li, 2004. "Active-Set Projected Trust-Region Algorithm for Box-Constrained Nonsmooth Equations," Journal of Optimization Theory and Applications, Springer, vol. 120(3), pages 601-625, March.
    3. Francisco Facchinei & Christian Kanzow, 2010. "Generalized Nash Equilibrium Problems," Annals of Operations Research, Springer, vol. 175(1), pages 177-211, March.
    4. Francisco Facchinei & Christian Kanzow & Sebastian Karl & Simone Sagratella, 2015. "The semismooth Newton method for the solution of quasi-variational inequalities," Computational Optimization and Applications, Springer, vol. 62(1), pages 85-109, September.
    5. Alexey Izmailov & Mikhail Solodov, 2014. "On error bounds and Newton-type methods for generalized Nash equilibrium problems," Computational Optimization and Applications, Springer, vol. 59(1), pages 201-218, October.
    6. Axel Dreves & Francisco Facchinei & Andreas Fischer & Markus Herrich, 2014. "A new error bound result for Generalized Nash Equilibrium Problems and its algorithmic application," Computational Optimization and Applications, Springer, vol. 59(1), pages 63-84, October.
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    Cited by:

    1. Leonardo Galli & Alessandro Galligari & Marco Sciandrone, 2020. "A unified convergence framework for nonmonotone inexact decomposition methods," Computational Optimization and Applications, Springer, vol. 75(1), pages 113-144, January.
    2. Axel Dreves & Simone Sagratella, 2020. "Nonsingularity and Stationarity Results for Quasi-Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 185(3), pages 711-743, June.

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