Computational Optimization and Applications

, Volume 69, Issue 3, pp 629–652 | Cite as

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

  • Leonardo GalliEmail author
  • Christian Kanzow
  • Marco Sciandrone


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.


Generalized Nash equilibrium problem Trust-region algorithm Nonmonotone strategy Global convergence Local superlinear convergence Quasi-variational inequalities 


  1. 1.
    Bellavia, S., Macconi, M., Morini, B.: An affine scaling trust-region approach to bound-constrained nonlinear systems. Appl. Numer. Math. 44, 180–257 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bellavia, S., Macconi, M., Morini, B.: STRSCNE: a scaled trust region solver for constrained nonlinear equations. Comput. Optim. Appl. 28, 31–50 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bertsekas, D.P.: Nonlinear Programming, 2nd edn. Athena Scientific, Belmont (1999)zbMATHGoogle Scholar
  4. 4.
    Calamai, P.H., Moré, J.J.: Projected gradient methods for linear constrained problems. Math. Program. 39, 93–116 (1987)CrossRefzbMATHGoogle Scholar
  5. 5.
    Dan, H., Yamashita, N., Fukushima, M.: Convergence properties of the inexact Levenberg–Marquardt method under local error bound conditions. Optim. Methods Softw. 17, 605–626 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dolan, E.D., Moré, J.J.: Benchmarking optimization software with performance profiles. Math. Program. 91, 201–213 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Dreves, A., Facchinei, F., Fischer, A., Herrich, M.: A new error bound result for generalized Nash equilibrium problems and its algorithmic application. Comput. Optim. Appl. 59, 63–84 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dreves, A., Facchinei, F., Kanzow, C., Sagratella, S.: On the solution of the KKT conditions of generalized Nash equilibrium problems. SIAM J. Optim. 21, 1082–1108 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Facchinei, F., Kanzow, C., Sagratella, S.: QVILIB: a library of quasi-variational inequality test problems. Pac. J. Optim. 9, 225–250 (2013)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Facchinei, F., Kanzow, C., Sagratella, S.: Solving quasi-variational inequalities via their KKT conditions. Math. Program. 114, 369–412 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Facchinei, F., Kanzow, C.: Generalized Nash equilibrium problems. Ann. Oper. Res. 175, 177–211 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Facchinei, F., Kanzow, C., Karl, S., Sagratella, S.: The semismooth Newton method for the solution of quasi-variational inequalities. Comput. Optim. Appl. 62, 85–109 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Fan, J., Yuan, Y.: On the quadratic convergence of the Levenberg–Marquardt method without nonsingularity assumption. Computing 74, 23–39 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Fischer, A., Herrich, M., Schönefeld, K.: Generalized Nash equilibrium problems—recent advances and challenges. Pesquisa Operacional 34, 521–558 (2014)CrossRefGoogle Scholar
  15. 15.
    Fischer, A., Shukla, P.K., Wang, M.: On the inexactness level of robust Levenberg–Marquardt methods. Optimization 59, 273–287 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Herrich, M.: Local convergence of Newton-type methods for nonsmooth constrained equations and applications. Ph.D. Thesis, Institute of Mathematics, Technical University of Dresden, Germany (2014)Google Scholar
  17. 17.
    Izmailov, A.F., Solodov, M.V.: On error bounds and Newton-type methods for generalized Nash equilibrium problems. Comput. Optim. Appl. 59, 201–218 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Kanzow, C., Steck, D.: Augmented Lagrangian methods for the solution of generalized Nash equilibrium problems. SIAM J. Optim. 26, 2034–2058 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Kanzow, C., Yamashita, N., Fukushima, M.: Levenberg–Marquardt methods with strong local convergence properties for solving nonlinear equations with convex constraints. J. Comput. Appl. Math. 172, 375–397 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Qi, L., Tong, X.J., Li, D.H.: Active-set projected trust-region algorithm for box-constrained nonsmooth equations. J. Optim. Theory Appl. 120, 601–625 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Toint, P.L.: Non-monotone trust-region algorithms for nonlinear optimization subject to convex constraints. Math. Program. 77, 69–94 (1997)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Tong, X.J., Qi, L.: On the convergence of a trust-region method for solving constrained nonlinear equations with degenerate solutions. J. Optim. Theory Appl. 123, 187–211 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Yamashita, N., Fukushima, M.: On the rate of convergence of the Levenberg–Marquardt method. Computing 15, 239–249 (2001)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of Information EngineeringUniversity of FlorenceFlorenceItaly
  2. 2.Institute of MathematicsUniversity of WürzburgWürzburgGermany

Personalised recommendations