Advertisement

Towards Tractable Constraint Qualifications for Parametric Optimisation Problems and Applications to Generalised Nash Games

  • Didier AusselEmail author
  • Anton Svensson
Article
  • 64 Downloads

Abstract

A generalised Nash game is a non-cooperative game in which each player is facing an optimisation problem where both the objective function and the feasible set depend on the variables of the other players. A classical way to treat numerically this difficult problem is to solve the nonlinear system composed of the concatenation of the Karush–Kuhn–Tucker optimality conditions of each player’s problem. The aim of this work is to provide constraint qualification conditions ensuring that both problems share the same set of solutions. Our main target here is to elaborate tractable conditions, that is, sets of conditions that are as simple as possible to fulfil. This is achieved through the analysis of “minimal” qualification conditions for parametric optimisation problems.

Keywords

Parametric optimisation Constraint qualifications KKT conditions GNEP Joint convexity 

Mathematics Subject Classification

90C31 90C33 90C46 91A40 

Notes

Acknowledgements

This research benefited from the support of the FMJH Program Gaspard Monge in optimisation and operation research, and from the support to this program from EDF. The second author was also benefited by a grant CONICYT-PFCHA/Doctorado Nacional/2018 N21180645.

References

  1. 1.
    Aussel, D., Svensson, A.: Some remarks about existence of equilibria, and the validity of the epcc-reformulation for multi-leader-follower games. J. Nonlinear Convex Anal. 19(7), 1141–1162 (2018)MathSciNetGoogle Scholar
  2. 2.
    Arrow, K.J., Debreu, G.: Existence of an equilibrium for a competitive economy. Econometrica: J. Econometric Soc. 22(3), 265–290 (1954)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Facchinei, F., Kanzow, C.: Generalized nash equilibrium problems. 4OR 5(3), 173–210 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Fischer, A., Herrich, M., Schönefeld, K.: Generalized nash equilibrium problems-recent advances and challenges. Pesquisa Oper. 34(3), 521–558 (2014)CrossRefGoogle Scholar
  5. 5.
    Stein, O., Sudermann-Merx, N.: The noncooperative transportation problem and linear generalized nash games. Eur. J. Oper. Res. 266(2), 543–553 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dreves, A., Gerdts, M.: A generalized nash equilibrium approach for optimal control problems of autonomous cars. Optim. Control Appl. Methods 39(1), 326–342 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Sudermann-Merx, N.G.: Linear generalized nash equilibrium problems. Ph.D. thesis, Karlsruher Instituts fur Technologie (2016)Google Scholar
  8. 8.
    Dreves, A.: Computing all solutions of linear generalized nash equilibrium problems. Math. Methods Oper. Res. 85(2), 207–221 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Rosen, J.B.: Existence and uniqueness of equilibrium points for concave n-person games. Econometrica 33(3), 520–534 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Aussel, D., Dutta, J.: Generalized nash equilibrium problem, variational inequality and quasiconvexity. Oper. Res. Lett. 36(4), 461–464 (2008). (Addendum: Oper. Res. Lett. 42 (2014), 398)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kanzow, C.: On the multiplier-penalty-approach for quasi-variational inequalities. Math. Program. 160(1–2), 33–63 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Bueno, L., Haeser, G., Rojas, F.: Optimality conditions and constraint qualifications for generalized nash equilibrium problems and their practical implications. SIAM J. Optim. pp. 31–54 (2019).  https://doi.org/10.1137/17M1162524
  13. 13.
    Kanzow, C., Steck, D.: Augmented lagrangian methods for the solution of generalized nash equilibrium problems. SIAM J. Optim. 26(4), 2034–2058 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Birgin, E.G., Haeser, G., Ramos, A.: Augmented lagrangians with constrained subproblems and convergence to second-order stationary points. Comput. Optim. Appl. 69(1), 51–75 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Gould, F., Tolle, J.W.: A necessary and sufficient qualification for constrained optimization. SIAM J. Appl. Math. 20(2), 164–172 (1971)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.University of Perpignan, Lab. PROMES UPR CNRS 8521PerpignanFrance
  2. 2.Universidad de ChileSantiagoChile

Personalised recommendations