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

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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.

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Notes

  1. This concept is not the joint convexity of a game as defined by Rosen. In general, they are not comparable, but our assumption is stronger when all players have the same constraint, i.e. \(g_j=g\), for all \(j\in J\).

References

  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)

    MathSciNet  Google Scholar 

  2. Arrow, K.J., Debreu, G.: Existence of an equilibrium for a competitive economy. Econometrica: J. Econometric Soc. 22(3), 265–290 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  3. Facchinei, F., Kanzow, C.: Generalized nash equilibrium problems. 4OR 5(3), 173–210 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. Fischer, A., Herrich, M., Schönefeld, K.: Generalized nash equilibrium problems-recent advances and challenges. Pesquisa Oper. 34(3), 521–558 (2014)

    Article  Google Scholar 

  5. Stein, O., Sudermann-Merx, N.: The noncooperative transportation problem and linear generalized nash games. Eur. J. Oper. Res. 266(2), 543–553 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  7. Sudermann-Merx, N.G.: Linear generalized nash equilibrium problems. Ph.D. thesis, Karlsruher Instituts fur Technologie (2016)

  8. Dreves, A.: Computing all solutions of linear generalized nash equilibrium problems. Math. Methods Oper. Res. 85(2), 207–221 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  9. Rosen, J.B.: Existence and uniqueness of equilibrium points for concave n-person games. Econometrica 33(3), 520–534 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kanzow, C.: On the multiplier-penalty-approach for quasi-variational inequalities. Math. Program. 160(1–2), 33–63 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  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. Kanzow, C., Steck, D.: Augmented lagrangian methods for the solution of generalized nash equilibrium problems. SIAM J. Optim. 26(4), 2034–2058 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  15. Gould, F., Tolle, J.W.: A necessary and sufficient qualification for constrained optimization. SIAM J. Appl. Math. 20(2), 164–172 (1971)

    Article  MathSciNet  MATH  Google Scholar 

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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.

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Authors and Affiliations

  1. University of Perpignan, Lab. PROMES UPR CNRS 8521, Perpignan, France

    Didier Aussel & Anton Svensson

  2. Universidad de Chile, Santiago, Chile

    Anton Svensson

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  1. Didier Aussel
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  2. Anton Svensson
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Correspondence to Didier Aussel.

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Aussel, D., Svensson, A. Towards Tractable Constraint Qualifications for Parametric Optimisation Problems and Applications to Generalised Nash Games. J Optim Theory Appl 182, 404–416 (2019). https://doi.org/10.1007/s10957-019-01529-4

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  • DOI: https://doi.org/10.1007/s10957-019-01529-4

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