Computing Locally Optimal Solutions of the Bilevel Optimization Problem Using the KKT Approach

  • Stephan DempeEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11548)


If the lower-level problem in a bilevel optimization problem is replaced by its Karush-Kuhn-Tucker conditions, a mathematical program with complementarity constraints is obtained. Solving this nonconvex optimization problem, locally optimal solutions are computed which do in general not correspond to locally optimal solutions of the bilevel problem. Using a relaxation of this problem in two constraints it can be shown that a sequence of locally optimal solutions of the relaxed problems converges to a point which is related to a locally optimal solution of the bilevel optimization problem. If the lower-level problem is a linear one, relaxation of only the complementarity constraint is sufficient.


Optimistic bilevel optimization KKT transformation Locally optimal solutions 


  1. 1.
    Allende, G.B., Still, G.: Solving bilevel programs with the KKT-approach. Math. Program. 138, 309–332 (2013)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bard, J.: Practical Bilevel Optimization: Algorithms and Applications. Kluwer Academic Publishers, Dordrecht (1998)CrossRefGoogle Scholar
  3. 3.
    Burtscheidt, J., Claus, M., Dempe, S.: Risk-averse models in bilevel stochastic linear programming. arXiv preprint arXiv:1901.11349 (2019)
  4. 4.
    Dempe, S.: A necessary and a sufficient optimality condition for bilevel programming problems. Optimization 25, 341–354 (1992)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Dempe, S.: Foundations of Bilevel Programming. Kluwer Academic Publishers, Dordrecht (2002)zbMATHGoogle Scholar
  6. 6.
    Dempe, S.: Bilevel optimization: theory, algorithms and applications (2018). Optimization
  7. 7.
    Dempe, S., Dutta, J.: Is bilevel programming a special case of a mathematical program with complementarity constraints? Math. Program. 131, 37–48 (2012)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Dempe, S., Franke, S.: Solution algorithm for an optimistic linear Stackelberg problem. Comput. Oper. Res. 41, 277–281 (2014)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Dempe, S., Franke, S.: On the solution of convex bilevel optimization problems. Comput. Optim. Appl. 63, 685–703 (2016)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Dempe, S., Franke, S.: Solution of bilevel optimization problems using the KKT approach. Optimization 1–19 (2019). online first publicationGoogle Scholar
  11. 11.
    Dempe, S., Kalashnikov, V., Pérez-Valdés, G., Kalashnykova, N.: Bilevel Programming Problems: Theory, Algorithms and Application to Energy Networks. Springer, Heidelberg (2015). Scholar
  12. 12.
    Dempe, S., Luo, G., Franke, S.: Pessimistic bilevel linear optimization. J. Nepal Math. Soc. 1, 1–10 (2018)Google Scholar
  13. 13.
    Hoheisel, T., Kanzow, C., Schwartz, A.: Theoretical and numerical comparison of relaxation methods for mathematical programs with complementarity constraints. Math. Program. 137(1–2), 257–288 (2013)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Kojima, M.: Strongly stable stationary solutions in nonlinear programs. In: Robinson, S. (ed.) Analysis and Computation of Fixed Points, pp. 93–138. Academic Press, New York (1980)CrossRefGoogle Scholar
  15. 15.
    Lampariello, L., Sagratella, S.: A bridge between bilevel programs and Nash games. J. Optim. Theory Appl. 174(2), 613–635 (2017). Scholar
  16. 16.
    Mersha, A.: Solution Methods for Bilevel Programming Problems. Ph.D. thesis, TU Bergakademie Freiberg (2008)Google Scholar
  17. 17.
    Mersha, A., Dempe, S.: Feasible direction method for bilevel programming problem. Optimization 61(4–6), 597–616 (2012)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Mirrlees, J.: The theory of moral hazard and unobservable bevaviour: part I. Rev. Econ. Stud. 66, 3–21 (1999)CrossRefGoogle Scholar
  19. 19.
    Outrata, J., Kočvara, M., Zowe, J.: Nonsmooth Approach to Optimization Problems with Equilibrium Constraints. Kluwer Academic Publishers, Dordrecht (1998)CrossRefGoogle Scholar
  20. 20.
    Outrata, J.: On the numerical solution of a class of Stackelberg problems. ZOR - Math. Meth. Oper. Res. 34, 255–277 (1990)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Ralph, D., Dempe, S.: Directional derivatives of the solution of a parametric nonlinear program. Math. Program. 70, 159–172 (1995)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Rog, R.: Lösungsalgorithmen für die KKT-Transformation von Zwei-Ebenen-Optimierungsaufgaben. Master’s thesis, TU Bergakademie Freiberg, Fakultät für Mathematik und Informatik (2017)Google Scholar
  23. 23.
    Scheel, H., Scholtes, S.: Mathematical programs with equilibrium constraints: stationarity, optimality, and sensitivity. Math. Oper. Res. 25, 1–22 (2000)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Scholtes, S.: Convergence properties of a regularization scheme for mathematical programs with complementarity constraints. SIAM J. Optim. 11, 918–936 (2001)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Ye, J., Zhu, D.: New necessary optimality conditions for bilevel programs by combining the MPEC and value function approaches. SIAM J. Optim. 20(4), 1885–1905 (2010)MathSciNetCrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Institute of Numerical Mathematics and OptimizationTU Bergakademie FreibergFreibergGermany

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