Is Pessimistic Bilevel Programming a Special Case of a Mathematical Program with Complementarity Constraints?

  • Didier AusselEmail author
  • Anton Svensson


One of the most commonly used methods for solving bilevel programming problems (whose lower level problem is convex) starts with reformulating it as a mathematical program with complementarity constraints. This is done by replacing the lower level problem by its Karush–Kuhn–Tucker optimality conditions. The obtained mathematical program with complementarity constraints is (locally) solved, but the question of whether a solution of the reformulation yields a solution of the initial bilevel problem naturally arises. The question was first formulated and answered negatively, in a recent work of Dempe and Dutta, for the so-called optimistic approach. We study this question for the pessimistic approach also in the case of a convex lower level problem with a similar answer. Some new notions of local solutions are defined for these minimax-type problems, for which the relations are shown. Some simple counterexamples are given.


Bilevel problem Mathematical programming with complementarity constraints Pessimistic approach 

Mathematics Subject Classification

90C30 90C33 90C47 



This research benefited from the support of the “FMJH Program Gaspard Monge in optimization and operation research”, and from the support to this program from EDF.


  1. 1.
    Dempe, S.: Foundations of Bilevel Programming. Springer, Berlin (2002)zbMATHGoogle Scholar
  2. 2.
    Dempe, S., Dutta, J.: Is bilevel programming a special case of a mathematical program with complementarity constraints? Math. Program. 131, 37–48 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Dempe, S., Mordukhovich, B.S., Zemkoho, A.B.: Sensitivity analysis for two-level value functions with applications to bilevel programming. SIAM J. Optim. 22, 1309–1343 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Ye, J.J., Zhu, D.: New necessary optimality conditions for bilevel programs by combining the MPEC and value function approaches. SIAM J. Optim. 20, 1885–1905 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Lignola, M.B., Morgan, J.: Topological existence and stability for Stackelberg problems. J. Optim. Theory Appl. 84(1), 145–169 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Loridan, P., Morgan, J.: Weak via strong Stackelberg problem: new results. J. Glob. Optim. 8, 263–287 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Dempe, S., Mordukhovich, B.S., Zemkoho, A.B.: Necessary optimality conditions in pessimistic bilevel programming. Optimization 63, 505–533 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dassanayaka, S.M.: Methods of variational analysis in pessimistic bilevel programming. PhD thesis, Wayne State University (2010)Google Scholar
  9. 9.
    Zheng, Y., Fang, D., Wan, Z.: A solution approach to the weak linear bilevel programming problems. Optimization 65, 1437–1449 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Zheng, Y., Fan, Y., Zhuo, X., Chen, J.: Pessimistic referential-uncooperative linear bilevel multi-follower decision making with an application to water resources optimal allocation (2016, unpublished)Google Scholar
  11. 11.
    Hiriart-Urruty, J.-B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms I: Fundamentals, vol. 305. Springer, Berlin (2013)zbMATHGoogle Scholar
  12. 12.
    Robinson, S.M.: Generalized Equations and Their Solutions, Part II: Applications to Nonlinear Programming. Springer, Berlin (1982)zbMATHGoogle Scholar
  13. 13.
    Bazaraa, M.S., Sherali, H.D., Shetty, C.M.: Nonlinear Programming: Theory and Algorithms. Wiley, Hoboken (2013)zbMATHGoogle Scholar

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

Authors and Affiliations

  1. 1.Université de PerpignanPerpignanFrance
  2. 2.Universidad de ChileSantiagoChile

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