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Optimality Conditions for Bilevel Programming: An Approach Through Variational Analysis

  • Joydeep Dutta
Chapter
Part of the Forum for Interdisciplinary Mathematics book series (FFIM)

Abstract

In this article, we focus on the study of bilevel programming problems where the feasible set in the lower-level problem is an abstract closed convex set and not described by any equalities and inequalities. In such a situation, we can view them as MPEC problems and develop necessary optimality conditions. We also relate various solution concepts in bilevel programming and establish some new connections. We study in considerable detail the notion of partial calmness and its application to derive necessary optimality conditions and also give some illustrative examples.

Notes

Acknowledgements

I am grateful to the organizers of the CIMPA school on Generalized Nash Equilibrium, Bilevel Programming and MPEC problems held in Delhi from 25 November to 6 December 2013 for giving me a chance to speak at the workshop which finally led to this article. I would also like to thank Didier Aussel for his thoughtful discussions on the article and his help in constructing Example 3.3. I am also indebted to the anonymous referee for the constructive suggestion which improved the presentation of this article.

References

  1. 1.
    Bazaraa, M.S., Sherali, H.D., Shetty, C.M.: Nonlinear programming. Theory and algorithms, 3rd edn. Wiley-Interscience, Wiley, Hoboken (2006)Google Scholar
  2. 2.
    Burke, J.V., Ferris, M.C.V.: Weak sharp minima in mathematical programming. SIAM J. Control Optim. 31, 1340–1359 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Clarke, F.: Optimization and Nonsmooth Analysis. Wiley-Interscience, Hoboken (1983)zbMATHGoogle Scholar
  4. 4.
    Clarke, F.: Functional Analysis, Calculus of Variations and Optimal Control. Graduate Texts in Mathematics, vol. 264. Springer, London, 2013Google Scholar
  5. 5.
    Craven, B.D.: Mathematical Programming and Control Theory. Chapman and Hall, London (1978)CrossRefzbMATHGoogle Scholar
  6. 6.
    Dempe, S.: Foundations of Bilevel Programing (Springer). Kluwer Academic Publishers, The Netherlands (2003)Google Scholar
  7. 7.
    Dempe, S., Dutta, J.: Is bilevel programming a special case of a mathematical program with complementarity constraints? Math. Program. Ser. A 131, 37–48 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Dempe, S., Dutta, J., Mordukhovich, B.S.: New necessary optimality conditions in optimistic bilevel programming. Optimization 56, 577–604 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Dempe, S., Zemkoho, A.B.: The bilevel programming problem: reformulations, constraint qualifications and optimality conditions. Math. Program. Ser. A 138(1–2), 447–473 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Dempe, S., Mordukhovich, B., Zemkoho, A.B.: Necessary optimality conditions in pessimistic bilevel programming. Optimization 63(4), 505–533 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Dempe, S., Dutta, J.: Bilevel programming with convex lower level problems. Optimization with multivalued mappings, pp. 51–71, Springer Optim. Appl., 2, Springer, New York, (2006)Google Scholar
  12. 12.
    Henrion, R., Surowiec, T.: On calmness conditions in convex bilevel programming. Appl. Anal. 90, 951–970 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Henrion, R., Outrata, J., Surowiec, T.: On the co-derivative of normal cone mappings to inequality systems. Nonlinear Anal. 71, 1213–1226 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Hiriart-Urruty, J.B., Lemarechal, C.: Fundemantals of Convex Analysis. Springer, Berlin (2003)Google Scholar
  15. 15.
    Jahn, J.: Vector Optimization. Applications, and Extensions. Springer-Verlag, Berlin, Theory (2004)CrossRefzbMATHGoogle Scholar
  16. 16.
    Mordukhovich, B.S., Levy, A.B.: Coderivatives in parametric optimization. Math. Program. Ser. A 99(2), 311–327 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Mordukhovich, B.S.: Variational analysis and generalized differentiation Vol 1 Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 331. Springer-Verlag, Berlin (2006)Google Scholar
  18. 18.
    Mordukhovich, B.S., Outrata, J.V.: On second-order subdifferentials and their applications. SIAM J. Optim. 12, 139–169 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Mordukhovich, B.S., Outrata, J.V.: Coderivative analysis of quasi-variational inequalities with applications to stability and optimization. SIAM J. Optim. 18, 389–412 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Rockafellar, R.T., Wets, R.J.B.: Variational Analysis. Springer, Berlin (1997)zbMATHGoogle Scholar
  21. 21.
    Ye, J.J., Zhu, D.: Optimality conditions for bilevel programming problems. Optimization 33, 9–27 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Ye, J.J.: Nondifferentiable multiplier rules for optimization and bilevel optimization problems. SIAM J. Optim. 15, 252–274 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Ye, J.J.: Constraint qualifications and necessary optimality conditions for optimization problems with variational inequality constraints. SIAM J. Optim. 10, 943–962 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    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)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Lin, G.H., Xu, M., Ye, J.J.: On solving simple bilevel programs with a nonconvex lower level program. Math. Program. Ser. A 144(1–2), 277–305 (2014)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2017

Authors and Affiliations

  1. 1.Economics Group, Department of Humanities and Social SciencesIndian Institute of Technology, KanpurKanpurIndia

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