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First-Order Necessary Optimality Conditions for General Bilevel Programming Problems

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Abstract

In Ref. 1, bilevel programming problems have been investigated using an equivalent formulation by use of the optimal value function of the lower level problem. In this comment, it is shown that Ref. 1 contains two incorrect results: in Proposition 2.1, upper semicontinuity instead of lower semicontinuity has to be used for guaranteeing existence of optimal solutions; in Theorem 5.1, the assumption that the abnormal part of the directional derivative of the optimal value function reduces to zero has to be replaced by the demand that a nonzero abnormal Lagrange multiplier does not exist.

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References

  1. YEZZA, A., First-Order Necessary Optimality Conditions for General Bilevel Programming Problems, Journal of Optimization Theory and Applications, Vol. 89, pp. 189–219, 1996.

    Google Scholar 

  2. LORIDAN, P., and MORGAN, J., Approximate Solutions for Two-Level Optimization Problems, Trends in Mathematical Optimization, Edited by K. Hoffman, J. Hiriart-Urruty, C. Lemarechal, and J. Zowe, International Series of Numerical Mathematics, Birkhäuser Verlag, Basel, Switzerland, Vol. 84, pp. 181–196, 1988.

    Google Scholar 

  3. LUCCHETTI, R., MIGNANEGO, F., and PIERI, G., Existence Theorem of Equilibrium Points in Stackelberg Games with Constraints, Optimization, Vol. 18, pp. 857–866, 1987.

    Google Scholar 

  4. HARKER, P. T., and PANG, J. S., Existence of Optimal Solutions to Mathematical Programs with Equilibrium Constraints, Operations Research Letters, Vol. 7, pp. 61–64, 1988.

    Google Scholar 

  5. DEMPE, S., and SCHMIDT, H., On an Algorithm Solving Two-Level Programming Problems with Nonunique Lower Level Solutions, Computational Optimization and Applications, Vol. 6, pp. 227–249, 1996.

    Google Scholar 

  6. LUDERER, B., Über der Äquivalenz nichtlinearer Optimierungsaufgaben, Technical Report, Technische Universität Karl-Marx-Stadt, Germany, 1983.

    Google Scholar 

  7. ROCKAFELLAR, R. T., Convex Analysis, Princeton University Press, Princeton, New Jersey, 1970.

    Google Scholar 

  8. CLARKE, F. H., Optimization and Nonsmooth Analysis, J. Wiley and Sons, New York, New York, 1983.

    Google Scholar 

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Authors and Affiliations

  1. Institute for Applied Mathematics II, Freiberg University of Mining and Technology, Freiberg, Germany

    S. Dempe (Associate Professor)

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  1. S. Dempe
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Dempe, S. First-Order Necessary Optimality Conditions for General Bilevel Programming Problems. Journal of Optimization Theory and Applications 95, 735–739 (1997). https://doi.org/10.1023/A:1022646611097

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  • DOI: https://doi.org/10.1023/A:1022646611097

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