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First-Order Necessary Optimality Conditions for General Bilevel Programming Problems

Author

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  • S. Dempe

    (Freiberg University of Mining and Technology)

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.

Suggested Citation

  • S. Dempe, 1997. "First-Order Necessary Optimality Conditions for General Bilevel Programming Problems," Journal of Optimization Theory and Applications, Springer, vol. 95(3), pages 735-739, December.
    • Handle: RePEc:spr:joptap:v:95:y:1997:i:3:d:10.1023_a:1022646611097
    DOI: 10.1023/A:1022646611097
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    Cited by:

    1. S. K. Suneja & B. Kohli, 2011. "Optimality and Duality Results for Bilevel Programming Problem Using Convexifactors," Journal of Optimization Theory and Applications, Springer, vol. 150(1), pages 1-19, July.
    2. Torbjorn Jansson, 2000. "Estimating Prices and Excess Demand and Trade Costs in a Spatial Price Equilibrium Model," Regional and Urban Modeling 283600040, EcoMod.
    3. Polyxeni-Margarita Kleniati & Claire Adjiman, 2014. "Branch-and-Sandwich: a deterministic global optimization algorithm for optimistic bilevel programming problems. Part I: Theoretical development," Journal of Global Optimization, Springer, vol. 60(3), pages 425-458, November.
    4. M. Lignola & Jacqueline Morgan, 2012. "Approximate values for mathematical programs with variational inequality constraints," Computational Optimization and Applications, Springer, vol. 53(2), pages 485-503, October.
    5. S. Dempe & N. Gadhi, 2010. "Second order optimality conditions for bilevel set optimization problems," Journal of Global Optimization, Springer, vol. 47(2), pages 233-245, June.
    6. Bhawna Kohli, 2012. "Optimality Conditions for Optimistic Bilevel Programming Problem Using Convexifactors," Journal of Optimization Theory and Applications, Springer, vol. 152(3), pages 632-651, March.

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