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Deriving the Properties of Linear Bilevel Programming via a Penalty Function Approach

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  • Z. K. Xu

    (Zhejiang Normal University)

Abstract

For the linear bilevel programming problem, we propose an assumption weaker than existing assumptions, while achieving similar results via a penalty function approach. The results include: equivalence between (i) existence of a solution to the problem, (ii) existence of an exact penalty function approach for solving the problem, and (iii) achievement of the optimal value of the equivalent form of the problem at some vertex of a certain polyhedral convex set. We prove that the assumption is both necessary and sufficient for the linear bilevel programming problem to admit an exact penalty function formulation, provided that the equivalent form of the problem has a feasible solution. A method is given for computing the minimal penalty function parameter value. This method can be executed by solving a set of linear programming problems. Lagrangian duality is also presented.

Suggested Citation

  • Z. K. Xu, 1999. "Deriving the Properties of Linear Bilevel Programming via a Penalty Function Approach," Journal of Optimization Theory and Applications, Springer, vol. 103(2), pages 441-456, November.
    • Handle: RePEc:spr:joptap:v:103:y:1999:i:2:d:10.1023_a:1021713105102
    DOI: 10.1023/A:1021713105102
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    Cited by:

    1. Herminia Calvete & Carmen Galé & Stephan Dempe & Sebastian Lohse, 2012. "Bilevel problems over polyhedra with extreme point optimal solutions," Journal of Global Optimization, Springer, vol. 53(3), pages 573-586, July.

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