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Optimality conditions for the bilevel programming problem

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  • Jonathan F. Bard

Abstract

The bilevel programming problem (BLPP) is a sequence of two optimization problems where the constraint region of the first is determined implicitly by the solution to the second. In this article it is first shown that the linear BLPP is equivalent to maximizing a linear function over a feasible region comprised of connected faces and edges of the original polyhedral constraint set. The solution is shown to occur at a vertex of that set. Next, under assumptions of differentiability, first‐order necessary optimality conditions are developed for the more general BLPP, and a potentially equivalent mathematical program is formulated. Finally, the relationship between the solution to this problem and Pareto optimality is discussed and a number of examples given.

Suggested Citation

  • Jonathan F. Bard, 1984. "Optimality conditions for the bilevel programming problem," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 31(1), pages 13-26, March.
    • Handle: RePEc:wly:navlog:v:31:y:1984:i:1:p:13-26
    DOI: 10.1002/nav.3800310104
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    Cited by:

    1. Benita, Francisco & López-Ramos, Francisco & Nasini, Stefano, 2019. "A bi-level programming approach for global investment strategies with financial intermediation," European Journal of Operational Research, Elsevier, vol. 274(1), pages 375-390.
    2. Adejuyigbe O. Fajemisin & Laura Climent & Steven D. Prestwich, 2021. "An analytics-based heuristic decomposition of a bilevel multiple-follower cutting stock problem," OR Spectrum: Quantitative Approaches in Management, Springer;Gesellschaft für Operations Research e.V., vol. 43(3), pages 665-692, September.
    3. Francisco López-Ramos & Stefano Nasini & Armando Guarnaschelli, 2019. "Road network pricing and design for ordinary and hazmat vehicles: Integrated model and specialized local search," Post-Print hal-02510066, HAL.
    4. P. A. Clark & A. W. Westerberg, 1988. "A note on the optimality conditions for the bilevel programming problem," Naval Research Logistics (NRL), John Wiley & Sons, vol. 35(5), pages 413-418, October.
    5. Abdul Sattar Safaei & Saba Farsad & Mohammad Mahdi Paydar, 2020. "Emergency logistics planning under supply risk and demand uncertainty," Operational Research, Springer, vol. 20(3), pages 1437-1460, September.
    6. Xiang Li & Tiesong Hu & Xin Wang & Ali Mahmoud & Xiang Zeng, 2023. "The New Solution Concept to Ill-Posed Bilevel Programming: Non-Antagonistic Pessimistic Solution," Mathematics, MDPI, vol. 11(6), pages 1-13, March.
    7. Chi-Bin Cheng & Hsu-Shih Shih & Boris Chen, 2017. "Subsidy rate decisions for the printer recycling industry by bi-level optimization techniques," Operational Research, Springer, vol. 17(3), pages 901-919, October.

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