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On the Quasiconcave Bilevel Programming Problem

Author

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  • H. I. Calvete

    (Universidad de Zaragoza)

  • C. Galé

    (Universidad de Zaragoza)

Abstract

Bilevel programming involves two optimization problems where the constraint region of the first-level problem is implicitly determined by another optimization problem. In this paper, we consider the case in which both objective functions are quasiconcave and the constraint region common to both levels is a polyhedron. First, it is proved that this problem is equivalent to minimizing a quasiconcave function over a feasible region comprised of connected faces of the polyhedron. Consequently, there is an extreme point of the polyhedron that solves the problem. Finally, it is shown that this model includes the most important case where the objective functions are ratios of concave and convex functions

Suggested Citation

  • H. I. Calvete & C. Galé, 1998. "On the Quasiconcave Bilevel Programming Problem," Journal of Optimization Theory and Applications, Springer, vol. 98(3), pages 613-622, September.
    • Handle: RePEc:spr:joptap:v:98:y:1998:i:3:d:10.1023_a:1022624029539
    DOI: 10.1023/A:1022624029539
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    References listed on IDEAS

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    1. Jonathan F. Bard, 1983. "An Algorithm for Solving the General Bilevel Programming Problem," Mathematics of Operations Research, INFORMS, vol. 8(2), pages 260-272, May.
    2. Wayne F. Bialas & Mark H. Karwan, 1984. "Two-Level Linear Programming," Management Science, INFORMS, vol. 30(8), pages 1004-1020, August.
    3. 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.
    4. James T. Moore & Jonathan F. Bard, 1990. "The Mixed Integer Linear Bilevel Programming Problem," Operations Research, INFORMS, vol. 38(5), pages 911-921, October.
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    Citations

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    Cited by:

    1. Calvete, Herminia I. & Gale, Carmen, 2004. "A note on `bilevel linear fractional programming problem'," European Journal of Operational Research, Elsevier, vol. 152(1), pages 296-299, January.
    2. Hecheng Li, 2015. "A genetic algorithm using a finite search space for solving nonlinear/linear fractional bilevel programming problems," Annals of Operations Research, Springer, vol. 235(1), pages 543-558, December.
    3. 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.
    4. Calvete, Herminia I. & Gale, Carmen & Mateo, Pedro M., 2008. "A new approach for solving linear bilevel problems using genetic algorithms," European Journal of Operational Research, Elsevier, vol. 188(1), pages 14-28, July.
    5. Shifali Bhargava, 2014. "Solving linear fractional multi-level programs," Operations Research and Decisions, Wroclaw University of Science and Technology, Faculty of Management, vol. 24(1), pages 5-21.
    6. Herminia Calvete & Carmen Galé & Pedro Mateo, 2009. "A genetic algorithm for solving linear fractional bilevel problems," Annals of Operations Research, Springer, vol. 166(1), pages 39-56, February.
    7. Calvete, Herminia I. & Gale, Carmen, 1999. "The bilevel linear/linear fractional programming problem," European Journal of Operational Research, Elsevier, vol. 114(1), pages 188-197, April.
    8. Jean Etoa, 2010. "Solving convex quadratic bilevel programming problems using an enumeration sequential quadratic programming algorithm," Journal of Global Optimization, Springer, vol. 47(4), pages 615-637, August.

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