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Bilevel Programming: Optimality Conditions and Duality

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Encyclopedia of Optimization

The bilevel programming problem (abbreviation: BPP) is a mathematical program in two variables x and θ, where x = x°(θ) is an optimal solution of another program. Specifically, BPP can be formulated in terms of two ordered objective functions φ and Ψ as follows:

(1)

where x = x°(θ) is an optimal solution of the program

(2)

Here the functions φ, Ψ, f i, g j : R n × R m → R, i ∈ P, j ∈ Q, are assumed to be continuous; x ∈ R n, θ ∈ R m; P, Q are finite index sets. Program (1) is often called the upper ( first level , outer , leader’s ) problem; then (2) is the lower ( second level , inner , follower’s ) problem. Many mathematical programs, such as minimax problems, linear integer, bilinear and quadratic programs, can be stated as special cases of bilevel programs. In view of the so-called Reduction Ansatz, developed in [18], [44], semi-infinite programs can be considered as special cases of bilevel programs. For stability and deformations of these see, e.g., [20], [21]. Problems...

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Author information

Authors and Affiliations

  1. Dept. Math. Statist. McGill Univ., Burnside Hall, 805 Sherbrooke Street West, Montreal, Quebec, Canada, H3A 2K6

    S. Zlobec

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  1. S. Zlobec
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Editor information

Editors and Affiliations

  1. Dept. Chemical Engin., Princeton Univ., Princeton, NJ, 08544-5263, USA

    Christodoulos A. Floudas

  2. Center for Applied Optim. Dept. Industrial and Systems Engin., Univ. Florida, Gainesville, FL, 32611, USA

    Panos M. Pardalos

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© 2001 Kluwer Academic Publishers

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Zlobec, S. (2001). Bilevel Programming: Optimality Conditions and Duality . In: Floudas, C.A., Pardalos, P.M. (eds) Encyclopedia of Optimization. Springer, Boston, MA. https://doi.org/10.1007/0-306-48332-7_39

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  • DOI: https://doi.org/10.1007/0-306-48332-7_39

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-0-7923-6932-5

  • Online ISBN: 978-0-306-48332-5

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