Bilevel program

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In mathematics, bilevel programs are optimization problems where one optimization problem is embedded in another one. Equivalently the constraints Bilevel programs are multilevel programs with two levels [1].

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[edit] Mathematical formulation of the problem

The optimistic formulation of bilevel programs problem can be stated simply as:

\min\limits_{x \in X, y \in Y}\;\; f^u(x,y)
\mbox{subject to: }\
g^u(x,y) \le 0, \;\;
y \in \arg\min\limits_{z \in Y} f^l(x,y)
g^l(x,z) \le 0, \;\;

where

f^u,f^l: R^{nx} \times R^{ny} \to R
g^u,g^l: R^{nx} \times R^{ny} \to R^{mu}
X \subseteq R^{nx}
Y \subseteq R^{ny}.

The variables z are dummy variables.

Similarly the pessimistic formulation is given by


[edit] Methods for solving the problem

[edit] Examples

[edit] See also

[edit] References

[edit] External links