Bilevel program

In mathematics, bilevel programs are optimization problems where one optimization problem is embedded in another one.Bilevel programs are multilevel programs with two levels [http://glossary.computing.society.informs.org/second.php?page=B.html#Bilevel_program] .

Mathematical formulation of the problem

The optimistic formulation of bilevel programs problem can be stated simply as::$minlimits\_\{x\; in\; X,\; y\; in\; Y\};;\; f^u(x,y)$

:$mbox\{subject\; to:\; \}$

::$g^u(x,y)\; le\; 0,\; ;;$

::$y\; in\; argminlimits\_\{z\; in\; Y\}\; f^l(x,z)$

::$g^l(x,z)\; le\; 0,\; ;;$

where:$f^u,f^l:\; R^\{nx\}\; imes\; R^\{ny\}\; o\; R$:$g^u,g^l:\; R^\{nx\}\; imes\; R^\{ny\}\; o\; R^\{mu\}$:$X\; subseteq\; R^\{nx\}$:$Y\; subseteq\; R^\{ny\}.$

The variables z are dummy variables.

Similarly the pessimistic formulation is given by :$minlimits\_\{x\; in\; X\}maxlimits\_\{y\; in\; Y\};;\; f^u(x,y)$

:$mbox\{subject\; to:\; \}$

::$g^u(x,y)\; le\; 0,\; ;;$

::$y\; in\; argminlimits\_\{z\; in\; Y\}\; f^l(x,z)$

::$g^l(x,z)\; le\; 0,\; ;;$

Methods for solving the problem

Examples

See also

* optimization

References

External links

* [http://glossary.computing.society.informs.org/ Mathematical Programming Glossary]