The bilevel programming (BP) problem is a hierarchical optimization problem where a subset of the variables is constrained to be a solution of a given optimization problem parameterized by the remaining variables. The BP problem is a multilevel programming problem with two levels. The hierarchical optimization structure appears naturally in many applications when lower level actions depend on upper level decisions. The applications of bilevel and multilevel programming include transportation (taxation, network design, trip demand estimation), management (coordination of multidivisional firms, network facility location, credit allocation), planning (agricultural policies, electric utility), and optimal design.
In mathematical terms, the BP problem consists of finding a solution to the upper level problem
where y, for each value of x, is the solution of the lower level problem:
with x ∈ R nx, y ∈ R ny, F, f : R nx + ny → R, g : R nx + ny → R nu, and h : R nx + ny → R nl (nx, ny, nu, and nl...
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Vicente, L.N. (2001). Bilevel Programming: Introduction, History and Overview . In: Floudas, C.A., Pardalos, P.M. (eds) Encyclopedia of Optimization. Springer, Boston, MA. https://doi.org/10.1007/0-306-48332-7_38
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