Bilevel Programming

Let us consider a sequential game where the first player (‘leader’) incorporates into his optimization process the optimal reaction vector y of the second player (‘follower’) to the leader’s decision vector x. This situation is described mathematically by the bilevel program

where it is understood that the leader is requested to select a vector x such that the parameterized set Y(x) is nonempty.

This formulation is extremely general in that it subsumes linear zero-one optimization , quadratic concave programming , disjoint bilinear programming, nonlinear complementarity , etc. If one denotes by y(x) the set of optimal answers to a given leader vector x, the above bilevel program can be recast as the’ standard’ mathematical program

The induced region of a bilevel program is defined as the feasible set of the above program. This set is usually nonconvex and might be disconnected. It is implicit that, whenever y(x) is not a singleton, the leader is free to select that element y ∈ y(x)...