Bilevel Programming: Global Optimization

A large number of mathematical programming problems have optimization problems in their constraints. Arising from the areas of game theory and multicriteria decision making, these bilevel programming problems (BPP) take the form:

(1)

where x ∈ R ⁿ ₁, y ∈ R ⁿ ₂ and the functions F(x, y), f(x, y), G(x, y) and g(x, y) are continuous and twice differentiable. It is generally assumed that these functions are convex; the case of nonconvex functions has not been considered in the literature so far (as of 2000).

Bilevel programming has its origins in Stackelberg game theory , in particular from models of two-person nonzero-sum games. In these games, two players make alternate moves in a pre-established order. The first player (the leader) selects a move, x, that optimizes his own cost function. The second player (the follower) then has to make a move ythat is constrained by the prior decision of the leader. The follower has access only to his own cost function, while the leader is aware of...