Bilevel Linear Programming: Complexity, Equivalence to Minmax, Concave Programs

A sequential optimization problem in which independent decision makers act in a noncooperative manner to minimize their individual costs, may be categorized as a Stackelberg game. The bilevel programming problem (BLPP) is a static, open loop version of this game where the leader controls the decision variables x ∈ X ⊆ R ⁿ, while the follower separately controls the decision variables y ∈ Y ⊆ R ^m (e.g., see [3], [9]).

In the model, it is common to assume that the leader goes first and chooses an x to minimize his objective function F(x, y). The follower then reacts by selecting a y to minimize his individual objective function f(x, y) without regard to the impact this choice has on the leader. Here, F: X × Y → R ¹ and f: X × Y → R ¹. The focus of this article is on the linear case introduced in Bilevel linear programming and given by:

(1)

(2)

(3)

(4)

where c ₁, c ₂ ∈ R ⁿ, d ₁, d ₂ ∈ R ^m, b ₁ ∈ R ^p, b ₂ ∈ R ^q, A ₁ ∈ R ^{p × n}, B ₁ ∈ R ^{p × m}, A ₂ ∈ R ^{q × n}, B ₂ ∈ R ^{q × m}. The sets X and Y...