Bilevel Programming Problems

Abstract

A problem where an optimization problem is constrained by another one is classified as a BiLevel Programming Problem, BLPP, and is of the general form:

$$ \begin{array}{l} \mathop {\min }\limits_{x,y} \quad F(x,y) \\ s.t. \\ \quad \quad \;G(x,y) \le 0 \\ \quad \quad \;H(x,y) = 0 \\ \quad \quad \;\mathop {\min }\limits_y f(x,y) \\ \quad \quad \;s.t. \\ \quad \quad \;\quad g(x,y) \le 0 \\ \quad \quad \;\quad h(x,y) = 0 \\ \quad \quad \;\quad x \in X \subseteq {R^{n1}},\;y \in Y \subseteq {R^{n2}} \\ \end{array} $$

where f, F : R ⁿ¹ × R ⁿ² →; R, g = [g ₁,.., g _J ] : R ⁿ¹ × R ⁿ² → R ^j, G = [G ₁,.., G _J ] : R ⁿ¹ × R ⁿ² → R ^j’, h = [h ₁,..,h _I ] : R ⁿ¹ × R ⁿ² → R ⁱ, H = [H ₁,.., H’ _I ] : R ⁿ¹ × R ⁿ² → R ^i’. F, G and H are the outer (planner’s or leader’s) problem objective function, inequality and equality constraints, and f, g, and h are the inner (behavorial or follower’s) problem objective, inequality and equality constraints, respectively. The decision variables of the outer problem are x and y and of the inner problem are y.