Bilevel Fractional Programming

Bilevel fractional programming, a class of bilevel programming ([4]; cf. also Bilevel programming: Introduction, history and overview), has been proposed as a generalization of standard fractional programming ([3]) for dealing with hierarchical systems with two decision levels. This problem assumes that objective functions of both levels are ratios of functions and the common constraint region to both levels, or feasible region, is a nonempty and compact polyhedron ([1]). It can be stated as follows:

where x ₁ ∈ R ⁿ ₁ and x ₂ ∈ R ⁿ ₂ are the variables controlled by the first level and the second level decision maker, respectively; h _i and g _i are continuous functions, h _i are nonnegative and concave and g _i are positive and convex on S; and S = {(x ₁, x ₂): A ¹ x ₁ + A ² x ₂ ≤ b, x ₁, x ₂ ≥ 0}, which is assumed to be nonempty and bounded. To assure that the BFP problem is well posed it is also assumed that, for each value of the first level variables x ₁, there is a unique solution to...