Special issue on “Applied bilevel programming”

min x1,x2 f1(x1, x2) subject to (x1, x2) ∈ X1, x2 ∈ S(x1), where S(x1) = argminx2∈X2(x1) f2(x1, x2). This situation arises whenever the upper level agent embeds within its optimization process the reaction of the lower level to its course of actions, and arises in fields as diverse as economics, telecommunications, transportation, engineering, or chemistry. In the past decade, the field has developed into an important area of mathematical programming, with several articles devoted to theory, models, and computational methods. Bilevel programs are closely related to leader–follower games, for which the Stackelberg duopoly game is the quintessential example. They are also related to the class of mathematical programs with equilibrium constraints (MPECs), where the elements of the set S(x1) are equilibrium states of a system parameterized in x1, frequently expressed in terms of the variational inequality