Bilevel Programming: Introduction, History and Overview

The bilevel programming (BP) problem is a hierarchical optimization problem where a subset of the variables is constrained to be a solution of a given optimization problem pa-rameterized by the remaining variables. The BP problem is a multilevel programming problem with two levels. The hierarchical optimization sctructure appears naturally in many applications when lower level actions depend on upper level decisions. The applications of bilevel and multilevel programming include transportation (taxation, network design, trip demand estimation), management (coordination of multi-divisional rms, network facility location, credit allocation), planning (agricultural policies, electric utility), and optimal design. In mathematical terms, the BP problem consists of nding a solution to the upper level problem minimize x;y F(x; y) subject to g(x; y) 0; where y, for each value of x, is the solution of the lower level problem: minimize y f(x; y) subject to h(x; y) 0; with x 2 < nl. The lower level problem is also referred as the fol-lower's problem or the inner problem. In a similar way, the upper level problem is also called the leader's problem or the outer problem. One could generalize the BP problem in diierent ways. For instance, if either x or y or both are restricted to take integer values we would obtain an integer BP problem 20]. Or, if we replace the lower level problem by a variational inequality we would get a generalized BP problem 14]. For each value of the upper level variables x, the lower level constraints h(x; y) 0 deene the constraint set (x) of the lower level problem: (x) = n y : h(x; y) 0 o : Then, the set M(x) of solutions for the lower level problem is given by minimizing the lower level function f(x; y) for all values in (x) of the lower level variables y: