An algorithm for the mixed-integer nonlinear bilevel programming problem

The bilevel programming problem (BLPP) is a two-person nonzero sum game in which play in sequential and cooperation is not permitted. In this paper, the authors examine a class of BLPPs where the leader controls a set of continuous and discrete variables and tries to minimize a convex nonlinear objective function. The follower’s objective function is a convex quadratic in a continuous decision space. All constraints are assumed to be linear. A branch and bound algorithm is developed that finds global optima. The main purpose of this paper is to identify efficient branching rules, and to determine the computational burden of the numeric procedures. Extensive test results are reported. The authors close by showing that it is not readily possible to extend the algorithm to the more general case involving integer follower variables.