A Value-Function-Based Exact Approach for the Bilevel Mixed-Integer Programming Problem

We examine bilevel mixed-integer programs whose constraints and objective functions depend on both upper- and lower-level variables. The class of problems we consider allows for nonlinear terms to appear in both the constraints and the objective functions, requires all upper-level variables to be integer, and allows a subset of the lower-level variables to be integer. This class of bilevel problems is difficult to solve because the upper-level feasible region is defined in part by optimality conditions governing the lower-level variables, which are difficult to characterize because of the nonconvexity of the follower problem. We propose an exact finite algorithm for these problems based on an optimal-value-function reformulation. We demonstrate how this algorithm can be tailored to accommodate either optimistic or pessimistic assumptions on the follower behavior. Computational experiments demonstrate that our approach outperforms a state-of-the-art algorithm for solving bilevel mixed-integer linear programs.