Global solution of nonlinear mixed-integer bilevel programs

An algorithm for the global optimization of nonlinear bilevel mixed-integer programs is presented, based on a recent proposal for continuous bilevel programs by Mitsos et al. (J Glob Optim 42(4):475–513, 2008). The algorithm relies on a convergent lower bound and an optional upper bound. No branching is required or performed. The lower bound is obtained by solving a mixed-integer nonlinear program, containing the constraints of the lower-level and upper-level programs; its convergence is achieved by also including a parametric upper bound to the optimal solution function of the lower-level program. This lower-level parametric upper bound is based on Slater-points of the lower-level program and subsets of the upper-level host sets for which this point remains lower-level feasible. Under suitable assumptions the KKT necessary conditions of the lower-level program can be used to tighten the lower bounding problem. The optional upper bound to the optimal solution of the bilevel program is obtained by solving an augmented upper-level problem for fixed upper-level variables. A convergence proof is given along with illustrative examples. An implementation is described and applied to a test set comprising original and literature problems. The main complication relative to the continuous case is the construction of the parametric upper bound to the lower-level optimal objective value, in particular due to the presence of upper-level integer variables. This challenge is resolved by performing interval analysis over the convex hull of the upper-level integer variables.