Parametric global optimisation for bilevel programming

We propose a global optimisation approach for the solution of various classes of bilevel programming problems based on recently developed parametric programming algorithms. We first describe how we can recast and solve the inner (follower's) problem of the bilevel formulation as a multi-parametric programming problem, with parameters being the (unknown) variables of the outer (leader's) problem. By inserting the obtained rational reaction sets in the upper level problem the overall problem is transformed into a set of independent quadratic, linear or mixed integer linear programming problems, which can be solved to global optimality. In particular, we solve bilevel quadratic and bilevel mixed integer linear problems, with or without right-hand-side uncertainty. A number of examples are presented to illustrate the steps and details of the proposed global optimisation strategy.