A note on linearized reformulations for a class of bilevel linear integer problems

We consider reformulations of a class of bilevel linear integer programs as equivalent linear mixed-integer programs (linear MIPs). The most common technique to reformulate such programs as a single-level problem is to replace the lower-level linear optimization problem by Karush-Kuhn-Tucker (KKT) optimality conditions. Employing the strong duality (SD) property of linear programs is an alternative method to perform such transformations. In this note, we describe two SD-based reformulations where the key idea is to exploit the binary expansion of upper-level integer variables. We compare the performance of an off-the-shelf MIP solver with the SD-based reformulations against the KKT-based one and show that the SD-based approaches can lead to orders of magnitude reduction in computational times for certain classes of instances.