  Inverse optimization, determining parameters of an optimization problem that
render a given solution optimal, has received increasing attention in recent
years. While significant inverse optimization literature exists for convex
optimization problems, there have been few advances for discrete problems,
despite the ubiquity of applications that fundamentally rely on discrete
decision-making. In this paper, we present a new set of theoretical insights
and algorithms for the general class of inverse mixed integer linear
optimization problems. Specifically, a general characterization of optimality
conditions is established and leveraged to design new cutting plane solution
algorithms. Through an extensive set of computational experiments, we show that
our methods provide substantial improvements over existing methods in solving
the largest and most difficult instances to date.
