On the Complexity of Inverse Mixed Integer Linear Optimization

Inverse optimization is the problem of determining the values of missing input parameters that are closest to given estimates and that will make a given target solution optimal. This study is concerned with inverse mixed integer linear optimization problems (MILPs) in which the missing parameters are objective function coefficients. This class generalizes the class studied by Ahuja and Orlin [2001], who showed that inverse linear optimization problems can be solved in polynomial time under mild conditions. We extend their result to the discrete case and show that the decision version of the inverse MILP is coNP–complete, while the optimal value verification problem is D–complete. We derive a cutting plane algorithm for solving inverse MILPs and show that there is a close relationship between the inverse problem and the well-known separation problem in both a complexity and an algorithmic sense.