  Inverse optimization is the problem of determining the values of missing
input parameters for an associated forward problem that are closest to given
estimates and that will make a given target vector optimal. This study is
concerned with the relationship of a particular inverse mixed integer linear
optimization problem (MILP) to both the forward problem and the separation
problem associated with its feasible region. We show that a decision version of
the inverse MILP in which a primal bound is verified is coNP-complete, whereas
primal bound verification for the associated forward problem is NP-complete,
and that the optimal value verification problems for both the inverse problem
and the associated forward problem are complete for the complexity class D^P.
We also describe a cutting-plane algorithm for solving inverse MILPs that
illustrates the close relationship between the separation problem for the
convex hull of solutions to a given MILP and the associated inverse problem.
The inverse problem is shown to be equivalent to the separation problem for the
radial cone defined by all inequalities that are both valid for the convex hull
of solutions to the forward problem and binding at the target vector. Thus, the
inverse, forward, and separation problems can be said to be equivalent.
