Classification of Dantzig–Wolfe reformulations for binary mixed integer programming problems

In this note, we provide a classification of Dantzig–Wolfe reformulations for Binary Mixed Integer Programming Problems. We specifically focus on modeling the binary conditions in the convexification approach to the Dantzig–Wolfe decomposition. For a general Binary Mixed Integer Programming problem, an extreme point of the overall problem does not necessarily correspond to an extreme point of the subproblem. Therefore, the binary conditions cannot in general be imposed on the new master problem variables but must be imposed on the original binary variables. In some cases, however, it is possible to impose the binary restrictions directly on the new master problem variables. The issue of imposing binary conditions on the original variables versus the master problem variables has not been discussed systematically for MIP problems in general in the literature and most of the research has been focused on the pure binary case. The classification indicates in which cases you can, and cannot, impose binary conditions on the new master problem variables.