A note on the MIR closure and basic relaxations of polyhedra

Anderson et al. (2005) show that for a polyhedral mixed integer set defined by a constraint system $Ax=b$ , along with integrality restrictions on some of the variables, any split cut is in fact a split cut for a basic relaxation, i.e., one defined by a subset of linearly independent constraints. This result implies that any split cut can be obtained as an intersection cut. Equivalence between split cuts obtained from simple disjunctions of the form $x_j=0$ or $x_j=1$ and intersection cuts was shown earlier for $0/1$ ‐mixed integer sets by Balas and Perregaard (2002) . We give a short proof of the result of Anderson, Cornuéjols and Li using the equivalence between mixed integer rounding (MIR) cuts and split cuts.