A note on the MIR closure and basic relaxations of polyhedra

Anderson, Cornuéjols and Li (2005) show that for a polyhedral mixed integer set defined by a constraint system Ax ≥ b, where x is n-dimensional, 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 xj ≤ 0 or xj ≥ 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.