A note on the MIR closure

In 1988, Nemhauser and Wolsey introduced the concept of MIR inequality for mixed integer linear programs. In 1998, Wolsey defined MIR inequalities differently. In some sense these definitions are equivalent. However, this note points out that the natural concepts of MIR closures derived from these two definitions are distinct. Dash, Günlük and Lodi made the same observation independently. Let S := {(x, y) ∈ Z+ × R p + : Ax + Gy ≤ b} be a mixed integer set. Here A ∈ Rm×n and G ∈ Rm×p are matrices and b ∈ R is a vector. Let P := {(x, y) ∈ R+ × R p + : Ax + Gy ≤ b} be the polyhedron that arises as the natural linear relaxation of S. We assume P 6= ∅. Nemhauser and Wolsey [6,7] define MIR inequalities by the following procedure. If cx+ hy ≤ c0 and cx+ hy ≤ c0 Email addresses: [email protected] (Pierre Bonami ), [email protected] (Gérard Cornuéjols ). 1 Supported in part by a grant from IBM. 2 Supported in part by NSF grant DMI-0352885 and ONR grant N00014-03-1-0188. Preprint submitted to Elsevier 13 August 2006 are valid inequalities for P , and π = c − c ∈ Z, π0 = bc0 − c0c and γ = c0 − c0 − π0, then πx+ ( cx+ hy − c0 ) /(1− γ) ≤ π0 is valid for S. Define the MIR closure as the intersection of all MIR inequalities. Nemhauser and Wolsey [7] proved that the MIR closure is identical to the split closure [1] and the Gomory mixed integer closure [4] (see [2] for another proof of the last identity). Later, Wolsey [8] (see also Marchand and Wolsey [5]) defined the MIR inequality as being generated from a single constraint ax + gy ≤ b where (x, y) ∈ Z+×R p +. Specifically, let f0 := b−bbc and fj := aj−bajc. The MIR inequality is