An analysis of mixed integer linear sets based on lattice point free convex sets

Split cuts are cutting planes for mixed integer programs whose validity is derived from maximal lattice point free polyhedra of the form $S:={x : \pi_0 \leq \pi^T x \leq \pi_0+1 }$ called split sets. The set obtained by adding all split cuts is called the split closure, and the split closure is known to be a polyhedron. A split set $S$ has max-facet-width equal to one in the sense that $\max \{{\pi }^{T}x:x\in S\}-\min \{{\pi }^{T}x:x\in S\}\le 1$. In this paper we consider using general lattice point free rational polyhedra to derive valid cuts for mixed integer linear sets. We say that lattice point free polyhedra with max-facet-width equal to $w$ have width size $w$. A split cut of width size $w$ is then a valid inequality whose validity follows from a lattice point free rational polyhedron of width size $w$. The $w$-th split closure is the set obtained by adding all valid inequalities of width size at most $w$. Our main result is a sufficient condition for the addition of a family of rational inequalities to result in a polyhedral relaxation. We then show that a corollary is that the $w$-th split closure is a polyhedron. Given this result, a natural question is which width size $w^{\ast }$ is required to design a finite cutting plane proof for the validity of an inequality. Specifically, for this value $w^{\ast }$, a finite cutting plane proof exists that uses lattice point free rational polyhedra of width size at most $w^{\ast }$, but no finite cutting plane proof that only uses lattice point free rational polyhedra of width size smaller than $w^{\ast }$. We characterize $w^{\ast }$ based on the faces of the linear relaxation.