Santanu S. Dey In this note, we present a simple geometric argument to determine a lower bound on the split rank of intersection cuts. As a first step of this argument, a polyhedral subset of the lattice-free convex set that is used to generate the intersection cut is constructed. We call this subset the restricted lattice-free set. It … Read more
Recently, Andersen et al.(2007), Borozan and Cornuejols (2007) and Cornuejols and Margot(2007) characterized extreme inequalities of a system of two rows with two free integer variables and nonnegative continuous variables. These inequalities are either split cuts or intersection cuts (Balas (1971)) derived using maximal lattice-free convex sets. In order to use these inequalities to obtain … Read more
In this paper, we introduce an operation that creates families of facet-defining inequalities for high-dimensional infinite group problems using facet-defining inequalities of lower-dimensional group problems. We call this family sequential-merge inequalities because they are produced by applying two group cuts one after the other and because the resultant inequality depends on the order of the … Read more
We propose an approximate lifting procedure for general integer programs. This lifting procedure uses information from multiple constraints of the problem formulation and can be used to strengthen formulations and cuts for mixed integer programs. In particular we demonstrate how it can be applied to improve Gomory’s fractional cut which is central to Glover’s primal … Read more
In this paper we derive new properties of extreme inequalities for infinite group problems. We develop tools to prove that given valid inequalities for the infinite group problem are extreme. These results show that integer infinite group problems have discontinuous extreme inequalities. These inequalities are strong when compared to related classes of continuous extreme inequalities. … Read more
In this paper, we lay the foundation for the study of the two-dimensional mixed integer infinite group problem (2DMIIGP). We introduce tools to determine if a given continuous and piecewise linear function over the two-dimensional infinite group is subadditive and to determine whether it defines a facet. We then present two different constructions that yield … Read more