Mixed-integer linear representability, disjunctions, and Chvatal functions --- modeling implications

TL;DR

This paper provides an algebraic characterization of MILP-representable sets using affine Chvatal inequalities, simplifying the modeling process and resolving a long-standing open question about elimination schemes without disjunctions.

Contribution

It introduces an algebraic characterization of MILP-representable sets via affine Chvatal inequalities, eliminating the need for disjunctions in modeling and solving open problems in the field.

Findings

MILP-R sets are exactly those describable by affine Chvatal inequalities with integer variables.

Projection of sets defined by affine Chvatal inequalities remains MILP-R.

A sequential elimination scheme yields the AC set characterization, answering Ryan's open question.

Abstract

Jeroslow and Lowe gave an exact geometric characterization of subsets of $\mathbb{R}^n$ that are projections of mixed-integer linear sets, also known as MILP-representable or MILP-R sets. We give an alternate algebraic characterization by showing that a set is MILP-R {\em if and only if} the set can be described as the intersection of finitely many {\em affine Chvatal inequalities} in continuous variables (termed AC sets). These inequalities are a modification of a concept introduced by Blair and Jeroslow. Unlike the case for linear inequalities, allowing for integer variables in Chvatal inequalities and projection does not enhance modeling power. We show that the MILP-R sets are still precisely those sets that are modeled as affine Chvatal inequalites with integer variables. Furthermore, the projection of a set defined by affine Chvatal inequalites with integer variables is still an MILP-R set. We give a sequential variable elimination scheme that, when applied to a MILP-R set yields the AC set characterization. This is related to the elimination scheme of Williams and Williams-Hooker, who describe projections of integer sets using \emph{disjunctions} of affine Chvatal systems. We show that disjunctions are unnecessary by showing how to find the affine Chvatal inequalities that cannot be discovered by the Williams-Hooker scheme. This allows us to answer a long-standing open question due to Ryan (1991) on designing an elimination scheme to represent finitely-generated integral monoids as a system of Chvatal inequalities \emph{without} disjunctions. Finally, our work can be seen as a generalization of the approach of Blair and Jeroslow, and Schrijver for constructing consistency testers for integer programs to general AC sets.