Mixed-integer linear representability, disjunctions, and variable elimination

TL;DR

This paper provides an algebraic characterization of MILP-representable sets using affine Chvátal inequalities, offering a variable elimination scheme that improves upon previous disjunctive methods and clarifies the limits of existing approaches.

Contribution

It introduces an algebraic characterization of MILP-representable sets via affine Chvátal inequalities and extends variable elimination techniques to improve upon prior disjunctive schemes.

Findings

Sets can be characterized by finitely many affine Chvátal inequalities.

Disjunctions are unnecessary for describing MILP-representable sets.

The new scheme can identify inequalities not found by previous disjunctive methods.

Abstract

Jeroslow and Lowe gave an exact geometric characterization of subsets of $\mathbb{R}^n$ that are projections of mixed-integer linear sets, a.k.a MILP-representable sets. We give an alternate algebraic characterization by showing that a set is MILP-representable {\em if and only if} the set can be described as the intersection of finitely many {\em affine Chv\'atal inequalities}. These inequalities are a modification of a concept introduced by Blair and Jeroslow. This gives a sequential variable elimination scheme that, when applied to the MILP representation of a set, explicitly gives the affine Chv\'atal inequalities characterizing the set. This is related to the elimination scheme of Wiliams and Williams-Hooker, who describe projections of integer sets using \emph{disjunctions} of Chv\'atal systems. Our scheme extends their work in two ways. First, we show that disjunctions are unnecessary, by showing how to find the affine Chv\'atal inequalities that cannot be discovered by the Williams-Hooker scheme. Second, disjunctions of Chv\'atal systems can give sets that are \emph{not} projections of mixed-integer linear sets; so the Williams-Hooker approach does not give an exact characterization of MILP representability.