A characterization of maximal homogeneous-quadratic-free sets

TL;DR

This paper characterizes maximal homogeneous-quadratic-free sets, extending the intersection cut framework to non-linear quadratic inequalities and providing a broader class of maximal sets through a new geometric characterization.

Contribution

It introduces a new characterization of maximal quadratic-free sets using non-expansive functions and sequences, broadening the understanding of $S$-free sets beyond linear cases.

Findings

Maximal quadratic-free sets are generated by non-expansive functions.

Every full-dimensional maximal quadratic-free set corresponds to some generating function.

The new characterization applies to general $S$-free sets beyond quadratic inequalities.

Abstract

The intersection cut framework was introduced by Balas in 1971 as a method for generating cutting planes in integer optimization. In this framework, one uses a full-dimensional convex $S$-free set, where $S$ is the feasible region of the integer program, to derive a cut separating $S$ from a non-integral vertex of a linear relaxation of $S$. Among all $S$-free sets, it is the inclusion-wise maximal ones that yield the strongest cuts. Recently, this framework has been extended beyond the integer case in order to obtain cutting planes in non-linear settings. In this work, we consider the specific setting when $S$ is defined by a homogeneous quadratic inequality. In this 'quadratic-free' setting, every function $\Gamma: D^m \to D^n$, where $D^k$ is the unit disk in $\mathbb{R}^k$, generates a representation of a quadratic-free set. While not every $\Gamma$ generates a maximal quadratic free set, it is the case that every full-dimensional maximal quadratic free set is generated by some $\Gamma$. Our main result shows that the corresponding quadratic-free set is full-dimensional and maximal if and only if $\Gamma$ is non-expansive and satisfies a technical condition. This result yields a broader class of maximal $S$-free sets than previously known. Our result stems from a new characterization of maximal $S$-free sets (for general $S$ beyond the quadratic setting) based on sequences that 'expose' inequalities defining the $S$-free set.