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arXiv · 2605.30602

A characterization of maximal inhomogeneous-quadratic-free sets

Gonzalo Muñoz·Joseph Paat·Felipe Serrano
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Abstract

The intersection cut framework is a versatile tool for generating valid inequalities in optimization. Its main ingredients are so-called $S$-free sets: convex sets whose interiors do not intersect a given set $S$. Among these, inclusion-wise maximal $S$-free sets are particularly important, as they yield the strongest intersection cuts. In the integer programming setting, maximal lattice-free sets are well studied and admit explicit characterizations. In the quadratic optimization context, Mu\~noz, Paat, and Serrano (2025) characterized maximal $S$-free sets when $S$ is defined by a homogeneous quadratic inequality. In this work, we characterize maximal $S$-free sets when $S$ is defined by an inhomogeneous quadratic inequality. As in the homogeneous case, our characterization is built using non-expansive functions. Together with the results in the homogeneous case, our results complete a characterization of $every$ maximal quadratic-free set via non-expansive functions.

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Gonzalo Muñoz, Joseph Paat, Felipe Serrano. 2026-05-28. A characterization of maximal inhomogeneous-quadratic-free sets. https://arxiv.org/abs/2605.30602

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