A characterization of maximal inhomogeneous-quadratic-free sets
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866910270877597696 |
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| author | Muñoz, Gonzalo Paat, Joseph Serrano, Felipe |
| author_facet | Muñoz, Gonzalo Paat, Joseph Serrano, Felipe |
| contents | 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, Muñoz, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_30602 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A characterization of maximal inhomogeneous-quadratic-free sets Muñoz, Gonzalo Paat, Joseph Serrano, Felipe Optimization and Control 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, Muñoz, 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. |
| title | A characterization of maximal inhomogeneous-quadratic-free sets |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2605.30602 |