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- W4298892313 abstract "Lattice-free sets (convex subsets of $mathbb{R}^d$ without interior integer points) and their applications for cutting-plane methods in mixed-integer optimization have been studied in recent literature. Notably, the family of all integral lattice-free polyhedra which are not properly contained in another integral lattice-free polyhedron has been of particular interest. We call these polyhedra $mathbb{Z}^d$-maximal. It is known that, for fixed $d$, the family $mathbb{Z}^d$-maximal integral lattice-free polyhedra is finite up to unimodular equivalence. In view of possible applications in cutting-plane theory, one would like to have a classification of this family. However, this turns out to be a challenging task already for small dimensions. In contrast, the subfamily of all integral lattice-free polyhedra which are not properly contained in any other lattice-free set, which we call $mathbb{R}^d$-maximal lattice-free polyhedra, allow a rather simple geometric characterization. Hence, the question was raised for which dimensions the notions of $mathbb{Z}^d$-maximality and $mathbb{R}^d$-maximality are equivalent. This was known to be the case for dimensions one and two. On the other hand, Nill and Ziegler (2011) showed that for dimension $d ge 4$, there exist polyhedra which are $mathbb{Z}^d$-maximal but not $mathbb{R}^d$-maximal. In this article, we consider the remaining case $d = 3$ and prove that for integral polyhedra the notions of $mathbb{R}^3$-maximality and $mathbb{Z}^3$-maximality are equivalent. As a consequence, the classification of all $mathbb{R}^3$-maximal integral polyhedra by Averkov, Wagner and Weismantel (2011) contains all $mathbb{Z}^3$-maximal integral polyhedra." @default.
- W4298892313 created "2022-10-02" @default.
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- W4298892313 date "2015-09-17" @default.
- W4298892313 modified "2023-10-18" @default.
- W4298892313 title "Notions of maximality for integral lattice-free polyhedra: the case of dimension three" @default.
- W4298892313 doi "https://doi.org/10.48550/arxiv.1509.05200" @default.
- W4298892313 hasPublicationYear "2015" @default.
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