Abstract
In 1971, Balas introduced intersection cuts as a method for generating cutting planes in integer optimization. These cuts are derived from convex S-free sets, and inclusion-wise maximal S-free sets yield the strongest intersection cuts. When S is a lattice, maximal S-free sets are well-studied. In this work, we provide a new characterization of maximal S-free sets, for arbitrary S, based on sequences that ‘expose’ inequalities defining the S-free set; these exposing sequences generalize the notion of blocking points when S is a lattice. We then apply our characterization to partially characterize maximal S-free polyhedra when S is defined by a homogeneous quadratic inequality. Our results generate new families of maximal quadratic-free sets and considerably generalize some of the constructions by Muñoz and Serrano (IPCO 2020), who first introduced maximal quadratic-free sets.
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Notes
- 1.
Note that Lemma 1 does not directly imply that \(\gamma (\boldsymbol{\beta })\) can be assumed to have unit norm. Moreover, one could produce a (not necessarily maximal) Q-free set with \(\gamma \) that satisfies \(\Vert \gamma (\boldsymbol{\beta })\Vert < 1\) for some \(\boldsymbol{\beta }\).
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Acknowledgements
The second author was supported by a Natural Sciences and Engineering Research Council of Canada (NSERC) Discovery Grant [RGPIN-2021-02475]. The authors would like to thank the three anonymous reviewers for their valuable feedback.
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Muñoz, G., Paat, J., Serrano, F. (2023). Towards a Characterization of Maximal Quadratic-Free Sets. In: Del Pia, A., Kaibel, V. (eds) Integer Programming and Combinatorial Optimization. IPCO 2023. Lecture Notes in Computer Science, vol 13904. Springer, Cham. https://doi.org/10.1007/978-3-031-32726-1_24
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