Maximal Quadratic-Free Sets
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The intersection cut paradigm is a powerful framework that facilitates the generation of valid linear inequalities, or cutting planes, for a potentially complex set S. The key ingredients in this construction are a simplicial conic relaxation of S and an S-free set: a convex zone whose interior does not intersect S. Ideally, such S-free set would be maximal inclusion-wise, as it would generate a deeper cutting plane. However, maximality can be a challenging goal in general. In this work, we show how to construct maximal S-free sets when S is defined as a general quadratic inequality. Our maximal S-free sets are such that efficient separation of a vertex in LP-based approaches to quadratically constrained problems is guaranteed. To the best of our knowledge, this work is the first to provide maximal quadratic-free sets.
KeywordsNon-convex quadratic Intersection cut S-free sets
We are indebted to Franziska Schlösser for several inspiring conversations. We would like to thank Stefan Vigerske, Antonia Chmiela, Ksenia Bestuzheva and Nils-Christian Kempke for helpful discussions. We would also like to thank the three anonymous reviewers for their valuable feedback. Lastly, we would like to acknowledge the support of the IVADO Institute for Data Valorization for their support through the IVADO Post-Doctoral Fellowship program and to the IVADO-ZIB academic partnership. The described research activities are funded by the German Federal Ministry for Economic Affairs and Energy within the project EnBA-M (ID: 03ET1549D). The work for this article has been (partly) conducted within the Research Campus MODAL funded by the German Federal Ministry of Education and Research (BMBF grant number 05M14ZAM).
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