On the strength of Burer’s lifted convex relaxation to quadratic programming with ball constraints
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| Titel: | On the strength of Burer’s lifted convex relaxation to quadratic programming with ball constraints |
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| Autoren: | Fatma Kılınç-Karzan, Shengding Sun |
| Weitere Verfasser: | Apollo - University of Cambridge Repository |
| Quelle: | Mathematical Programming. |
| Publication Status: | Preprint |
| Verlagsinformationen: | Springer Science and Business Media LLC, 2025. |
| Publikationsjahr: | 2025 |
| Schlagwörter: | Optimization and Control (math.OC), 4901 Applied Mathematics, 4903 Numerical and Computational Mathematics, 49 Mathematical Sciences, FOS: Mathematics, 10 Reduced Inequalities, Mathematics - Optimization and Control |
| Beschreibung: | We study quadratic programs with m ball constraints, and the strength of a lifted convex relaxation for it recently proposed by Burer (2024). Burer shows this relaxation is exact when $$m=2$$ m = 2 . For general m, Burer (2024) provides numerical evidence that this lifted relaxation is tighter than the Kronecker product based Reformulation Linearization Technique (RLT) inequalities introduced by Anstreicher (2017), and conjectures that this must be theoretically true as well. In this note, we provide an affirmative answer to this question and formally prove that this lifted relaxation indeed implies the Kronecker inequalities in the original space. Our proof is based on a decomposition of non-rank-one extreme rays of the lifted relaxation for each pair of ball constraints. Burer (2024) also numerically observes that for this lifted relaxation, an RLT-based inequality proposed by Zhen et al. (2021) is redundant, and conjectures this to be theoretically true as well. We also provide a formal proof that Zhen et al. (2021)’s as well as Jiang and Li (2019)’s SST inequalities are redundant for this lifted relaxation. In addition, we establish that Burer’s lifted relaxation is a particular case of the moment-sum-of-squares hierarchy. |
| Publikationsart: | Article |
| Dateibeschreibung: | application/pdf |
| Sprache: | English |
| ISSN: | 1436-4646 0025-5610 |
| DOI: | 10.1007/s10107-025-02278-1 |
| DOI: | 10.48550/arxiv.2407.14992 |
| Zugangs-URL: | http://arxiv.org/abs/2407.14992 |
| Rights: | CC BY |
| Dokumentencode: | edsair.doi.dedup.....b975dc3c6e27ae5eebd09bab005e9e9e |
| Datenbank: | OpenAIRE |
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Nájsť tento článok vo Web of Science | Abstract: | We study quadratic programs with m ball constraints, and the strength of a lifted convex relaxation for it recently proposed by Burer (2024). Burer shows this relaxation is exact when $$m=2$$ m = 2 . For general m, Burer (2024) provides numerical evidence that this lifted relaxation is tighter than the Kronecker product based Reformulation Linearization Technique (RLT) inequalities introduced by Anstreicher (2017), and conjectures that this must be theoretically true as well. In this note, we provide an affirmative answer to this question and formally prove that this lifted relaxation indeed implies the Kronecker inequalities in the original space. Our proof is based on a decomposition of non-rank-one extreme rays of the lifted relaxation for each pair of ball constraints. Burer (2024) also numerically observes that for this lifted relaxation, an RLT-based inequality proposed by Zhen et al. (2021) is redundant, and conjectures this to be theoretically true as well. We also provide a formal proof that Zhen et al. (2021)’s as well as Jiang and Li (2019)’s SST inequalities are redundant for this lifted relaxation. In addition, we establish that Burer’s lifted relaxation is a particular case of the moment-sum-of-squares hierarchy. |
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| ISSN: | 14364646 00255610 |
| DOI: | 10.1007/s10107-025-02278-1 |