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Mathematical Programming

, Volume 170, Issue 2, pp 541–553 | Cite as

Quadratic programs with hollows

  • Boshi Yang
  • Kurt Anstreicher
  • Samuel Burer
Short Communication Series A

Abstract

Let \(\mathcal {F}\) be a quadratically constrained, possibly nonconvex, bounded set, and let \(\mathcal {E}_1, \ldots , \mathcal {E}_l\) denote ellipsoids contained in \(\mathcal {F}\) with non-intersecting interiors. We prove that minimizing an arbitrary quadratic \(q(\cdot )\) over \(\mathcal {G}:= \mathcal {F}{\setminus } \cup _{k=1}^\ell {{\mathrm{int}}}(\mathcal {E}_k)\) is no more difficult than minimizing \(q(\cdot )\) over \(\mathcal {F}\) in the following sense: if a given semidefinite-programming (SDP) relaxation for \(\min \{ q(x) : x \in \mathcal {F}\}\) is tight, then the addition of l linear constraints derived from \(\mathcal {E}_1, \ldots , \mathcal {E}_l\) yields a tight SDP relaxation for \(\min \{ q(x) : x \in \mathcal {G}\}\). We also prove that the convex hull of \(\{ (x,xx^T) : x \in \mathcal {G}\}\) equals the intersection of the convex hull of \(\{ (x,xx^T) : x \in \mathcal {F}\}\) with the same l linear constraints. Inspired by these results, we resolve a related question in a seemingly unrelated area, mixed-integer nonconvex quadratic programming.

Keywords

Nonconvex quadratic programming Semidefinite programming Convex hull 

Mathematics Subject Classification

90C20 90C22 90C25 90C26 90C30 

Notes

Acknowledgements

The authors would like to thank two anonymous referees and the associate editor for helpful suggestions and insights.

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Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2017

Authors and Affiliations

  1. 1.Department of Mathematical SciencesClemson UniversityClemsonUSA
  2. 2.Department of Management SciencesUniversity of IowaIowa CityUSA

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