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Quadratic programs with hollows

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.

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Notes

  1. 1.

    More formally, \({{\mathrm{proj}}}_x(\mathcal {S}(\mathcal {F}))\) is a relaxation of \(\mathcal {F}\), where \({{\mathrm{proj}}}_x(\cdot )\) denotes projection onto the x coordinates. We ignore this distinction between \(\mathcal {S}(\mathcal {F})\) and \({{\mathrm{proj}}}_x(\mathcal {S}(\mathcal {F}))\) to reduce notation.

  2. 2.

    To be usable in practice, a valid SDP relaxation \(\mathcal {R}(\mathcal {F})\) should have a known positive semidefinite (PSD) representation [16, Section 6.4]. However, it is convenient in this note to consider \(\mathcal {R}(\mathcal {F})\) to be a valid SDP relaxation regardless of whether or not an explicit PSD representation for \(\mathcal {R}(\mathcal {F})\) is known. We also apply this terminology to \(\mathcal {C}(\mathcal {F})\), which in fact may not have an explicit PSD representation—although the PSD constraint is always valid for \(\mathcal {C}(\mathcal {F})\).

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Acknowledgements

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

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Correspondence to Boshi Yang.

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Yang, B., Anstreicher, K. & Burer, S. Quadratic programs with hollows. Math. Program. 170, 541–553 (2018). https://doi.org/10.1007/s10107-017-1157-0

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Keywords

  • Nonconvex quadratic programming
  • Semidefinite programming
  • Convex hull

Mathematics Subject Classification

  • 90C20
  • 90C22
  • 90C25
  • 90C26
  • 90C30