### Abstract

Original language | English |
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Pages (from-to) | 1475–1486 |

Number of pages | 12 |

Journal | Optimization Letters |

Volume | 12 |

Issue number | 7 |

DOIs | |

Publication status | Published - 1 Oct 2018 |

### Cite this

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**A spectrahedral representation of the first derivative relaxation of the positive semidefinite cone.** / Saunderson, James Francis.

Research output: Contribution to journal › Article › Research › peer-review

TY - JOUR

T1 - A spectrahedral representation of the first derivative relaxation of the positive semidefinite cone

AU - Saunderson, James Francis

PY - 2018/10/1

Y1 - 2018/10/1

N2 - If X is an n × n symmetric matrix, then the directional derivative of X →det(X) in the direction I is the elementary symmetric polynomial of degree n − 1 in the eigenvalues of X. This is a polynomial in the entries of X with the property that it is hyperbolic with respect to the direction I. The corresponding hyperbolicity cone is a relaxation of the positive semidefinite (PSD) cone known as the first derivative relaxation (or Renegar derivative) of the PSD cone. A spectrahedal cone is a convex cone that has a representation as the intersection of a subspace with the cone of PSD matrices in some dimension. We show that the first derivative relaxation of the PSD cone is a spectrahedral cone, and give an explicit spectrahedral description of sizen+12− 1. The construction provides a new explicit example of a hyperbolicity cone that is also a spectrahedron. This is consistent with the generalized Lax conjecture,which conjectures that every hyperbolicity cone is a spectrahedron.

AB - If X is an n × n symmetric matrix, then the directional derivative of X →det(X) in the direction I is the elementary symmetric polynomial of degree n − 1 in the eigenvalues of X. This is a polynomial in the entries of X with the property that it is hyperbolic with respect to the direction I. The corresponding hyperbolicity cone is a relaxation of the positive semidefinite (PSD) cone known as the first derivative relaxation (or Renegar derivative) of the PSD cone. A spectrahedal cone is a convex cone that has a representation as the intersection of a subspace with the cone of PSD matrices in some dimension. We show that the first derivative relaxation of the PSD cone is a spectrahedral cone, and give an explicit spectrahedral description of sizen+12− 1. The construction provides a new explicit example of a hyperbolicity cone that is also a spectrahedron. This is consistent with the generalized Lax conjecture,which conjectures that every hyperbolicity cone is a spectrahedron.

U2 - 10.1007/s11590-018-1246-x

DO - 10.1007/s11590-018-1246-x

M3 - Article

VL - 12

SP - 1475

EP - 1486

JO - Optimization Letters

JF - Optimization Letters

SN - 1862-4472

IS - 7

ER -