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

Research output: Contribution to journalArticleResearchpeer-review

Abstract

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.
Original languageEnglish
Pages (from-to)1475–1486
Number of pages12
JournalOptimization Letters
Volume12
Issue number7
DOIs
Publication statusPublished - 1 Oct 2018

Cite this

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

In: Optimization Letters, Vol. 12, No. 7, 01.10.2018, p. 1475–1486.

Research output: Contribution to journalArticleResearchpeer-review

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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.

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