A spectrahedral representation of the first derivative relaxation of the positive semidefinite cone
Abstract If X is an $$n\times n$$ symmetric matrix, then the directional derivative of $$X \mapsto \det (X)$$ 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 res...
Ausführliche Beschreibung
Autor*in: |
Saunderson, James [verfasserIn] |
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Artikel |
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Sprache: |
Englisch |
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2018 |
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Anmerkung: |
© Springer-Verlag GmbH Germany, part of Springer Nature 2018 |
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Übergeordnetes Werk: |
Enthalten in: Optimization letters - Springer Berlin Heidelberg, 2007, 12(2018), 7 vom: 05. März, Seite 1475-1486 |
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Übergeordnetes Werk: |
volume:12 ; year:2018 ; number:7 ; day:05 ; month:03 ; pages:1475-1486 |
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DOI / URN: |
10.1007/s11590-018-1246-x |
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Katalog-ID: |
OLC2055616404 |
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520 | |a Abstract If X is an $$n\times n$$ symmetric matrix, then the directional derivative of $$X \mapsto \det (X)$$ 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 size $$\left( {\begin{array}{c}n+1\\ 2\end{array}}\right) -1$$. 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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10.1007/s11590-018-1246-x doi (DE-627)OLC2055616404 (DE-He213)s11590-018-1246-x-p DE-627 ger DE-627 rakwb eng 510 VZ 510 VZ 31.48$jVariationsrechnung bkl 31.80$jAngewandte Mathematik bkl Saunderson, James verfasserin (orcid)0000-0002-5456-0180 aut A spectrahedral representation of the first derivative relaxation of the positive semidefinite cone 2018 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag GmbH Germany, part of Springer Nature 2018 Abstract If X is an $$n\times n$$ symmetric matrix, then the directional derivative of $$X \mapsto \det (X)$$ 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 size $$\left( {\begin{array}{c}n+1\\ 2\end{array}}\right) -1$$. 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. Spectrahedron Hyperbolic polynomial Hyperbolicity cone Elementary symmetric polynomial Enthalten in Optimization letters Springer Berlin Heidelberg, 2007 12(2018), 7 vom: 05. März, Seite 1475-1486 (DE-627)527562920 (DE-600)2274663-8 (DE-576)272713724 1862-4472 nnns volume:12 year:2018 number:7 day:05 month:03 pages:1475-1486 https://doi.org/10.1007/s11590-018-1246-x lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-WIW GBV_ILN_26 GBV_ILN_70 GBV_ILN_267 GBV_ILN_2018 31.48$jVariationsrechnung VZ 106408143 (DE-625)106408143 31.80$jAngewandte Mathematik VZ 106419005 (DE-625)106419005 AR 12 2018 7 05 03 1475-1486 |
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10.1007/s11590-018-1246-x doi (DE-627)OLC2055616404 (DE-He213)s11590-018-1246-x-p DE-627 ger DE-627 rakwb eng 510 VZ 510 VZ 31.48$jVariationsrechnung bkl 31.80$jAngewandte Mathematik bkl Saunderson, James verfasserin (orcid)0000-0002-5456-0180 aut A spectrahedral representation of the first derivative relaxation of the positive semidefinite cone 2018 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag GmbH Germany, part of Springer Nature 2018 Abstract If X is an $$n\times n$$ symmetric matrix, then the directional derivative of $$X \mapsto \det (X)$$ 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 size $$\left( {\begin{array}{c}n+1\\ 2\end{array}}\right) -1$$. 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. Spectrahedron Hyperbolic polynomial Hyperbolicity cone Elementary symmetric polynomial Enthalten in Optimization letters Springer Berlin Heidelberg, 2007 12(2018), 7 vom: 05. März, Seite 1475-1486 (DE-627)527562920 (DE-600)2274663-8 (DE-576)272713724 1862-4472 nnns volume:12 year:2018 number:7 day:05 month:03 pages:1475-1486 https://doi.org/10.1007/s11590-018-1246-x lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-WIW GBV_ILN_26 GBV_ILN_70 GBV_ILN_267 GBV_ILN_2018 31.48$jVariationsrechnung VZ 106408143 (DE-625)106408143 31.80$jAngewandte Mathematik VZ 106419005 (DE-625)106419005 AR 12 2018 7 05 03 1475-1486 |
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10.1007/s11590-018-1246-x doi (DE-627)OLC2055616404 (DE-He213)s11590-018-1246-x-p DE-627 ger DE-627 rakwb eng 510 VZ 510 VZ 31.48$jVariationsrechnung bkl 31.80$jAngewandte Mathematik bkl Saunderson, James verfasserin (orcid)0000-0002-5456-0180 aut A spectrahedral representation of the first derivative relaxation of the positive semidefinite cone 2018 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag GmbH Germany, part of Springer Nature 2018 Abstract If X is an $$n\times n$$ symmetric matrix, then the directional derivative of $$X \mapsto \det (X)$$ 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 size $$\left( {\begin{array}{c}n+1\\ 2\end{array}}\right) -1$$. 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. Spectrahedron Hyperbolic polynomial Hyperbolicity cone Elementary symmetric polynomial Enthalten in Optimization letters Springer Berlin Heidelberg, 2007 12(2018), 7 vom: 05. März, Seite 1475-1486 (DE-627)527562920 (DE-600)2274663-8 (DE-576)272713724 1862-4472 nnns volume:12 year:2018 number:7 day:05 month:03 pages:1475-1486 https://doi.org/10.1007/s11590-018-1246-x lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-WIW GBV_ILN_26 GBV_ILN_70 GBV_ILN_267 GBV_ILN_2018 31.48$jVariationsrechnung VZ 106408143 (DE-625)106408143 31.80$jAngewandte Mathematik VZ 106419005 (DE-625)106419005 AR 12 2018 7 05 03 1475-1486 |
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10.1007/s11590-018-1246-x doi (DE-627)OLC2055616404 (DE-He213)s11590-018-1246-x-p DE-627 ger DE-627 rakwb eng 510 VZ 510 VZ 31.48$jVariationsrechnung bkl 31.80$jAngewandte Mathematik bkl Saunderson, James verfasserin (orcid)0000-0002-5456-0180 aut A spectrahedral representation of the first derivative relaxation of the positive semidefinite cone 2018 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag GmbH Germany, part of Springer Nature 2018 Abstract If X is an $$n\times n$$ symmetric matrix, then the directional derivative of $$X \mapsto \det (X)$$ 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 size $$\left( {\begin{array}{c}n+1\\ 2\end{array}}\right) -1$$. 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. Spectrahedron Hyperbolic polynomial Hyperbolicity cone Elementary symmetric polynomial Enthalten in Optimization letters Springer Berlin Heidelberg, 2007 12(2018), 7 vom: 05. März, Seite 1475-1486 (DE-627)527562920 (DE-600)2274663-8 (DE-576)272713724 1862-4472 nnns volume:12 year:2018 number:7 day:05 month:03 pages:1475-1486 https://doi.org/10.1007/s11590-018-1246-x lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-WIW GBV_ILN_26 GBV_ILN_70 GBV_ILN_267 GBV_ILN_2018 31.48$jVariationsrechnung VZ 106408143 (DE-625)106408143 31.80$jAngewandte Mathematik VZ 106419005 (DE-625)106419005 AR 12 2018 7 05 03 1475-1486 |
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10.1007/s11590-018-1246-x doi (DE-627)OLC2055616404 (DE-He213)s11590-018-1246-x-p DE-627 ger DE-627 rakwb eng 510 VZ 510 VZ 31.48$jVariationsrechnung bkl 31.80$jAngewandte Mathematik bkl Saunderson, James verfasserin (orcid)0000-0002-5456-0180 aut A spectrahedral representation of the first derivative relaxation of the positive semidefinite cone 2018 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag GmbH Germany, part of Springer Nature 2018 Abstract If X is an $$n\times n$$ symmetric matrix, then the directional derivative of $$X \mapsto \det (X)$$ 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 size $$\left( {\begin{array}{c}n+1\\ 2\end{array}}\right) -1$$. 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. Spectrahedron Hyperbolic polynomial Hyperbolicity cone Elementary symmetric polynomial Enthalten in Optimization letters Springer Berlin Heidelberg, 2007 12(2018), 7 vom: 05. März, Seite 1475-1486 (DE-627)527562920 (DE-600)2274663-8 (DE-576)272713724 1862-4472 nnns volume:12 year:2018 number:7 day:05 month:03 pages:1475-1486 https://doi.org/10.1007/s11590-018-1246-x lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-WIW GBV_ILN_26 GBV_ILN_70 GBV_ILN_267 GBV_ILN_2018 31.48$jVariationsrechnung VZ 106408143 (DE-625)106408143 31.80$jAngewandte Mathematik VZ 106419005 (DE-625)106419005 AR 12 2018 7 05 03 1475-1486 |
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A spectrahedral representation of the first derivative relaxation of the positive semidefinite cone |
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Saunderson, James |
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title_sort |
a spectrahedral representation of the first derivative relaxation of the positive semidefinite cone |
title_auth |
A spectrahedral representation of the first derivative relaxation of the positive semidefinite cone |
abstract |
Abstract If X is an $$n\times n$$ symmetric matrix, then the directional derivative of $$X \mapsto \det (X)$$ 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 size $$\left( {\begin{array}{c}n+1\\ 2\end{array}}\right) -1$$. 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. © Springer-Verlag GmbH Germany, part of Springer Nature 2018 |
abstractGer |
Abstract If X is an $$n\times n$$ symmetric matrix, then the directional derivative of $$X \mapsto \det (X)$$ 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 size $$\left( {\begin{array}{c}n+1\\ 2\end{array}}\right) -1$$. 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. © Springer-Verlag GmbH Germany, part of Springer Nature 2018 |
abstract_unstemmed |
Abstract If X is an $$n\times n$$ symmetric matrix, then the directional derivative of $$X \mapsto \det (X)$$ 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 size $$\left( {\begin{array}{c}n+1\\ 2\end{array}}\right) -1$$. 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. © Springer-Verlag GmbH Germany, part of Springer Nature 2018 |
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title_short |
A spectrahedral representation of the first derivative relaxation of the positive semidefinite cone |
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https://doi.org/10.1007/s11590-018-1246-x |
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2024-07-04T02:38:23.616Z |
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