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

Gespeichert in:

Autor*in:	Saunderson, James [verfasserIn]

Format:	E-Artikel
Sprache:	Englisch

Erschienen:	2018

Schlagwörter:	Spectrahedron Hyperbolic polynomial Hyperbolicity cone Elementary symmetric polynomial

Anmerkung:	© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Übergeordnetes Werk:	Enthalten in: Optimization letters - Berlin : Springer, 2007, 12(2018), 7 vom: 05. März, Seite 1475-1486
Übergeordnetes Werk:	volume:12 ; year:2018 ; number:7 ; day:05 ; month:03 ; pages:1475-1486

Links:	Volltext

DOI / URN:	10.1007/s11590-018-1246-x

Katalog-ID:	SPR020966385

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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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allfields	10.1007/s11590-018-1246-x doi (DE-627)SPR020966385 (SPR)s11590-018-1246-x-e DE-627 ger DE-627 rakwb eng 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 Computermedien c rdamedia Online-Ressource cr 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 (dpeaa)DE-He213 Hyperbolic polynomial (dpeaa)DE-He213 Hyperbolicity cone (dpeaa)DE-He213 Elementary symmetric polynomial (dpeaa)DE-He213 Enthalten in Optimization letters Berlin : Springer, 2007 12(2018), 7 vom: 05. März, Seite 1475-1486 (DE-627)534676499 (DE-600)2374345-1 1862-4480 nnns volume:12 year:2018 number:7 day:05 month:03 pages:1475-1486 https://dx.doi.org/10.1007/s11590-018-1246-x lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 12 2018 7 05 03 1475-1486
spelling	10.1007/s11590-018-1246-x doi (DE-627)SPR020966385 (SPR)s11590-018-1246-x-e DE-627 ger DE-627 rakwb eng 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 Computermedien c rdamedia Online-Ressource cr 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 (dpeaa)DE-He213 Hyperbolic polynomial (dpeaa)DE-He213 Hyperbolicity cone (dpeaa)DE-He213 Elementary symmetric polynomial (dpeaa)DE-He213 Enthalten in Optimization letters Berlin : Springer, 2007 12(2018), 7 vom: 05. März, Seite 1475-1486 (DE-627)534676499 (DE-600)2374345-1 1862-4480 nnns volume:12 year:2018 number:7 day:05 month:03 pages:1475-1486 https://dx.doi.org/10.1007/s11590-018-1246-x lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 12 2018 7 05 03 1475-1486
allfields_unstemmed	10.1007/s11590-018-1246-x doi (DE-627)SPR020966385 (SPR)s11590-018-1246-x-e DE-627 ger DE-627 rakwb eng 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 Computermedien c rdamedia Online-Ressource cr 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 (dpeaa)DE-He213 Hyperbolic polynomial (dpeaa)DE-He213 Hyperbolicity cone (dpeaa)DE-He213 Elementary symmetric polynomial (dpeaa)DE-He213 Enthalten in Optimization letters Berlin : Springer, 2007 12(2018), 7 vom: 05. März, Seite 1475-1486 (DE-627)534676499 (DE-600)2374345-1 1862-4480 nnns volume:12 year:2018 number:7 day:05 month:03 pages:1475-1486 https://dx.doi.org/10.1007/s11590-018-1246-x lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 12 2018 7 05 03 1475-1486
allfieldsGer	10.1007/s11590-018-1246-x doi (DE-627)SPR020966385 (SPR)s11590-018-1246-x-e DE-627 ger DE-627 rakwb eng 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 Computermedien c rdamedia Online-Ressource cr 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 (dpeaa)DE-He213 Hyperbolic polynomial (dpeaa)DE-He213 Hyperbolicity cone (dpeaa)DE-He213 Elementary symmetric polynomial (dpeaa)DE-He213 Enthalten in Optimization letters Berlin : Springer, 2007 12(2018), 7 vom: 05. März, Seite 1475-1486 (DE-627)534676499 (DE-600)2374345-1 1862-4480 nnns volume:12 year:2018 number:7 day:05 month:03 pages:1475-1486 https://dx.doi.org/10.1007/s11590-018-1246-x lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 12 2018 7 05 03 1475-1486
allfieldsSound	10.1007/s11590-018-1246-x doi (DE-627)SPR020966385 (SPR)s11590-018-1246-x-e DE-627 ger DE-627 rakwb eng 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 Computermedien c rdamedia Online-Ressource cr 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 (dpeaa)DE-He213 Hyperbolic polynomial (dpeaa)DE-He213 Hyperbolicity cone (dpeaa)DE-He213 Elementary symmetric polynomial (dpeaa)DE-He213 Enthalten in Optimization letters Berlin : Springer, 2007 12(2018), 7 vom: 05. März, Seite 1475-1486 (DE-627)534676499 (DE-600)2374345-1 1862-4480 nnns volume:12 year:2018 number:7 day:05 month:03 pages:1475-1486 https://dx.doi.org/10.1007/s11590-018-1246-x lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 12 2018 7 05 03 1475-1486
language	English
source	Enthalten in Optimization letters 12(2018), 7 vom: 05. März, Seite 1475-1486 volume:12 year:2018 number:7 day:05 month:03 pages:1475-1486
sourceStr	Enthalten in Optimization letters 12(2018), 7 vom: 05. März, Seite 1475-1486 volume:12 year:2018 number:7 day:05 month:03 pages:1475-1486
format_phy_str_mv	Article
institution	findex.gbv.de
topic_facet	Spectrahedron Hyperbolic polynomial Hyperbolicity cone Elementary symmetric polynomial
isfreeaccess_bool	false
container_title	Optimization letters
authorswithroles_txt_mv	Saunderson, James @@aut@@
publishDateDaySort_date	2018-03-05T00:00:00Z
hierarchy_top_id	534676499
id	SPR020966385
language_de	englisch
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author	Saunderson, James
spellingShingle	Saunderson, James misc Spectrahedron misc Hyperbolic polynomial misc Hyperbolicity cone misc Elementary symmetric polynomial A spectrahedral representation of the first derivative relaxation of the positive semidefinite cone
authorStr	Saunderson, James
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topic_title	A spectrahedral representation of the first derivative relaxation of the positive semidefinite cone Spectrahedron (dpeaa)DE-He213 Hyperbolic polynomial (dpeaa)DE-He213 Hyperbolicity cone (dpeaa)DE-He213 Elementary symmetric polynomial (dpeaa)DE-He213
topic	misc Spectrahedron misc Hyperbolic polynomial misc Hyperbolicity cone misc Elementary symmetric polynomial
topic_unstemmed	misc Spectrahedron misc Hyperbolic polynomial misc Hyperbolicity cone misc Elementary symmetric polynomial
topic_browse	misc Spectrahedron misc Hyperbolic polynomial misc Hyperbolicity cone misc Elementary symmetric polynomial
format_facet	Elektronische Aufsätze Aufsätze Elektronische Ressource
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title	A spectrahedral representation of the first derivative relaxation of the positive semidefinite cone
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title_full	A spectrahedral representation of the first derivative relaxation of the positive semidefinite cone
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title_sort	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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container_issue	7
title_short	A spectrahedral representation of the first derivative relaxation of the positive semidefinite cone
url	https://dx.doi.org/10.1007/s11590-018-1246-x
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doi_str	10.1007/s11590-018-1246-x
up_date	2024-07-03T19:24:33.379Z
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A spectrahedral representation of the first derivative relaxation of the positive semidefinite cone

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