Smooth hyperbolicity cones are spectrahedral shadows

被引：6

作者：

Netzer, Tim ^{[1
]}

Sanyal, Raman ^{[2
]}

机构：

[1] Univ Leipzig, D-04109 Leipzig, Germany

[2] Free Univ Berlin, Fachbereich Math & Informat, Berlin, Germany

来源：

MATHEMATICAL PROGRAMMING | 2015年 / 153卷 / 01期

基金：

欧洲研究理事会;

关键词：

Hyperbolic polynomials; Hyperbolicity cones; Spectrahedra; Spectrahedral shadows; sdp-representable; POLYNOMIALS; REPRESENTATION; INEQUALITY;

D O I：

10.1007/s10107-014-0744-6

中图分类号：

TP31 [计算机软件];

学科分类号：

081202 ; 0835 ;

摘要：

Hyperbolicity cones are convex algebraic cones arising from hyperbolic polynomials. A well-understood subclass of hyperbolicity cones is that of spectrahedral cones and it is conjectured that every hyperbolicity cone is spectrahedral. In this paper we prove a weaker version of this conjecture by showing that every smooth hyperbolicity cone is the linear projection of a spectrahedral cone, that is, a spectrahedral shadow.

引用

页码：213 / 221

页数：9

共 19 条

[1] Hyperbolic polynomials and convex analysis
Bauschke, HH
Güler, O
Lewis, AS
Sendov, HS
[J]. CANADIAN JOURNAL OF MATHEMATICS-JOURNAL CANADIEN DE MATHEMATIQUES, 2001, 53 (03): : 470 - 488
[2] Hyperbolicity cones of elementary symmetric polynomials are spectrahedral
Branden, Petter
[J]. OPTIMIZATION LETTERS, 2014, 8 (05) : 1773 - 1782
[3] Obstructions to determinantal representability
Branden, Petter
[J]. ADVANCES IN MATHEMATICS, 2011, 226 (02) : 1202 - 1212
[4] Homogeneous multivariate polynomials with the half-plane property
Choe, YB
Oxley, JG
Sokal, AD
Wagner, DG
[J]. ADVANCES IN APPLIED MATHEMATICS, 2004, 32 (1-2) : 88 - 187
[5] GARDING L, 1959, J MATH MECH, V8, P957
[6] Hyperbolic polynomials and interior point methods for convex programming
Guler, O
[J]. MATHEMATICS OF OPERATIONS RESEARCH, 1997, 22 (02) : 350 - 377
[7] Gurvits L, 2008, ELECTRON J COMB, V15
[8] Linear matrix inequality representation of sets
Helton, J. William
Vinnikov, Victor
[J]. COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS, 2007, 60 (05) : 654 - 674
[9] SUFFICIENT AND NECESSARY CONDITIONS FOR SEMIDEFINITE REPRESENTABILITY OF CONVEX HULLS AND SETS
Helton, J. William
Nie, Jiawang
[J]. SIAM JOURNAL ON OPTIMIZATION, 2009, 20 (02) : 759 - 791
[10] Semidefinite representation of convex sets
Helton, J. William
Nie, Jiawang
[J]. MATHEMATICAL PROGRAMMING, 2010, 122 (01) : 21 - 64

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