Smooth hyperbolicity cones are spectrahedral shadows

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.

[1]  James Renegar,et al.  Hyperbolic Programs, and Their Derivative Relaxations , 2006, Found. Comput. Math..

[2]  J. Helton,et al.  Linear matrix inequality representation of sets , 2003, math/0306180.

[3]  David G. Wagner,et al.  Homogeneous multivariate polynomials with the half-plane property , 2004, Adv. Appl. Math..

[4]  Waa Nuij.  A Note on Hyperbolic Polynomials. , 1968 .

[5]  A. Lewis,et al.  The lax conjecture is true , 2003, math/0304104.

[6]  James Renegar,et al.  Central Swaths , 2010, Found. Comput. Math..

[7]  Raman Sanyal,et al.  On the derivative cones of polyhedral cones , 2011, 1105.2924.

[8]  José Carlos Goulart de Siqueira,et al.  Differential Equations , 1919, Nature.

[9]  Leonid Gurvits,et al.  Van der Waerden/Schrijver-Valiant like Conjectures and Stable (aka Hyperbolic) Homogeneous Polynomials: One Theorem for all , 2007, Electron. J. Comb..

[10]  Petter Brand'en,et al.  Obstructions to determinantal representability , 2010, 1004.1382.

[11]  Heinz H. Bauschke,et al.  Hyperbolic Polynomials and Convex Analysis , 2001, Canadian Journal of Mathematics.

[12]  L. Gårding.  An Inequality for Hyperbolic Polynomials , 1959 .

[13]  Fuji Zhang,et al.  Rainbow H-Factors of Complete s-Uniform r-Partite Hypergraphs , 2008, Electron. J. Comb..

[14]  J. William Helton,et al.  Semidefinite representation of convex sets , 2007, Math. Program..

[15]  Petter Brändén,et al.  Hyperbolicity cones of elementary symmetric polynomials are spectrahedral , 2012, Optim. Lett..

[16]  J. William Helton,et al.  Sufficient and Necessary Conditions for Semidefinite Representability of Convex Hulls and Sets , 2007, SIAM J. Optim..

[17]  Philip D. Plowright,et al.  Convexity , 2019, Optimization for Chemical and Biochemical Engineering.

[18]  Victor Vinnikov,et al.  LMI Representations of Convex Semialgebraic Sets and Determinantal Representations of Algebraic Hypersurfaces: Past, Present, and Future , 2012, 1205.2286.

[19]  Rainer Sinn,et al.  A Note on the Convex Hull of Finitely Many Projections of Spectrahedra , 2009 .

[20]  P. Lax,et al.  Differential Equations, Difference Equations and Matrix Theory , 2015 .

[21]  Osman Güler,et al.  Hyperbolic Polynomials and Interior Point Methods for Convex Programming , 1997, Math. Oper. Res..