Abstract
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.
Similar content being viewed by others
References
Bauschke, H., Güler, O., Lewis, A., Sendov, H.: Hyperbolic polynomials and convex analysis. Can. J. Math. 53(3), 470–488 (2001)
Brändén, P.: Hyperbolicity cones of elementary symmetric polynomials are spectrahedral. Optim. Lett. (2012). doi:10.1007/s11590-013-0694-6
Brändén, P.: Obstructions to determinantal representability. Adv. Math. 226(2), 1202–1212 (2011)
Choe, Y.B., Oxley, J.G., Sokal, A.D., Wagner, D.G.: Homogeneous multivariate polynomials with the half-plane property. Adv. Appl. Math. 32(1–2), 88–187 (2004). Special issue on the Tutte polynomial
Gårding, L.: An inequality for hyperbolic polynomials. J. Math. Mech. 8, 957–965 (1959)
Güler, O.: Hyperbolic polynomials and interior point methods for convex programming. Math. Oper. Res. 22(2), 350–377 (1997)
Gurvits, L.: Van der Waerden/Schrijver-Valiant like conjectures and stable (aka hyperbolic) homogeneous polynomials: one theorem for all. Electron. J. Combin. 15(1), (2008) Research Paper 66, 26
Helton, J.W., Nie, J.: Semidefinite representation of convex sets. Math. Program. 122(1), Ser. A, 21–64 (2010)
Helton, J.W., Vinnikov, V.: Linear matrix inequality representation of sets. Commun. Pure Appl. Math. 60(5), 654–674 (2007)
Helton, J.W., Nie, J.: Sufficient and necessary conditions for semidefinite representability of convex hulls and sets. SIAM J. Optim. 20(2), 759–791 (2009)
Lax, P.D.: Differential equations, difference equations and matrix theory. Commun. Pure Appl. Math. 11, 175–194 (1958)
Lewis, A.S., Parrilo, P.A., Ramana, M.V.: The Lax conjecture is true. Proc. Am. Math. Soc. 133(9), 2495–2499 (2005) (electronic)
Netzer, T., Sinn, R.: A note on the convex hull of finitely many projections of spectrahedra. arXiv:0908.3386
Nuij, W.: A note on hyperbolic polynomials. Math. Scand. 23(1968), 69–72 (1969)
Renegar, J.: Hyperbolic programs, and their derivative relaxations. Found. Comput. Math. 6(1), 59–79 (2006)
Renegar, J.: Central swaths: a generalization of the central path. Found. Comput. Math. 13(3), 405–454 (2013)
Sanyal, R.: On the derivative cones of polyhedral cones. Adv. Geom. 13(2), 315–321 (2013). doi:10.1515/advgeom-2011-051
Vinnikov, V.: LMI representations of convex semialgebraic sets and determinantal representations of algebraic hypersurfaces: past, present, and future. In: Dym, H., de Oliveira, M.C., Putinar, M. (eds.) Mathematical Methods in Systems, Optimization, and Control. Operator Theory: Advances and Applications, vol. 222, Springer Basel, Heidelberg, pp. 325–349 (2012). doi:10.1007/978-3-0348-0411-0_23
Webster, R.: Convexity. Oxford Science Publications, The Clarendon Press Oxford University Press, New York (1994)
Acknowledgments
We would like to thank Daniel Plaumann for many inspiring discussions on the topic and the referees for helpful suggestions.
Author information
Authors and Affiliations
Corresponding author
Additional information
Raman Sanyal has been supported by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement no 247029.
Rights and permissions
About this article
Cite this article
Netzer, T., Sanyal, R. Smooth hyperbolicity cones are spectrahedral shadows. Math. Program. 153, 213–221 (2015). https://doi.org/10.1007/s10107-014-0744-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10107-014-0744-6