Mathematical Programming

, Volume 153, Issue 2, pp 309–331 | Cite as

Polynomial-sized semidefinite representations of derivative relaxations of spectrahedral cones

  • James Saunderson
  • Pablo A. Parrilo
Full Length Paper Series A


We give explicit polynomial-sized (in \(n\) and \(k\)) semidefinite representations of the hyperbolicity cones associated with the elementary symmetric polynomials of degree \(k\) in \(n\) variables. These convex cones form a family of non-polyhedral outer approximations of the non-negative orthant that preserve low-dimensional faces while successively discarding high-dimensional faces. More generally we construct explicit semidefinite representations (polynomial-sized in \(k,m\), and \(n\)) of the hyperbolicity cones associated with \(k\)th directional derivatives of polynomials of the form \(p(x)=\det (\sum _{i=1}^{n}A_i x_i)\) where the \(A_i\) are \(m\times m\) symmetric matrices. These convex cones form an analogous family of outer approximations to any spectrahedral cone. Our representations allow us to use semidefinite programming to solve the linear cone programs associated with these convex cones as well as their (less well understood) dual cones.


Hyperbolic polynomial Hyperbolicity cone Elementary symmetric polynomial Semidefinite representation 

Mathematics Subject Classification

90C22 90C25 52A41 52A20 



The authors would like to thank the anonymous referees for many helpful suggestions that led to substantial improvements in the presentation of the paper. This research was funded by the Air Force Office of Scientific Research under Grants FA9550-11-1-0305 and FA9550-12-1-0287.


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Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2014

Authors and Affiliations

  1. 1.Laboratory for Information and Decision Systems, Department of Electrical Engineering and Computer ScienceMassachusetts Institute of TechnologyCambridgeUSA

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