Polynomial-sized semidefinite representations of derivative relaxations of spectrahedral cones

James Saunderson, Pablo A. Parrilo

Research output: Contribution to journalArticleResearchpeer-review

8 Citations (Scopus)


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 kth directional derivatives of polynomials of the form p(x) = det(Σni=1 Ai xi ) where the Ai are m × 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.

Original languageEnglish
Pages (from-to)309-331
Number of pages23
JournalMathematical Programming
Issue number2
Publication statusPublished - 22 Nov 2015
Externally publishedYes


  • Elementary symmetric polynomial
  • Hyperbolic polynomial
  • Hyperbolicity cone
  • Semidefinite representation

Cite this