## Abstract

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(Σ^{n}_{i}=_{1} A_{i} *x*_{i} ) where the A_{i} 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 language | English |
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Pages (from-to) | 309-331 |

Number of pages | 23 |

Journal | Mathematical Programming |

Volume | 153 |

Issue number | 2 |

DOIs | |

Publication status | Published - 22 Nov 2015 |

Externally published | Yes |

## Keywords

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