  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.
