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We give explicit polynomial-sized (in $n$ and $k$) semidefiniterepresentations of the hyperbolicity cones associated with the elementarysymmetric polynomials of degree $k$ in $n$ variables. These convex cones form afamily of non-polyhedral outer approximations of the non-negative orthant thatpreserve low-dimensional faces while successively discarding high-dimensionalfaces. 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 $m\times m$ symmetricmatrices. These convex cones form an analogous family of outer approximationsto any spectrahedral cone. Our representations allow us to use semidefiniteprogramming to solve the linear cone programs associated with these convexcones as well as their (less well understood) dual cones.

Polynomial-sized Semidefinite Representations of Derivative Relaxations of Spectrahedral Cones