Exponential lower bounds on spectrahedral representations of hyperbolicity cones

Abstract

Hyperbolic programming is a generalization of semidefinite programming in which one optimizes over linear sections of hyperbolicity cones rather than semidefinite ones. It is not known whether this generalization is strict: the Generalized Lax Conjecture asks whether every hyperbolicity cone is a section of a semidefinite cone of sufficiently high dimension. We study a quantitative version of this question, and prove that the space of hyperbolicity cones of hyperbolic polynomials of degree d in n variables contains (n/d)Ω(d) pairwise distant cones in the Hausdorff metric, implying that any semidefinite representation of such cones must have dimension at least (n/d)Ω(d) (even allowing a small approximation error). The cones are perturbations of the hyperbolicity cones of elementary symmetric polynomials. Our proof contains several ingredients of independent interest, including the identification of a large subspace in which the elementary symmetric polynomials lie in the relative interior of the set of hyperbolic polynomials, and a quantitative generalization of the fact that a real-rooted polynomial with two consecutive zero coefficients must have a high multiplicity root at zero.