Polynomial-sized semidefinite representations of derivative relaxations of spectrahedral cones

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 kth directional derivatives of polynomials of the form p(x) = det( 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.

A spectrahedral representation of the first derivative relaxation of the positive semidefinite cone

“…+ 1. Explicit projected spectrahedral representations of the cones S n,(k) + of size O(n 2 min{k, n − k}) were given by Saunderson and Parrilo [SP15], leaving open (except in the cases k = n − 2, n − 1) the question of whether these cones are spectrahedra. The main result of this paper is that S n,(1) + is a spectrahedron.…”

“…• The spectrahedral representation of S n,(1) + in Theorem 1 has size d = n+1 2 − 1 = 1 2 (n + 2)(n − 1). This is about half the size of the smallest previously known projected spectrahedral representation of S n,(1) + , i.e., representation as the image of a spectrahedral cone under a linear map [SP15].…”

Symbolic computation in hyperbolic programming

“…The derivative D (1) e (f ) defines a singular quartic, still with four nodes and hyperbolic with respect to e (see Figure 1), which is not representable as a determinant of a symmetric pencil. Let Λ + (D e (f ), e) (multiplicity one), whose coordinates are given as elements of certified rational intervals, with 10 significant decimal digits, is: [26]). We consider the semidefinite representation of the 3-ellipse, as computed in [17].…”

Definite determinantal representations of multivariate polynomials

“…For example, dehomogenized polynomial of Vamos cube V 8 is a RZ polynomial without a definite determinantal representation [Bra11]. This leads to the generalized Lax conjecture, for details see [Vin12] [ KPV15], [SP15], [NS15]. Hyperbolic polynomials which play an important role in partial differential equations are the homegenized version of RZ polynomials [Brä10].…”