44 research outputs found
Hyperbolicity cones of elementary symmetric polynomials are spectrahedral
We prove that the hyperbolicity cones of elementary symmetric polynomials are
spectrahedral, i.e., they are slices of the cone of positive semidefinite
matrices. The proof uses the matrix--tree theorem, an idea already present in
Choe et al.Comment: 9 pages. Some typos corrected. Details added. To appear in
Optimization Letter
Exponential lower bounds on spectrahedral representations of hyperbolicity cones
The Generalized Lax Conjecture asks whether every hyperbolicity cone is a
section of a semidefinite cone of sufficiently high dimension. We prove that
the space of hyperbolicity cones of hyperbolic polynomials of degree in
variables contains pairwise distant cones in a certain
metric, and therefore that any semidefinite representation of such cones must
have dimension at least (even if a small approximation is
allowed). The 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 quantitative versions of several basic facts about real rooted
polynomials.Comment: Fixed a mistake in the proof of Lemma 6. The statement is unchanged
except for constant factors, and the main theorem is unaffected. Wrote a
slightly stronger statement for the main theorem, emphasizing approximate
representations (the proof is the same). Added one figur
A Note on the Hyperbolicity Cone of the Specialized V\'amos Polynomial
The specialized V\'amos polynomial is a hyperbolic polynomial of degree four
in four variables with the property that none of its powers admits a definite
determinantal representation. We will use a heuristical method to prove that
its hyperbolicity cone is a spectrahedron.Comment: Notable easier arguments and minor correction
Hyperbolic Polynomials and Generalized Clifford Algebras
We consider the problem of realizing hyperbolicity cones as spectrahedra,
i.e. as linear slices of cones of positive semidefinite matrices. The
generalized Lax conjecture states that this is always possible. We use
generalized Clifford algebras for a new approach to the problem. Our main
result is that if -1 is not a sum of hermitian squares in the Clifford algebra
of a hyperbolic polynomial, then its hyperbolicity cone is spectrahedral. Our
result also has computational applications, since this sufficient condition can
be checked with a single semidefinite program
Polynomial-sized Semidefinite Representations of Derivative Relaxations of Spectrahedral Cones
We give explicit polynomial-sized (in and ) semidefinite
representations of the hyperbolicity cones associated with the elementary
symmetric polynomials of degree in 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 , and ) of the hyperbolicity cones associated with
th directional derivatives of polynomials of the form where the are 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.Comment: 20 pages, 1 figure. Minor changes, expanded proof of Lemma
Smooth Hyperbolicity Cones are Spectrahedral Shadows
Hyperbolicity cones are convex algebraic cones arising from hyperbolic
polynomials. A well-understood subclass of hyperbolicity cones is that of
spectrahedral cones and it is conjectured that every hyperbolicity cone is
spectrahedral. In this paper we prove a weaker version of this conjecture by
showing that every smooth hyperbolicity cone is the linear projection of a
spectrahedral cone, that is, a spectrahedral shadow