# Spectrahedral relaxations of hyperbolicity cones

## TL;DR

This paper constructs spectrahedral relaxations for hyperbolicity cones defined by real zero polynomials, providing new tools for hyperbolic programming and insights towards the generalized Lax conjecture.

## Contribution

It introduces a spectrahedral relaxation based on the cubic part of the polynomial, offering a new approach to approximate hyperbolicity cones and addressing the generalized Lax conjecture.

## Key findings

- Constructed a spectrahedron containing the convex set defined by a real zero polynomial.
- Provided conditions under which the spectrahedral relaxation equals the original convex set.
- Developed hierarchies of semidefinite relaxations for hyperbolic programs.

## Abstract

Let $p$ be a real zero polynomial in $n$ variables. Then $p$ defines a rigidly convex set $C(p)$. We construct a linear matrix inequality of size $n+1$ in the same $n$ variables that depends only on the cubic part of $p$ and defines a spectrahedron $S(p)$ containing $C(p)$. The proof of the containment uses the characterization of real zero polynomials in two variables by Helton and Vinnikov. We exhibit many cases where $C(p)=S(p)$. In terms of optimization theory, we introduce a small semidefinite relaxation of a potentially huge hyperbolic program. If the hyperbolic program is a linear program, we introduce even a finitely convergent hierachy of semidefinite relaxations. With some extra work, we discuss the homogeneous setup where real zero polynomials correspond to homogeneous polynomials and rigidly convex sets correspond to hyperbolicity cones. The main aim of our construction is to attack the generalized Lax conjecture saying that $C(p)$ is always a spectrahedron. We show that the ``weak real zero amalgamation conjecture'' of Sawall and the author would imply the following partial result towards the generalized Lax conjecture: Given finitely many planes in $\mathbb R^n$, there is a spectrahedron containing $C(p)$ that coincides with $C(p)$ on each of these planes. This uses again the result of Helton and Vinnikov.