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

TL;DR

This paper proves that the first derivative relaxation of the positive semidefinite cone is a spectrahedral cone and provides an explicit description, supporting the generalized Lax conjecture.

Contribution

It demonstrates that the first derivative relaxation of the PSD cone is spectrahedral and offers an explicit spectrahedral representation.

Findings

The first derivative relaxation of the PSD cone is spectrahedral.

An explicit spectrahedral description of size rom the binomial coefficient is provided.

Supports the generalized Lax conjecture that all hyperbolicity cones are spectrahedra.

Abstract

If $X$ is an $n\times n$ symmetric matrix, then the directional derivative of $X \mapsto \det(X)$ in the direction $I$ is the elementary symmetric polynomial of degree $n-1$ in the eigenvalues of $X$. This is a polynomial in the entries of $X$ with the property that it is hyperbolic with respect to the direction $I$. The corresponding hyperbolicity cone is a relaxation of the positive semidefinite (PSD) cone known as the first derivative relaxation (or Renegar derivative) of the PSD cone. A spectrahedal cone is a convex cone that has a representation as the intersection of a subspace with the cone of PSD matrices in some dimension. We show that the first derivative relaxation of the PSD cone is a spectrahedral cone, and give an explicit spectrahedral description of size $\binom{n+1}{2}-1$. The construction provides a new explicit example of a hyperbolicity cone that is also a spectrahedron. This is consistent with the generalized Lax conjecture, which conjectures that every hyperbolicity cone is a spectrahedron.