Automorphisms of rank-one generated hyperbolicity cones and their derivative relaxations

TL;DR

This paper studies the automorphisms of rank-one generated hyperbolicity cones and their derivative relaxations, revealing their structure and automorphism groups, with applications to nonnegative orthant and positive semidefinite cones.

Contribution

It characterizes the automorphisms of ROG hyperbolicity cones and their derivatives, extending understanding beyond spectrahedral cones and relating automorphisms to underlying permutation-invariant sets.

Findings

Automorphisms of derivative relaxations fix a certain direction of the original cone.

Complete determination of automorphisms for derivative relaxations of nonnegative orthant and PSD cones.

Relations established between automorphisms of spectral cones and permutation-invariant sets.

Abstract

A hyperbolicity cone is said to be rank-one generated (ROG) if all its extreme rays have rank one, where the rank is computed with respect to the underlying hyperbolic polynomial. This is a natural class of hyperbolicity cones which are strictly more general than the ROG spectrahedral cones. In this work, we present a study of the automorphisms of ROG hyperbolicity cones and their derivative relaxations. One of our main results states that the automorphisms of the derivative relaxations are exactly the automorphisms of the original cone fixing a certain direction. As an application, we completely determine the automorphisms of the derivative relaxations of the nonnegative orthant and of the cone of positive semidefinite matrices. More generally, we also prove relations between the automorphisms of a spectral cone and the underlying permutation-invariant set, which might be of independent interest.