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 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 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.

## Article

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