Automorphisms of Rank-One Generated Hyperbolicity Cones and Their Derivative Relaxations

A hyperbolicity cone is said to be rank-one generated (ROG) if all its extreme rays have rank 1, where the computed with respect underlying hyperbolic polynomial. This a natural class of cones which are strictly more general than ROG spectrahedral cones. In this work, we present study automorphisms and their derivative relaxations. One our main results states that relaxations exactly original fixing certain direction. As an application, completely determine nonnegative orthant positive semidefinite matrices. More generally, also prove relations between spectral permutation-invariant set, might independent interest.