Minimal hyperbolic polynomials and ranks of homogeneous cones

Abstract

The starting point of this paper is the computation of minimal hyperbolic polynomials of duals of cones arising from chordal sparsity patterns. From that, we investigate the relation between ranks of homogeneous cones and their minimal polynomials. Along the way, we answer in the negative a question posed in an earlier paper and show examples of homogeneous cones that cannot be realized as rank-one generated (ROG) hyperbolicity cones.

arXiv comment21 pages, 1 figure. To appear in the Journal of Convex Analysis. The question at the end was solved by Hideto Nakashima and we included some comments on the solution. Dedicated to Prof. R.T. Rockafellar on the occasion of his 90th birthday