Self-concordant barriers for hyperbolic means

The geometric mean and the function (det(·))^l/m (on the m-by-m positive definite matrices) are examples of ‘hyperbolic means’: functions of the form p^l/m, where p is a hyperbolic polynomial of degree m. (A homogeneous polynomial p is ‘hyperbolic’ with respect to a vector d if the polynomial t ↦ p(x+td) has only real roots for every vector x.) Any hyperbolic mean is positively homogeneous and concave (on a suitable domain): we present a self-concordant barrier for its hypograph, with barrier parameter O(m²). Our approach is direct, and shows, for example, that the function −m log(det(·)−1) is an m²-self-concordant barrier on a natural domain. Such barriers suggest novel interior point approaches to convex programs involving hyperbolic means.