Self-concordant barriers for hyperbolic means

The geometric mean and the function (det(·))1/m (on the m-by-m positive definite matrices) are examples of "hyperbolic means": functions of the form p1/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.