Hyperbolic Polynomials and Convex Analysis

Abstract

A homogeneous polynomial p(x) is hyperbolic with respect to a given vector d if the real polynomial t 7! p(x + td) has all real roots for all vectors x. We show that any symmetric convex function of these roots is a convex function of x, generalizing a fundamental result of Garding. Consequently we are able to prove a number of deep results about hyperbolic polynomials with ease. In particular, our result subsumes von Neumann's characterization of unitarily invariant matrix norms, and Davis's characterization of convex functions of the eigenvalues of Hermitian matrices. We then develop various convex-analytic tools for such symmetric functions, of interest in interior-point methods for optimization problems posed over related cones. Department of Combinatorics & Optimization, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada. Email: [email protected]. Research supported by an NSERC Postdoctoral Fellowship and by the Department of Combinatorics & Optimization, Univ..