Hyperbolic polynomials have their origins in partial di erential equations. We show in this paper that they have applications in interior point methods for convex programming. Each homogeneous hyperbolic polynomial p has an associated open and convex cone called its hyperbolicity cone. The function F (x) = ? log p(x) is a logarithmically homogeneous self{concordant barrier function for the hyperbolicity cone with barrier parameter equal to the degree of p. The function F (x) possesses striking additional properties that are useful in designing long{step interior point methods. For example, we show that the long{step primal potential reduction methods of Nesterov and Todd and the surface{following methods of Nesterov and Nemirovskii extend to hyperbolic barrier functions. We also show that there exists a hyperbolic barrier function on every homogeneous cone.
From a computational perspective, Güler REF showed that if p has degree m and is hyperbolic with respect to e then log p is a self-concordant barrier function (with barrier parameter m) for the hyperbolicity cone Λ + (p, e).
Hyperbolic Polynomials and Interior Point Methods for Convex Programming
Mathematics of Operations Research
Mathematics Computer Science