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Hyperbolic polynomials have their origins in partial diierential 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… (More)

We show that the universal barrier function of a convex cone introduced by Nesterov and Nemirovskii is the logarithm of the characteristic function of the cone. This interpretation demonstrates the invariance of the universal barrier under the automorphism group of the underlying cone. This provides a simple method for calculating the universal barrier for… (More)

We characterize the barrier parameter of the optimal self{concordant barriers for homogeneous cones. In particular, we prove that for homogeneous convex cones this parameter is the same as the rank of the corresponding Siegel domain. We also provide lower bounds on the barrier parameter in terms of the Carath eodory number of the cone. The bounds are tight… (More)

Let K be a regular convex cone in R n and F(x) its universal barrier function. Let D k F(x))h; : : : ; h] be kth order directional derivative at the point x 2 K 0 and direction h 2 R n. We show that for every m 3 there exists a constant c(m) > 0 depending only on m such that jD m F(x))h; : : : ; h]j c(m) D 2 F(x))h; h] m=2. For m = 3, this is the… (More)