Hyperbolic Polynomials and Interior Point Methods for Convex Programming

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Abstract
Hyperbolic polynomials have their origins in partial differential 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. We give an explicit representation of this cone in terms of polynomial inequalities. 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.

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