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# Hyperbolic Polynomials and Convex Analysis

Published:2001-06-01
Printed: Jun 2001
• Heinz H. Bauschke
• Osman Güler
• Adrian S. Lewis
• Hristo S. Sendov
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## Abstract

A homogeneous real polynomial $p$ is {\em hyperbolic} with respect to a given vector $d$ if the univariate polynomial $t \mapsto p(x-td)$ has all real roots for all vectors $x$. Motivated by partial differential equations, G{\aa}rding proved in 1951 that the largest such root is a convex function of $x$, and showed various ways of constructing new hyperbolic polynomials. We present a powerful new such construction, and use it to generalize G{\aa}rding's result to arbitrary symmetric functions of the roots. Many classical and recent inequalities follow easily. We develop various convex-analytic tools for such symmetric functions, of interest in interior-point methods for optimization problems over related cones.
 Keywords: convex analysis, eigenvalue, G{\aa}rding's inequality, hyperbolic barrier function, hyperbolic polynomial, hyperbolicity cone, interior-point method, semidefinite program, singular value, symmetric function
 MSC Classifications: 90C25 - Convex programming 15A45 - Miscellaneous inequalities involving matrices 52A41 - Convex functions and convex programs [See also 26B25, 90C25]

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