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Hyperbolic Polynomials and Convex Analysis |
Canad. J. Math. 53(2001), 470-488
Published:2001-06-01
Printed: Jun 2001
Printed: Jun 2001
- Heinz H. Bauschke
- Osman Güler
- Adrian S. Lewis
- Hristo S. Sendov
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] |