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Hyperbolic Polynomials and Convex Analysis

Published online by Cambridge University Press:  20 November 2018

Heinz H. Bauschke ,
Osman Güler ,
Adrian S. Lewis  and
Hristo S. Sendov
Heinz H. Bauschke
Affiliation:
Department of Mathematics and Statistics Okanagan University College Kelowna, British Columbia V1V 1V7, email: [email protected]
Osman Güler
Affiliation:
Department of Mathematics and Statistics University of Maryland Baltimore County Baltimore, Maryland 21228, USA, email: [email protected]
Adrian S. Lewis
Affiliation:
Department of Combinatorics & Optimization University of Waterloo Waterloo, Ontario N2L 3G1, email: [email protected]
Hristo S. Sendov
Affiliation:
Department of Combinatorics & Optimization University of Waterloo Waterloo, Ontario N2L 3G1, email: [email protected]
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Abstract

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A homogeneous real polynomial $p$ is 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, Gårding 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 Gårding’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

90C2515A4552A41convex analysiseigenvalueGårding’s inequalityhyperbolic barrier functionhyperbolic polynomialhyperbolicity coneinterior-point methodsemidefinite programsingular valuesymmetric functionunitarily invariant norm
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 53 , Issue 3 , 01 June 2001 , pp. 470 - 488
Copyright
Copyright © Canadian Mathematical Society 2001

References

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