CMS/SMC
Canadian Mathematical Society
www.cms.math.ca
Canadian Mathematical Society
  location:  PublicationsjournalsCJM
Abstract view

Hyperbolic Polynomials and Convex Analysis

  Published:2001-06-01
 Printed: Jun 2001
  • Heinz H. Bauschke
  • Osman Güler
  • Adrian S. Lewis
  • Hristo S. Sendov
Format:   HTML   LaTeX   MathJax   PDF   PostScript  

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 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, 15A45, 52A41 show english descriptions Convex programming
Miscellaneous inequalities involving matrices
Convex functions and convex programs [See also 26B25, 90C25]
90C25 - Convex programming
15A45 - Miscellaneous inequalities involving matrices
52A41 - Convex functions and convex programs [See also 26B25, 90C25]
 

© Canadian Mathematical Society, 2014 : https://cms.math.ca/