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Information Center for Mathematical Science

논문검색

Information Center for Mathematical Science

논문검색

Canadian Journal of Mathematics
( Vol. 52 NO.3 / (2000))
Hyperbolic Polynomials and Convex Analysis
Heinz H. Bauschke, Osman Guler, Adrian S. Lewis, Hristo S. Sendov,
Pages. 470-488
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.
Contents 1. Introduction
2. Tools
3. Composition and convexity
4. Euclidean structure
5. Duality
6. Further examples
Key words Convex analysis, eigenvalue, G{\aa}rding's inequality, hyperbolic barrier function, hyperbolic polynomial, hyperbolicity cone, interior-point method, semidefinite program, singular value, symmetric function, unitarily invariant norm
Mathmatical Subject Classification 90C46, 15A45, 52A41