Hyperbolic Polynomials and Convex Analysis

作者：Heinz H. Bauschke, Osman Güler, Adrian S. Lewis, Hristo S. Sendov · 发表于：Canadian Journal of Mathematics · 年份：2001 · DOI：10.4153/cjm-2001-020-6 · 被引用次数：105 · 研究领域：Matrix Theory and Algorithms、Advanced Optimization Algorithms Research、Mathematical Inequalities and Applications

Abstract A homogeneous real polynomialpishyperbolicwith respect to a given vector d if the univariate polynomial t ⟼ p(x − td) has all real roots for all vectorsx. Motivated by partial differential equations, Gårding proved in 1951 that the largest such root is a convex function ofx, 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.