Variational Analysis of Functions of the Roots of Polynomials

James Burke (burkemath.washington.edu)
Adrian Lewis (aslewissfu.ca)
Michael Overton (overtoncs.nyu.edu)

Abstract: The Gauss-Lucas Theorem on the roots of polynomials nicely simplifies calculating the subderivative and regular subdifferential of the abscissa mapping on polynomials (the maximum of the real parts of the roots). This paper extends this approach to more general functions of the roots. By combining the Gauss-Lucas methodology with an analysis of the splitting behavior of the roots, we obtain characterizations of the subderivative and regular subdifferential for these functions as well. In particular, we completely characterize the subderivative and regular subdifferential of the radius mapping (the maximum of the moduli of the roots). The abscissa and radius mappings are important for the study of continuous and discrete time linear dynamical systems.

Keywords: Nonsmooth analysis, subdifferential, subgradient

Category 1: Convex and Nonsmooth Optimization (Nonsmooth Optimization )

Citation: http://www.cs.nyu.edu/overton/papers/pdffiles/gausslucas.pdf Submitted to Math Programming.

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Entry Submitted: 06/24/2004
Entry Accepted: 06/26/2004
Entry Last Modified: 07/28/2004

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