Variational analysis of the abscissa mapping for polynomials via the Gauss-Lucas theorem

James V. Burke, Adrian S. Lewis, Michael L. Overton

Research output: Contribution to journalArticle

Abstract

Consider the linear space ℘n of polynomials of degree n or less over the complex field. The abscissa mapping on ℘n is the mapping that takes a polynomial to the maximum real part of its roots. This mapping plays a key role in the study of stability properties for linear systems. Burke and Overton have shown that the abscissa mapping is everywhere subdifferentially regular in the sense of Clarke on the manifold ℳn of polynomials of degree n. In addition, they provide a formula for the subdifferential. The result is surprising since the abscissa mapping is not Lipschitzian on ℳn. A key supporting lemma uses a proof technique due to Levantovskii for determining the tangent cone to the set of stable polynomials. This proof is arduous and opaque. It is a major obstacle to extending the variational theory to other functions of the roots of polynomials. In this note, we provide an alternative proof based on the Gauss-Lucas Theorem. This new proof is both insightful and elementary.

Original languageEnglish (US)
Pages (from-to)259-268
Number of pages10
JournalJournal of Global Optimization
Volume28
Issue number3-4
DOIs
StatePublished - Apr 2004

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Abscissa
Variational Analysis
Gauss
Polynomials
Polynomial
Theorem
Stable Polynomials
Roots
Tangent Cone
Subdifferential
Linear Space
Linear systems
Cones
Lemma
Linear Systems
analysis
Alternatives

ASJC Scopus subject areas

  • Applied Mathematics
  • Control and Optimization
  • Management Science and Operations Research
  • Global and Planetary Change

Cite this

Variational analysis of the abscissa mapping for polynomials via the Gauss-Lucas theorem. / Burke, James V.; Lewis, Adrian S.; Overton, Michael L.

In: Journal of Global Optimization, Vol. 28, No. 3-4, 04.2004, p. 259-268.

Research output: Contribution to journalArticle

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