Variational analysis of the Crouzeix ratio

Anne Greenbaum, Adrian S. Lewis, Michael Overton

Research output: Contribution to journalArticle

Abstract

Let W(A) denote the field of values (numerical range) of a matrix A. For any polynomial p and matrix A, define the Crouzeix ratio to have numerator (Formula presented.) and denominator (Formula presented.). Crouzeix’s 2004 conjecture postulates that the globally minimal value of the Crouzeix ratio is 1 / 2, over all polynomials p of any degree and matrices A of any order. We derive the subdifferential of this ratio at pairs (p, A) for which the largest singular value of p(A) is simple. In particular, we show that at certain candidate minimizers (p, A), the Crouzeix ratio is (Clarke) regular and satisfies a first-order nonsmooth optimality condition, and hence that its directional derivative is nonnegative there in every direction in polynomial-matrix space. We also show that pairs (p, A) exist at which the Crouzeix ratio is not regular.

Original languageEnglish (US)
Pages (from-to)1-15
Number of pages15
JournalMathematical Programming
DOIs
StateAccepted/In press - Nov 2 2016

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Variational Analysis
Polynomials
Field of Values
Polynomial Matrices
Numerator
Polynomial
Numerical Range
Directional derivative
Subdifferential
Denominator
Postulate
Singular Values
Optimality Conditions
Minimizer
Derivatives
Non-negative
Denote
First-order

ASJC Scopus subject areas

  • Software
  • Mathematics(all)

Cite this

Variational analysis of the Crouzeix ratio. / Greenbaum, Anne; Lewis, Adrian S.; Overton, Michael.

In: Mathematical Programming, 02.11.2016, p. 1-15.

Research output: Contribution to journalArticle

Greenbaum, Anne ; Lewis, Adrian S. ; Overton, Michael. / Variational analysis of the Crouzeix ratio. In: Mathematical Programming. 2016 ; pp. 1-15.
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