### 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 language | English (US) |
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Pages (from-to) | 1-15 |

Number of pages | 15 |

Journal | Mathematical Programming |

DOIs | |

State | Accepted/In press - Nov 2 2016 |

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### ASJC Scopus subject areas

- Software
- Mathematics(all)

### Cite this

*Mathematical Programming*, 1-15. https://doi.org/10.1007/s10107-016-1083-6

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

Research output: Contribution to journal › Article

*Mathematical Programming*, pp. 1-15. https://doi.org/10.1007/s10107-016-1083-6

}

TY - JOUR

T1 - Variational analysis of the Crouzeix ratio

AU - Greenbaum, Anne

AU - Lewis, Adrian S.

AU - Overton, Michael

PY - 2016/11/2

Y1 - 2016/11/2

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84994338825&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84994338825&partnerID=8YFLogxK

U2 - 10.1007/s10107-016-1083-6

DO - 10.1007/s10107-016-1083-6

M3 - Article

AN - SCOPUS:84994338825

SP - 1

EP - 15

JO - Mathematical Programming

JF - Mathematical Programming

SN - 0025-5610

ER -