### 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 ℳ^{n} 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 ℳ^{n}. 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 language | English (US) |
---|---|

Pages (from-to) | 259-268 |

Number of pages | 10 |

Journal | Journal of Global Optimization |

Volume | 28 |

Issue number | 3-4 |

DOIs | |

State | Published - Apr 2004 |

### Fingerprint

### ASJC Scopus subject areas

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

### Cite this

*Journal of Global Optimization*,

*28*(3-4), 259-268. https://doi.org/10.1023/B:JOGO.0000026448.63457.51

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

Research output: Contribution to journal › Article

*Journal of Global Optimization*, vol. 28, no. 3-4, pp. 259-268. https://doi.org/10.1023/B:JOGO.0000026448.63457.51

}

TY - JOUR

T1 - Variational analysis of the abscissa mapping for polynomials via the Gauss-Lucas theorem

AU - Burke, James V.

AU - Lewis, Adrian S.

AU - Overton, Michael L.

PY - 2004/4

Y1 - 2004/4

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=3543079847&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=3543079847&partnerID=8YFLogxK

U2 - 10.1023/B:JOGO.0000026448.63457.51

DO - 10.1023/B:JOGO.0000026448.63457.51

M3 - Article

VL - 28

SP - 259

EP - 268

JO - Journal of Global Optimization

JF - Journal of Global Optimization

SN - 0925-5001

IS - 3-4

ER -