Some regularity results for the pseudospectral abscissa and pseudospectral radius of a matrix

Mert G̈urb̈uzbalaban, Michael L. Overton

Research output: Contribution to journalArticle

Abstract

The ε-pseudospectral abscissa αε and radius ρε of an n×n matrix are, respectively, the maximal real part and the maximal modulus of points in its ε-pseudospectrum, defined using the spectral norm. It was proved in [A.S. Lewis and C.H.J. Pang, SIAM J. Optim., 19 (2008), pp. 1048-1072] that for fixed ε > 0, αε and ρε are Lipschitz continuous at a matrix A except when αε and ρε are attained at a critical point of the norm of the resolvent (in the nonsmooth sense), and it was conjectured that the points where αε and ρε are attained are not resolvent-critical. We prove this conjecture, which leads to the new result that αε and ρε are Lipschitz continuous, and also establishes the Aubin property with respect to both ε and A of the ε-pseudospectrum for the points z ε ℂ where αε and ρε are attained. Finally, we give a proof showing that the pseudospectrum can never be "pointed outwards."

Original languageEnglish (US)
Pages (from-to)281-285
Number of pages5
JournalSIAM Journal on Optimization
Volume22
Issue number2
DOIs
StatePublished - 2012

Fingerprint

Pseudospectra
Abscissa
Regularity
Radius
Resolvent
Lipschitz
Spectral Norm
Critical point
Modulus
Norm

Keywords

  • Aubin property
  • Eigenvalue perturbation
  • Lipschitz multifunction
  • Pseudospectral abscissa
  • Pseudospectral radius
  • Pseudospectrum

ASJC Scopus subject areas

  • Software
  • Theoretical Computer Science

Cite this

Some regularity results for the pseudospectral abscissa and pseudospectral radius of a matrix. / G̈urb̈uzbalaban, Mert; Overton, Michael L.

In: SIAM Journal on Optimization, Vol. 22, No. 2, 2012, p. 281-285.

Research output: Contribution to journalArticle

@article{cd73ea13f8d147459a1aee052f62faac,
title = "Some regularity results for the pseudospectral abscissa and pseudospectral radius of a matrix",
abstract = "The ε-pseudospectral abscissa αε and radius ρε of an n×n matrix are, respectively, the maximal real part and the maximal modulus of points in its ε-pseudospectrum, defined using the spectral norm. It was proved in [A.S. Lewis and C.H.J. Pang, SIAM J. Optim., 19 (2008), pp. 1048-1072] that for fixed ε > 0, αε and ρε are Lipschitz continuous at a matrix A except when αε and ρε are attained at a critical point of the norm of the resolvent (in the nonsmooth sense), and it was conjectured that the points where αε and ρε are attained are not resolvent-critical. We prove this conjecture, which leads to the new result that αε and ρε are Lipschitz continuous, and also establishes the Aubin property with respect to both ε and A of the ε-pseudospectrum for the points z ε ℂ where αε and ρε are attained. Finally, we give a proof showing that the pseudospectrum can never be {"}pointed outwards.{"}",
keywords = "Aubin property, Eigenvalue perturbation, Lipschitz multifunction, Pseudospectral abscissa, Pseudospectral radius, Pseudospectrum",
author = "Mert G̈urb̈uzbalaban and Overton, {Michael L.}",
year = "2012",
doi = "10.1137/110822840",
language = "English (US)",
volume = "22",
pages = "281--285",
journal = "SIAM Journal on Optimization",
issn = "1052-6234",
publisher = "Society for Industrial and Applied Mathematics Publications",
number = "2",

}

TY - JOUR

T1 - Some regularity results for the pseudospectral abscissa and pseudospectral radius of a matrix

AU - G̈urb̈uzbalaban, Mert

AU - Overton, Michael L.

PY - 2012

Y1 - 2012

N2 - The ε-pseudospectral abscissa αε and radius ρε of an n×n matrix are, respectively, the maximal real part and the maximal modulus of points in its ε-pseudospectrum, defined using the spectral norm. It was proved in [A.S. Lewis and C.H.J. Pang, SIAM J. Optim., 19 (2008), pp. 1048-1072] that for fixed ε > 0, αε and ρε are Lipschitz continuous at a matrix A except when αε and ρε are attained at a critical point of the norm of the resolvent (in the nonsmooth sense), and it was conjectured that the points where αε and ρε are attained are not resolvent-critical. We prove this conjecture, which leads to the new result that αε and ρε are Lipschitz continuous, and also establishes the Aubin property with respect to both ε and A of the ε-pseudospectrum for the points z ε ℂ where αε and ρε are attained. Finally, we give a proof showing that the pseudospectrum can never be "pointed outwards."

AB - The ε-pseudospectral abscissa αε and radius ρε of an n×n matrix are, respectively, the maximal real part and the maximal modulus of points in its ε-pseudospectrum, defined using the spectral norm. It was proved in [A.S. Lewis and C.H.J. Pang, SIAM J. Optim., 19 (2008), pp. 1048-1072] that for fixed ε > 0, αε and ρε are Lipschitz continuous at a matrix A except when αε and ρε are attained at a critical point of the norm of the resolvent (in the nonsmooth sense), and it was conjectured that the points where αε and ρε are attained are not resolvent-critical. We prove this conjecture, which leads to the new result that αε and ρε are Lipschitz continuous, and also establishes the Aubin property with respect to both ε and A of the ε-pseudospectrum for the points z ε ℂ where αε and ρε are attained. Finally, we give a proof showing that the pseudospectrum can never be "pointed outwards."

KW - Aubin property

KW - Eigenvalue perturbation

KW - Lipschitz multifunction

KW - Pseudospectral abscissa

KW - Pseudospectral radius

KW - Pseudospectrum

UR - http://www.scopus.com/inward/record.url?scp=84865691297&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84865691297&partnerID=8YFLogxK

U2 - 10.1137/110822840

DO - 10.1137/110822840

M3 - Article

VL - 22

SP - 281

EP - 285

JO - SIAM Journal on Optimization

JF - SIAM Journal on Optimization

SN - 1052-6234

IS - 2

ER -