-

 

 

 




Optimization Online





 

SOME REGULARITY RESULTS FOR THE PSEUDOSPECTRAL ABSCISSA AND PSEUDOSPECTRAL RADIUS OF A MATRIX

Mert Gurbuzbalaban(mert***at***cims.nyu.edu)
Michael L. Overton(overton***at***cs.nyu.edu)

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

Keywords: Pseudospectrum, pseudospectral abscissa, pseudospectral radius,eigenvalue perturbation, Lipschitz multifunction, Aubin property

Category 1: Convex and Nonsmooth Optimization (Nonsmooth Optimization )

Category 2: Global Optimization (Theory )

Category 3: Nonlinear Optimization

Citation: Submitted. Jan 2011.

Download: [PDF]

Entry Submitted: 02/14/2011
Entry Accepted: 02/14/2011
Entry Last Modified: 02/14/2011

Modify/Update this entry


  Visitors Authors More about us Links
  Subscribe, Unsubscribe
Digest Archive
Search, Browse the Repository

 

Submit
Update
Policies
Coordinator's Board
Classification Scheme
Credits
Give us feedback
Optimization Journals, Sites, Societies
Mathematical Programming Society