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Mert Gurbuzbalaban(mertcims.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, 19:10481072, 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 resolventcritical. 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 Modify/Update this entry  
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