Some Regularity Results for the Pseudospectral Abscissa and Pseudospectral Radius of a Matrix

Abstract

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 ∈ C where αε and ρε are attained. Finally, we give a proof showing that the pseudospectrum can never be "pointed outwards."

Robust Stability of Differential-Algebraic Equations

“…Remark 4. Numerical algorithms for computing the stability radii for ODEs are proposed in a number of works, e.g., see [4,6,12,34,38,39,40,42,45,71,72,73,81]. Some extensions to DAEs are discussed in [5].…”

A BFGS-SQP method for nonsmooth, nonconvex, constrained optimization and its evaluation using relative minimization profiles

“…Like the spectral radius, the pseudospectral radius is nonconvex, nonsmooth, and continuously differentiable almost everywhere. However, in contrast to the spectral radius, the pseudospectral radius is locally Lipschitz [GO12] and is thus potentially an easier function to optimize. For example, the known convergence rates for gradient sampling hold for minimizing the pseudospectral radius but not the spectral radius.…”