Structured pseudospectra and structured sensitivity of eigenvalues
Journal of Computational and Applied Mathematics
A successive SDP-NSDP approach to a robust optimization problem in finance
Computational Optimization and Applications
Automatica (Journal of IFAC)
Lipschitz Behavior of the Robust Regularization
SIAM Journal on Control and Optimization
Schur Decompositions of a Matrix and the Boundary of Its Pseudospectrum
SIAM Journal on Matrix Analysis and Applications
Structured Pseudospectra for Small Perturbations
SIAM Journal on Matrix Analysis and Applications
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The $\epsilon$-pseudospectrum of a matrix A is the subset of the complex plane consisting of all eigenvalues of all complex matrices within a distance $\epsilon$ of A. We are interested in two aspects of "optimization and pseudospectra." The first concerns maximizing the function "real part" over an $\epsilon$-pseudospectrum of a fixed matrix: this defines a function known as the $\epsilon$-pseudospectral abscissa of a matrix. We present a bisection algorithm to compute this function. Our second interest is in minimizing the $\epsilon$-pseudospectral abscissa over a set of feasible matrices. A prerequisite for local optimization of this function is an understanding of its variational properties, the study of which is the main focus of the paper. We show that, in a neighborhood of any nonderogatory matrix, the $\epsilon$-pseudospectral abscissa is a nonsmooth but locally Lipschitz and subdifferentially regular function for sufficiently small $\epsilon$; in fact, it can be expressed locally as the maximum of a finite number of smooth functions. Along the way we obtain an eigenvalue perturbation result: near a nonderogatory matrix, the eigenvalues satisfy a Hölder continuity property on matrix space---a property that is well known when only a single perturbation parameter is considered. The pseudospectral abscissa is a powerful modeling tool: not only is it a robust measure of stability, but it also reveals the transient (as opposed to asymptotic) behavior of associated dynamical systems.