Emre Mengi

Numerical Optimization of Eigenvalues of Hermitian Matrix Functions

This work concerns the global minimization of a prescribed eigenvalue or a weighted sum of prescribed eigenvalues of a Hermitian matrix-valued function depending on its parameters analytically in a box. We describe how the analytical properties of eigenvalue functions can be put into use to derive piecewise quadratic functions that underestimate the eigenvalue functions. These piecewise quadratic underestimators lead us to a global minimization algorithm, originally due to Breiman and Cutler. We prove the global convergence of the algorithm and show that it can be effectively used for the minimization of extreme eigenvalues, e.g., the largest eigenvalue or the sum of the largest specified number of eigenvalues. This is particularly facilitated by the analytical formulas for the first derivatives of eigenvalues, as well as analytical lower bounds on the second derivatives that can be deduced for extreme eigenvalue functions. The applications that we have in mind also include the

On the Estimation of the Distance to Uncontrollability for Higher Order Systems

{rm H}_infty

Locating a nearest matrix with an eigenvalue of prespecified algebraic multiplicity

-norm of a ...

Large-Scale Computation of

A higher order dynamical system of order

\mathcal{L}_\infty

k

-Norms by a Greedy Subspace Method

is called controllable if the trajectory of the system as well as its first

Structure-preserving eigenvalue solvers for robust stability and controllability estimates

k-1

Nonlinear eigenvalue problems with specified eigenvalues

derivatives can be adjusted to pass through any given point at a finite time by choosing the input appropriately. The distance to uncontrollability is the norm of the smallest perturbation yielding an uncontrollable system. We derive a singular value minimization characterization for the distance to uncontrollability and present a trisection algorithm exploiting the singular value characterization. The algorithm is devised for low accuracy and depends on the extraction of the imaginary eigenvalues of even-odd matrix polynomials of degree

Nearest Linear Systems with Highly Deficient Reachable Subspaces

2k

Generalized eigenvalue problems with specified eigenvalues

and size