Numerical Optimization of Eigenvalues of Hermitian Matrix Functions
It provides a theoretically grounded global optimization method for non-convex eigenvalue problems, benefiting researchers in control theory and numerical linear algebra.
This paper presents a global minimization algorithm for prescribed eigenvalues of Hermitian matrix functions, using piece-wise quadratic underestimators derived from analytical eigenvalue properties. The algorithm achieves global convergence and is effective for minimizing extreme eigenvalues, such as the largest eigenvalue or sum of largest eigenvalues.
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 piece-wise quadratic functions that underestimate the eigenvalue functions. These piece-wise quadratic under-estimators 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 ${\rm H}_\infty$-norm of a linear dynamical system, numerical radius, distance to uncontrollability and various other non-convex eigenvalue optimization problems, for which, generically, the eigenvalue function involved is simple at all points.