Numerical Optimization of Eigenvalues of Hermitian Matrix Functions

TL;DR

This paper develops a global optimization algorithm for eigenvalues of Hermitian matrix functions, leveraging analytical properties to efficiently minimize extreme eigenvalues and related non-convex problems.

Contribution

It introduces a piece-wise quadratic under-estimator approach and proves its global convergence for minimizing eigenvalues of Hermitian matrix functions.

Findings

Effective minimization of extreme eigenvalues demonstrated

Algorithm converges globally under specified conditions

Applicable to various non-convex eigenvalue problems

Abstract

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.