Numerical optimization of eigenvalues of Hermitian matrix functions

Emre Mengi, E. Alper Yildirim, Mustafa Kilic

Research output: Contribution to journalArticlepeer-review

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 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 H-norm of a linear dynamical system, numerical radius, distance to uncontrollability, and various other nonconvex eigenvalue optimization problems, for which, generically, the eigenvalue function involved is simple at all points.

Original languageEnglish
Pages (from-to)699-724
Number of pages26
JournalSIAM Journal on Matrix Analysis and Applications
Volume35
Issue number2
DOIs
Publication statusPublished - 5 Jun 2014

Keywords

  • Analytic
  • Global optimization
  • Hermitian eigenvalues
  • Perturbation of eigenvalues
  • Quadratic programming

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