Variational Analysis of Convexly Generated Spectral Max Functions

Abstract

The spectral abscissa is the largest real part of an eigenvalue of a matrix and the spectral radius is the largest modulus. Both are examples of spectral max functions---the maximum of a real-valued function over the spectrum of a matrix. These mappings arise in the control and stabilization of dynamical systems. In 2001, Burke and Overton characterized the regular subdifferential of the spectral abscissa and showed that the spectral abscissa is subdifferentially regular in the sense of Clarke when all active eigenvalues are nonderogatory. In this paper we develop new techniques to obtain these results for the more general class of convexly generated spectral max functions. In particular, we extend the Burke-Overton subdifferential regularity result to this class. These techniques allow us to obtain new variational results for the spectral radius.