Variational Analysis of Convexly Generated Spectral Max Functions

TL;DR

This paper develops new variational analysis techniques for convexly generated spectral max functions, extending existing results on spectral abscissa and radius, with applications in control theory.

Contribution

It introduces novel methods to analyze the subdifferential properties of convexly generated spectral max functions, generalizing prior results on spectral abscissa.

Findings

Extended subdifferential regularity results to convexly generated spectral max functions.

Derived new variational formulas for the spectral radius.

Provided theoretical foundations for control and stabilization applications.

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.