Parabolic Regularity of Spectral Functions

TL;DR

This paper investigates the second-order variational properties of spectral functions, establishing conditions for parabolic regularity and deriving second subderivatives to aid in matrix optimization.

Contribution

It characterizes parabolic regularity for spectral functions based on their symmetric components and computes second subderivatives for convex spectral functions.

Findings

Parabolic regularity of spectral functions equals that of their symmetric functions.

Second subderivatives of convex spectral functions are explicitly calculated.

Second-order optimality conditions for matrix optimization are derived.

Abstract

This paper is devoted to the study of the second-order variational analysis of spectral functions. It is well-known that spectral functions can be expressed as a composite function of symmetric functions and eigenvalue functions. We establish several second-order properties of spectral functions when their associated symmetric functions enjoy these properties. Our main attention is given to characterize parabolic regularity for this class of functions. It was observed recently that parabolic regularity can play a central rule in ensuring the validity of important second-order variational properties such as twice epi-differentiability. We demonstrates that for convex spectral functions, their parabolic regularity amounts to that of their symmetric functions. As an important consequence, we calculate the second subderivative of convex spectral functions, which allows us to establish second-order optimality conditions for a class of matrix optimization problems.