# Generalized subdifferentials of spectral functions over Euclidean Jordan algebras

## TL;DR

This paper develops formulas for various generalized subdifferentials of spectral functions on Euclidean Jordan algebras, extending previous results and applying them to eigenvalue functions, with implications for nonsmooth optimization.

## Contribution

It provides new formulas for regular, approximate, horizon, and Clarke subdifferentials of spectral functions, extending existing theory and analyzing the KL property in this context.

## Key findings

- Formulas for regular, approximate, and horizon subdifferentials of spectral functions.
- Extension of Clarke subdifferential formula under local lower semicontinuity.
- Analysis of the Kurdyka-Lojasiewicz property and transfer of KL-exponent for spectral functions.

## Abstract

This paper is devoted to the study of generalized subdifferentials of spectral functions over Euclidean Jordan algebras. Spectral functions appear often in optimization problems playing the role of "regularizer", "barrier", "penalty function" and many others. We provide formulae for the regular, approximate and horizon subdifferentials of spectral functions. In addition, under local lower semicontinuity, we also furnish a formula for the Clarke subdifferential, thus extending an earlier result by Baes. As application, we compute the generalized subdifferentials of the function that maps an element to its k-th largest eigenvalue. Furthermore, in connection with recent approaches for nonsmooth optimization, we present a study of the Kurdyka-Lojasiewicz (KL) property for spectral functions and prove a transfer principle for the KL-exponent. In our proofs, we make extensive use of recent tools such as the commutation principle of Ram\'irez, Seeger and Sossa and majorization principles developed by Gowda.