Generalized subdifferentials of spectral functions over Euclidean Jordan algebras

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 field playing the role of "regularizer", "barrier", "penalty function" and many others. We provide formulae for the regular, approximating 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 Ramírez, Seeger and Sossa and majorization principles developed by Gowda.

Interpretable and Scalable Graphical Models for Complex Spatio-temporal Processes

“…It remains to prove that − log det j diag(Ψ j ) is also a KL function with an exponent of 1 2 . By Theorem 30 in Lourenço and Takeda (2019), if we have f : R r → R a symmetric function and F : E → R the corresponding spectral function, the followings hold (i) F satisfies the KL property at X iff f satisfies the KL property at λ(X), i.e., the eigenvalues of X.…”