Generalized subdifferentials of spectral functions over Euclidean Jordan algebras

Bruno F. Lourenco (brunoism.ac.jp)
Akiko Takeda (takedamist.i.u-tokyo.ac.jp)

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 Ramirez, Seeger and Sossa and majorization principles developed by Gowda.

Keywords: spectral functions, generalized subdifferential, approximating subdifferential, Euclidean Jordan algebra, Kurdyka-Lojasiewicz inequality

Category 1: Convex and Nonsmooth Optimization (Nonsmooth Optimization )

Category 2: Nonlinear Optimization

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Entry Submitted: 01/25/2019
Entry Accepted: 01/25/2019
Entry Last Modified: 11/01/2020

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