Commutation Principles for Nonsmooth Variational Problems on Euclidean Jordan Algebras
Juyoung Jeong, David Sossa · SIAM Journal on Optimization · 2025
Abstract. The commutation principle proved by Ramírez, Seeger, and Sossa ( SIAM J. Optim. 23 (2013), pp. 687–694) in the setting of Euclidean Jordan algebras says that for a Fréchet differentiable function [Formula: see text] and a spectral function [Formula: see text], any local minimizer or maximizer [Formula: see text] of [Formula: see text] over a spectral set [Formula: see text] operator commutes with the gradient of [Formula: see text] at [Formula: see text]. In this paper, we improve this commutation principle by allowing [Formula: see text] to be nonsmooth. For example, for the case of local minimizer, we show that [Formula: see text] operator commutes with some element of the limiting (Mordukhovich) subdifferential of [Formula: see text] at [Formula: see text] provided that [Formula: see text] is subdifferentially regular at [Formula: see text] satisfying a qualification condition. For the case of local maximizer, we prove that [Formula: see text] operator commutes with each element of the (Fenchel) subdifferential of [Formula: see text] at [Formula: see text] whenever this subdifferential is nonempty. As an application, we characterize local optimizers of shifted strictly convex spectral functions and norms over automorphism invariant sets.