Commutation principles for nonsmooth variational problems on Euclidean Jordan algebras

The commutation principle proved by Ram\'irez, Seeger, and Sossa (SIAM J Optim 23:687-694, 2013) in the setting of Euclidean Jordan algebras says that for a Fr\'echet differentiable function Θ and a spectral function F, any local minimizer or maximizer a of Θ+F over a spectral set E operator commutes with the gradient of Θ at a. In this paper, we improve this commutation principle by allowing Θ to be nonsmooth with mild regularity assumptions over it. For example, for the case of local minimizer, we show that a operator commutes with some element of the limiting (Mordukhovich) subdifferential of Θ at a provided that Θ is subdifferentially regular at a satisfying a qualification condition. For the case of local maximizer, we prove that a operator commutes with each element of the (Fenchel) subdifferential of Θ at a whenever this subdifferential is nonempty. As an application, we characterize the local optimizers of shifted strictly convex spectral functions and norms over automorphism invariant sets.