Commutation principles for optimization problems on spectral sets in Euclidean Jordan algebras

TL;DR

This paper extends the commutation principle in Euclidean Jordan algebras, showing that solutions to certain optimization problems strongly operator commute with subgradients, broadening the understanding of spectral set optimization.

Contribution

It introduces strong operator commutativity relations for solutions of spectral optimization problems using subgradients, generalizing previous results.

Findings

Strong operator commutativity with subgradients established

Results apply to convex spectral problems and FTvN-systems

Improves known commutativity relations for linear functions

Abstract

The commutation principle of Ramirez, Seeger, and Sossa proved in the setting of Euclidean Jordan algebras says that when the sum of a real valued function $h$ and a spectral function $\Phi$ is minimized/maximized over a spectral set $E$, any local optimizer $a$ at which $h$ is Fr\'{e}chet differentiable operator commutes with the derivative $h^{\prime}(a)$. In this paper, assuming the existence of a subgradient in place the derivative (of $h$), we establish `strong operator commutativity' relations: If $a$ solves the problem $\underset{E}{\max}\,(h+\Phi)$, then $a$ strongly operator commutes with every element in the subdifferential of $h$ at $a$; If $E$ and $h$ are convex and $a$ solves the problem $\underset{E}{\min}\,h$, then $a$ strongly operator commutes with the negative of some element in the subdifferential of $h$ at $a$. These results improve known (operator) commutativity relations for linear $h$ and for solutions of variational inequality problems. We establish these results via a geometric commutation principle that is valid not only in Euclidean Jordan algebras, but also in the broader setting of FTvN-systems.