Commutation principles for optimization problems involving strictly Schur-convex functions in Euclidean Jordan algebras
Pedro G. Massey, Noelia B. Rios, David Sossa
TL;DR
This paper develops commutation principles for optimizers of shifted spectral functions in Euclidean Jordan algebras by proving that, under strict Schur-convexity of F and appropriate spectral (or weakly spectral) domain structure, optimizers must commute with the shift element a in a strong sense. The authors leverage the Lie structure of Aut(𝒱) and majorization of eigenvalues to derive global and local optimality results without requiring differentiability of F, and they extend these principles to eigenvalue-orbit settings and simple EJAs. A key application is to condition-number minimization in EJAs, where a vectorized eigenvalue-spread κ and its strictly Schur-convex norm κ_{||·||} yield concrete commutation relations and reduced-form optimization on eigenvalues. The results generalize and unify several prior commutation principles for spectral functions and provide sharp, structure-dependent conclusions. The methodology blends majorization theory, automorphism group analysis, and spectral decompositions to connect local optima with exact eigenstructure constraints.
Abstract
In this work we establish several commutation principles for optimizers of shifts of spectral functions in the context of Euclidean Jordan Algebras (EJAs). For instance, we show that under certain assumptions, if $\bar x$ is a (local) optimizer of $F(x-a)$ for $x\inΩ$, where $Ω\subset \mathcal V$ is a spectral set of an EJA $\mathcal V$, $a\in \mathcal V$ and $F:\mathcal V\rightarrow \mathbb R$ is a strictly Schur-convex spectral function, then $a$ and $\bar x$ operator commute. We make no further assumption on the smoothness of $F$; instead, we take advantage of the smoothness (Lie structure) of the Automorphism group of $\mathcal V$ and make use of majorization techniques for the eigenvalues of elements in EJAs. Our approach allows us to deal with several problems considered in the literature, related to strictly convex spectral functions and strictly convex spectral norms. In particular, we use our commutation principles to analyze the problem of minimizing the condition number in EJAs.