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

Abstract

The commutation principle of Ramírez, Seeger, and Sossa [13] proved in the setting of Euclidean Jordan algebras says that when the sum of a real valued function h and a spectral function Φ is minimized/maximized over a spectral set E, any local optimizer a at which h is Fréchet differentiable operator commutes with the derivative h ′ (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 max E (h+Φ), 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 min E 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.

Euclidean Jordan Algebras, Symmetric Association Schemes, Strongly Regular Graphs, and Modified Krein Parameters of a Strongly Regular Graph

“…To apply matrix theory to engineering, mathematics, and cryptography, we refer to References [5,6]. Several mathematicians and engineers have developed their investigation into several science areas of mathematics, working in the environment of Euclidean Jordan algebras (see, for instance, References [7][8][9]). Euclidean Jordan algebras have also become a good tool for analyzing discrete structures' eigenvalues like strongly regular graphs (see [10]).…”