Commutation principles for optimization problems on spectral sets in Euclidean Jordan algebras
Links to Fileshttps://arxiv.org/abs/2009.04874
MetadataShow full item record
Type of Work10 pages
conference papers and proceedings preprints
Citation of Original PublicationMuddappa Gowda, Commutation principles for optimization problems on spectral sets in Euclidean Jordan algebras, https://arxiv.org/abs/2009.04874
RightsThis item is likely protected under Title 17 of the U.S. Copyright Law. Unless on a Creative Commons license, for uses protected by Copyright Law, contact the copyright holder or the author.
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 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 maxE(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 minEh, 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.