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

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.Comment: 10 page

On the connectedness of spectral sets and irreducibility of spectral cones in Euclidean Jordan algebras

Let V be a Euclidean Jordan algebra of rank n. The eigenvalue map from V to R^n takes any element x in V to the vector of eigenvalues of x written in the decreasing order. A spectral set in V is the inverse image of a permutation set in R^n under the eigenvalue map. If the permutation set is also a convex cone, the spectral set is said to be a spectral cone. This paper deals with connectedness and arcwise connectedness properties of spectral sets. By relying on the result that in a simple Euclidean Jordan algebra, every eigenvalue orbit is arcwise connected, we show that if a permutation invariant set is connected (arcwise connected), then the corresponding spectral set is connected (respectively, arcwise connected). A related result is that in a simple Euclidean Jordan algebra, every pointed spectral cone is irreducible

Optimizing certain combinations of spectral and linear//distance functions over spectral sets

In the settings of Euclidean Jordan algebras, normal decomposition systems (or Eaton triples), and structures induced by complete isometric hyperbolic polynomials, we consider the problem of optimizing a certain combination (such as the sum) of spectral and linear$/$distance functions over a spectral set. To present a unified theory, we introduce a new system called Fan-Theobald-von Neumann system which is a triple $(V,W,{\lambda})$, where $V$ and $W$ are real inner product spaces and ${\lambda}:V\rightarrow W$ is a norm preserving map satisfying a Fan-Theobald-von Neumann type inequality together with a condition for equality. In this general setting, we show that optimizing a certain combination of spectral and linear$/$distance functions over a set of the form $E={\lambda}^{-1}(Q)$ in $V$, where $Q$ is a subset of $W$, is equivalent to optimizing a corresponding combination over the set ${\lambda}(E)$ and relate the attainment of the optimal value to a commutativity concept. We also study related results for convex functions in place of linear$/$distance functions. Particular instances include the classical results of Fan and Theobald, von Neumann, results of Tam, Lewis, and Bauschke et al., and recent results of Ramirez et al. As an application, we present a commutation principle for variational inequality problems over such a system.Comment: 35 pages, some typos have been fixed in the latest versio

Generalized subdifferentials of spectral functions over Euclidean Jordan algebras

This paper is devoted to the study of generalized subdifferentials of spectral functions over Euclidean Jordan algebras. Spectral functions appear often in optimization problems playing the role of "regularizer", "barrier", "penalty function" and many others. We provide formulae for the regular, approximate and horizon subdifferentials of spectral functions. In addition, under local lower semicontinuity, we also furnish a formula for the Clarke subdifferential, thus extending an earlier result by Baes. As application, we compute the generalized subdifferentials of the function that maps an element to its k-th largest eigenvalue. Furthermore, in connection with recent approaches for nonsmooth optimization, we present a study of the Kurdyka-Lojasiewicz (KL) property for spectral functions and prove a transfer principle for the KL-exponent. In our proofs, we make extensive use of recent tools such as the commutation principle of Ram\'irez, Seeger and Sossa and majorization principles developed by Gowda.Comment: 26 pages. Some minor fixes and trimming. Accepted for publication at the SIAM Journal on Optimizatio