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.