  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.
