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

M. Seetharama Gowda
### Technical Report Number: trGOW19-01

### Abstract:

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,λ),
where V and W are real inner product spaces and λ from V to 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=λ^{-1}(Q) in V, where Q is a subset of W, is equivalent to optimizing a corresponding combination over the set λ(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.

pdf (Feb. 15, 2019, revised March 8, 2019))

pdf (further revised Nov. 21, 2019)