[Paper Review] Optimizing certain combinations of spectral and linear$/$distance functions over spectral sets

M. Seetharama Gowda|arXiv (Cornell University)|Feb 18, 2019

Advanced Optimization Algorithms Research31 references4 citations

TL;DR

This paper introduces the Fan-Theobald-von Neumann (FTvN) system, a unified framework for optimizing combinations of spectral, linear, and distance functions over spectral sets in diverse algebraic structures. It establishes that such optimization problems over spectral sets are equivalent to corresponding problems over the image of the eigenvalue map, with optimal values attained under a commutativity condition, generalizing classical results in Jordan algebras, normal decomposition systems, and hyperbolic polynomials.

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 $λ:V ightarrow W$ 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.

Motivation & Objective

• To unify and generalize optimization results across Euclidean Jordan algebras, normal decomposition systems, and hyperbolic polynomial structures.
• To identify a minimal set of axioms—captured in the Fan-Theobald-von Neumann system—that ensure equivalence between optimization over spectral sets and their image under the eigenvalue map.
• To extend classical results on spectral optimization to combinations of spectral functions with linear, distance, and convex functions.
• To establish a commutativity condition that characterizes attainment of optimal values in such optimization problems.
• To provide a commutation principle for variational inequality problems in this generalized setting.

Proposed method

• Introduces the FTvN system as a triple $({\cal V}, {\cal W}, \lambda)$, where $\lambda: {\cal V} \to {\cal W}$ is a norm-preserving map satisfying a Fan-Theobald-von Neumann-type inequality and an equality condition.
• Defines spectral sets as preimages $E = \lambda^{-1}(Q)$ for subsets $Q \subseteq {\cal W}$, and spectral functions as compositions $\Phi = \phi \circ \lambda$.
• Derives equivalence theorems: $\sup_E(f + \Phi) = \sup_{\lambda(E)}(f^* + \phi)$ and $\inf_E(g + \Phi) = \inf_{\lambda(E)}(g_* + \phi)$, where $f^*$ and $g_*$ are induced functions on ${\cal W}$.
• Establishes that optimal values are attained if and only if the optimizer $x$ commutes with the parameter $c$ in the sense $\langle c, x \rangle = \langle \lambda(c), \lambda(x) \rangle$.
• Applies the framework to recover and generalize known results from Euclidean Jordan algebras, Eaton triples, and hyperbolic polynomials.
• Constructs a novel example of an FTvN system that is neither a normal decomposition system nor a Euclidean Jordan algebra, using a non-identity isometric linear map on a real inner product space.

Experimental results

Research questions

• RQ1 Can the equivalence between optimization over spectral sets and their image under the eigenvalue map be generalized beyond classical settings like Jordan algebras and Eaton triples?
• RQ2 What minimal axiomatic structure ensures that optimization of a sum of spectral and linear/distance functions over a spectral set reduces to optimization over the image space?
• RQ3 How can the attainment of optimal values be characterized in terms of a commutativity condition between the decision variable and the problem parameter?
• RQ4 To what extent can convex functions replace linear or distance functions in such optimization equivalences?
• RQ5 Are there non-trivial examples of FTvN systems that are not isomorphic to known structures like Euclidean Jordan algebras or normal decomposition systems?

Key findings

• The optimization of $f + \Phi$ over a spectral set $E = \lambda^{-1}(Q)$ is equivalent to optimizing $f^* + \phi$ over $\lambda(E)$, with $\sup_E(f + \Phi) = \sup_{\lambda(E)}(f^* + \phi)$ and $\inf_E(g + \Phi) = \inf_{\lambda(E)}(g_* + \phi)$.
• Optimal values are attained if and only if the optimizer $x$ satisfies $\langle c, x \rangle = \langle \lambda(c), \lambda(x) \rangle$, defining a commutativity condition in the FTvN system.
• The framework recovers and generalizes classical results: Fan's inequality, von Neumann's trace inequality, Theobald's trace minimization, and Tam's results on symmetric cones.
• The paper constructs a non-trivial FTvN system $({\mathbb{R}}^2, {\mathbb{R}}^2, S)$, where $S$ is a $90^\circ$ rotation, which is not a normal decomposition system nor a Euclidean Jordan algebra.
• The commutation principle for variational inequality problems is established: solutions to VI problems over spectral sets correspond to solutions over the image space under $\lambda$, under the same commutativity condition.
• The equivalence results hold not only for sums but also for other combinations of functions, including convex functions in place of linear or distance functions.