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Commutation principles for nonsmooth variational problems on Euclidean Jordan algebras

Juyoung Jeong, David Sossa

TL;DR

The article generalizes the commutation principle for spectral problems on Euclidean Jordan algebras to nonsmooth $\Theta$, deriving optimality-based commutativity results using Mordukhovich and Fenchel subdifferentials. It develops the necessary nonsmooth variational machinery, leverages the Lie-group structure of automorphisms, and employs chain rules on spectral mappings to connect local optimality with commutativity relations. The main contributions include a maximization principle under weakly spectral data, a minimization principle under Clarke regularity, and applications to local optima of shifted spectral functions and norms over automorphism-invariant sets. These results broaden the scope of commutativity techniques to a wider class of nonsmooth problems with spectral structure, enabling sharper local optimality analysis in variational settings. The findings have potential implications for stability and sensitivity analysis in spectral optimization and for deriving structure of local solutions in automorphism-invariant contexts.

Abstract

The commutation principle proved by Ramírez, Seeger, and Sossa (SIAM J Optim 23:687-694, 2013) in the setting of Euclidean Jordan algebras says that for a Fréchet differentiable function $Θ$ and a spectral function $F$, any local minimizer or maximizer $a$ of $Θ+F$ over a spectral set $\mathcal{E}$ operator commutes with the gradient of $Θ$ at $a$. In this paper, we improve this commutation principle by allowing $Θ$ to be nonsmooth with mild regularity assumptions over it. For example, for the case of local minimizer, we show that $a$ operator commutes with some element of the limiting (Mordukhovich) subdifferential of $Θ$ at $a$ provided that $Θ$ is subdifferentially regular at $a$ satisfying a qualification condition. For the case of local maximizer, we prove that $a$ operator commutes with each element of the (Fenchel) subdifferential of $Θ$ at $a$ whenever this subdifferential is nonempty. As an application, we characterize the local optimizers of shifted strictly convex spectral functions and norms over automorphism invariant sets.

Commutation principles for nonsmooth variational problems on Euclidean Jordan algebras

TL;DR

The article generalizes the commutation principle for spectral problems on Euclidean Jordan algebras to nonsmooth , deriving optimality-based commutativity results using Mordukhovich and Fenchel subdifferentials. It develops the necessary nonsmooth variational machinery, leverages the Lie-group structure of automorphisms, and employs chain rules on spectral mappings to connect local optimality with commutativity relations. The main contributions include a maximization principle under weakly spectral data, a minimization principle under Clarke regularity, and applications to local optima of shifted spectral functions and norms over automorphism-invariant sets. These results broaden the scope of commutativity techniques to a wider class of nonsmooth problems with spectral structure, enabling sharper local optimality analysis in variational settings. The findings have potential implications for stability and sensitivity analysis in spectral optimization and for deriving structure of local solutions in automorphism-invariant contexts.

Abstract

The commutation principle proved by Ramírez, Seeger, and Sossa (SIAM J Optim 23:687-694, 2013) in the setting of Euclidean Jordan algebras says that for a Fréchet differentiable function and a spectral function , any local minimizer or maximizer of over a spectral set operator commutes with the gradient of at . In this paper, we improve this commutation principle by allowing to be nonsmooth with mild regularity assumptions over it. For example, for the case of local minimizer, we show that operator commutes with some element of the limiting (Mordukhovich) subdifferential of at provided that is subdifferentially regular at satisfying a qualification condition. For the case of local maximizer, we prove that operator commutes with each element of the (Fenchel) subdifferential of at whenever this subdifferential is nonempty. As an application, we characterize the local optimizers of shifted strictly convex spectral functions and norms over automorphism invariant sets.
Paper Structure (7 sections, 18 theorems, 93 equations)

This paper contains 7 sections, 18 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. Notations and mathematical backgrounds
  3. Tools from nonsmooth variational analysis
  4. Euclidean Jordan algebras
  5. The limiting normal cone of a smooth manifold
  6. Nonsmooth commutation principles
  7. Optimization of shifted spectral functions/norms over weakly spectral sets

Key Result

Theorem 1

Let $\mathcal{V}$ be a Euclidean Jordan algebra and let $\Theta : \mathcal{V} \to \mathbb{R}$ be a Fréchet differentiable function. Let $F : \mathcal{V} \to \mathbb{R}$ be a spectral function and let $\mathcal{E} \subseteq \mathcal{V}$ be a spectral set. If $a$ is a local minimizer/maximizer of the then $a$ and $\nabla \Theta(a)$ operator commute.

Theorems & Definitions (35)

  • Theorem 1
  • Example 1
  • Example 2
  • Definition 1
  • Proposition 1
  • Theorem 2
  • Definition 2
  • Proposition 2
  • proof
  • Proposition 3
  • ...and 25 more