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Some commutation principles for optimization problems over transformation groups and semi-FTvN systems

M. Seetharama Gowda, David Sossa

TL;DR

This work introduces two notions of commutativity in optimization: commutativity relative to a transformation group ${\cal G}$ and strong commutativity within semi-FTvN systems. It proves that strong commutativity implies commutativity and shows that, in Euclidean Jordan algebras, commutativity relative to algebra automorphisms reduces to operator commutativity, unifying several classical notions. The authors establish that complete hyperbolic polynomials induce semi-FTvN systems, linking spectral mappings to optimization structure, and they derive a suite of commutation principles that generalize prior FTvN-system results to broader group-based and hyperbolic-polynomial contexts. The results provide a versatile algebraic-analytic framework for obtaining optimality conditions in structured optimization problems, with concrete instances in Jordan algebras and hyperbolic-polynomial systems. Collectively, the paper extends commutativity principles to new settings and highlights their role as fundamental optimality conditions across transformation-group and spectral-geometry frameworks.

Abstract

We introduce the concepts of commutativity relative to a transformation group and strong commutativity in the setting of a semi-FTvN system and show their appearance as optimality conditions in certain optimization problems. In the setting of a semi-FTvN system (in particular, in an FTvN system), we show that strong commutativity implies commutativity and observe that in the special case of Euclidean Jordan algebra, commutativity and strong commutativity concepts reduce, respectively, to those of operator and strong operator commutativity. We demonstrate that every complete hyperbolic polynomial induces a semi-FTvN system. By way of an application, we describe several commutation principles.

Some commutation principles for optimization problems over transformation groups and semi-FTvN systems

TL;DR

This work introduces two notions of commutativity in optimization: commutativity relative to a transformation group and strong commutativity within semi-FTvN systems. It proves that strong commutativity implies commutativity and shows that, in Euclidean Jordan algebras, commutativity relative to algebra automorphisms reduces to operator commutativity, unifying several classical notions. The authors establish that complete hyperbolic polynomials induce semi-FTvN systems, linking spectral mappings to optimization structure, and they derive a suite of commutation principles that generalize prior FTvN-system results to broader group-based and hyperbolic-polynomial contexts. The results provide a versatile algebraic-analytic framework for obtaining optimality conditions in structured optimization problems, with concrete instances in Jordan algebras and hyperbolic-polynomial systems. Collectively, the paper extends commutativity principles to new settings and highlights their role as fundamental optimality conditions across transformation-group and spectral-geometry frameworks.

Abstract

We introduce the concepts of commutativity relative to a transformation group and strong commutativity in the setting of a semi-FTvN system and show their appearance as optimality conditions in certain optimization problems. In the setting of a semi-FTvN system (in particular, in an FTvN system), we show that strong commutativity implies commutativity and observe that in the special case of Euclidean Jordan algebra, commutativity and strong commutativity concepts reduce, respectively, to those of operator and strong operator commutativity. We demonstrate that every complete hyperbolic polynomial induces a semi-FTvN system. By way of an application, we describe several commutation principles.

Paper Structure

This paper contains 8 sections, 20 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Commutativity relative to a transformation group
  3. Semi-FTvN systems
  4. Automorphisms and commutativity
  5. The center and unit element
  6. The FTvN system induced by a Euclidean Jordan algebra
  7. The semi-FTvN system induced by a complete hyperbolic polynomial
  8. Commutation principles

Key Result

Theorem 1.1

Consider ${\cal V}$ and ${\cal G}$ as above. Let $\Theta:{\cal V}\rightarrow \mathbb{R}$ be a Fréchet differentiable function and $F:{\cal V}\rightarrow \mathbb{R}$ be a ${\cal G}$-invariant function. Let $E$ be a set in ${\cal V}$ that is ${\cal G}$-invariant. If $a$ is a local optimizer of the pro then $a$ commutes with $\Theta^{\prime}(a)$ relative to ${\cal G}.$

Theorems & Definitions (49)

  • Theorem 1.1
  • Definition 2.1
  • Theorem 2.2
  • Example 2.3
  • Example 2.4
  • Example 2.5
  • Example 2.6
  • Example 2.7
  • Theorem 2.8
  • Corollary 2.9
  • ...and 39 more