Eigenvalue analysis of constrained minimization problem for homogeneous polynomial

TL;DR

This paper introduces Pareto eigenvalues for symmetric tensors, linking them to constrained polynomial minimization, and establishes conditions for copositivity based on these eigenvalues.

Contribution

It defines Pareto $H$- and $Z$-eigenvalues, proves their existence, and connects them to the minimum of constrained polynomial problems, providing new solution methods.

Findings

Existence of Pareto eigenvalues for symmetric tensors

Minimum Pareto eigenvalues equal constrained polynomial minima

Copositivity characterized by non-negative Pareto eigenvalues

Abstract

In this paper, the concepts of Pareto $H$-eigenvalue and Pareto $Z$-eigenvalue are introduced for studying constrained minimization problem and the necessary and sufficient conditions of such eigenvalues are given. It is proved that a symmetric tensor has at least one Pareto $H$-eigenvalue (Pareto $Z$-eigenvalue). Furthermore, the minimum Pareto $H$-eigenvalue (or Pareto $Z$-eigenvalue) of a symmetric tensor is exactly equal to the minimum value of constrained minimization problem of homogeneous polynomial deduced by such a tensor, which gives an alternative methods for solving the minimum value of constrained minimization problem. In particular, a symmetric tensor $\mathcal{A}$ is copositive if and only if every Pareto $H$-eigenvalue ($Z-$eigenvalue) of $\mathcal{A}$ is non-negative.