# Locally Optimal Eigenvectors of Regular Simplex Tensors

## TL;DR

This paper investigates the local optimality of eigenvectors of regular simplex tensors, a newly introduced class of symmetric tensors, through first and second-order necessary conditions in a constrained nonconvex optimization framework.

## Contribution

It introduces the concept of regular simplex tensors and analyzes their local optimality conditions, providing a foundation for future research on eigenpair verification and tensor properties.

## Key findings

- Established first and second-order optimality conditions for regular simplex tensor eigenvectors.
- Discussed potential directions for robust eigenpair verification.
- Highlighted future research avenues in tensor analysis.

## Abstract

Identifying locally optimal solutions is an important issue given an optimization model. In this paper, we focus on a special class of symmetric tensors termed regular simplex tensors, which is a newly-emerging concept, and investigate its local optimality of the related constrained nonconvex optimization model. This is proceeded by checking the first-order and second-order necessary condition sequentially. Some interesting directions concerning the regular simplex tensors, including the robust eigenpairs checking and other potential issues, are discussed in the end for future work.