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Locally Optimal Eigenvectors of Regular Simplex Tensors

Lei Wang

TL;DR

This paper focuses on a special class of symmetric tensors termed regular simplex tensors, which is a newly-emerging concept, and investigates its local optimality of the related constrained nonconvex optimization model.

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.

Locally Optimal Eigenvectors of Regular Simplex Tensors

TL;DR

This paper focuses on a special class of symmetric tensors termed regular simplex tensors, which is a newly-emerging concept, and investigates its local optimality of the related constrained nonconvex optimization model.

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.
Paper Structure (13 sections, 90 equations, 2 figures)

This paper contains 13 sections, 90 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Preliminaries
  4. Optimization theories of Tensor eigenpairs
  5. Regular simplex frame and tensor
  6. Problem reformulation for regular simplex tensor eigenpairs
  7. Structure of all stationary points of (\ref{['optmodel']})
  8. Structure of all locally optimal points of (\ref{['optmodel']})
  9. Odd $m$ case
  10. Even $m$ case with (\ref{['u_classifyeven1']})
  11. Even $m$ case with (\ref{['u_classifyeven2']})
  12. Future Work
  13. Appendix

Figures (2)

  • Figure 1: An intuitive sketch map concerning the reformulation from model (\ref{['optmodelori']}) to (\ref{['optmodel']}) for the case of $n=3$: transforming the regular triangle from 2D to 3D space.
  • Figure 2: The sketch figures for the curve of the function $f(x) = x^{m-1} -\alpha x - \beta$ for odd and even case: a) odd $m$ case: $m=3, \alpha=1, \beta =2$; b) even $m$ case: $m=4, \alpha=7, \beta =4$.

Theorems & Definitions (5)

  • proof
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