JAMC

Article http://dx.doi.org/10.26855/jamc.2026.03.001

Almost Copositive Tensors and Related Research

Download PDF How to cite this paper

TOTAL VIEWS: 584

Mingjun Sheng

Chongqing Normal University, Chongqing 401331, China.

*Corresponding author:Mingjun Sheng

This work was supported by the Chongqing Graduate Student Research Innovation Project (Grant No. CYB25249).

Published: February 26,2026

Abstract

This paper extends the theory of almost copositive matrices to higher-order symmetric tensors. For even-order real symmetric tensors, we first introduce definitions for almost copositive tensors, almost copositive-plus tensors, and the concept of a key index. These definitions provide a unified framework for analyzing copositivity-related properties in tensor spaces. Subsequently, by formulating a constrained optimization model and applying the Karush-Kuhn-Tucker (KKT) conditions, we derive properties of the eigenvalues of almost copositive tensors. Finally, we prove the existence of key indices for almost copositive-plus tensors and demonstrate that their number is at most two. Overall, this work generalizes important matrix results to higher-order tensors and provides new theoretical tools for future studies in tensor optimization and related applications.

Keywords

Almost copositive tensor; copositive-plus tensor; symmetric tensor; eigenvalue

References

[1] Väliaho H. Almost copositive matrices. Linear Algebra Appl. 1989;116:121-134.

[2] Qi L. Symmetric nonnegative tensors and copositive tensors. Linear Algebra Appl. 2013;439(1):228-238.

[3] Qi L, Luo Z. Tensor Analysis: Spectral Theory and Special Tensors. Society for Industrial and Applied Mathematics (SIAM); 2017.

[4] Chen H, Huang ZH, Qi L. Copositive tensor detection and its applications in physics and hypergraphs. Comput Optim Appl. 2018;69(1):133-158.

[5] Wang Y, Shen J, Bu C. Hypergraph characterizations of copositive tensors. Front Math China. 2021;16(3):815-824.

[6] Song Y, Qi L. Analytical expressions of copositivity for fourth-order symmetric tensors. Anal Appl. 2021;19(5):779-800.

[7] Song Y, Li X. Copositivity for a class of fourth-order symmetric tensors given by scalar dark matter. J Optim Theory Appl. 2022;195(1):334-346.

[8] Wang C, Chen H, Wang Y, Yan H. An alternating shifted inverse power method for the extremal eigenvalues of fourth-order partially symmetric tensors. Appl. Math. Lett. 2023 Jul 1;141:108601.

How to cite this paper

Almost Copositive Tensors and Related Research

How to cite this paper: Mingjun Sheng. (2026) Almost Copositive Tensors and Related Research. Journal of Applied Mathematics and Computation, 10(1), 1-6.

DOI: http://dx.doi.org/10.26855/jamc.2026.03.001