Hypergraph characterizations of copositive tensors

doi:10.1007/s11464-021-0931-8

Front. Math. China

2021, Vol. 16

Issue (3) : 815-824 https://doi.org/10.1007/s11464-021-0931-8

RESEARCH ARTICLE

Hypergraph characterizations of copositive tensors

Yue WANG, Jihong SHEN(

), Changjiang BU

College of Intelligent Systems Science and Engineering, Harbin Engineering University, Harbin 150001, China

Download: PDF(335 KB)
Export: BibTeX | EndNote | Reference Manager | ProCite | RefWorks

Abstract

A real symmetric tensor $A = (a i 1 ⋯ i m) ∈ ℝ [m n]$ is copositive (resp., strictly copositive) if $A x m ≥ 0$ (resp., $A x m > 0$ ) for any nonzero nonnegative vector $x ∈ ℝ n$ : By using the associated hypergraph of $A$ , we give necessary and sufficient conditions for the copositivity of $A$ : For a real symmetric tensor $A$ satisfying the associated negative hypergraph $H_(A)$ and associated positive hypergraph $H + (A)$ are edge disjoint subhypergraphs of a supertree or cored hypergraph, we derive criteria for the copositivity of $A$ : We also use copositive tensors to study the positivity of tensor systems.

Keywords Copositive tensor hypergraph positive system

Corresponding Author(s): Jihong SHEN

Issue Date: 14 July 2021

Cite this article:

Yue WANG,Jihong SHEN,Changjiang BU. Hypergraph characterizations of copositive tensors[J]. Front. Math. China, 2021, 16(3): 815-824.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-021-0931-8
https://academic.hep.com.cn/fmc/EN/Y2021/V16/I3/815

1	Z Bartosiewicz. Positive nonlinear systems, response maps and realizations. In: Proc of 54th IEEE Conference on Decision and Control, December 15-18, 2015, Osaka.2015, 6379–6384 https://doi.org/10.1109/CDC.2015.7403224
2	A Bretto. Hypergraph Theory: An Introduction. Berlin: Springer, 2013 https://doi.org/10.1007/978-3-319-00080-0
3	C J Bu, H F Li, J Zhou. Inverse Perron values and connectivity of a uniform hyper-graph. Electron J Combin, 2018, 25: P4.28 https://doi.org/10.37236/7410
4	C J Bu, Y M Wei, L Z Sun, J Zhou. Brualdi-type eigenvalue inclusion sets of tensors. Linear Algebra Appl, 2015, 480: 168–175 https://doi.org/10.1016/j.laa.2015.04.034
5	H B Chen, Z H Huang, L Q Qi. Copositive detection of tensors: theory and algorithm. J Optim Theory Appl, 2017, 174: 746–761 https://doi.org/10.1007/s10957-017-1131-2
6	H B Chen, Z H Huang, L Q Qi. Copositive tensor detection and its applications in physics and hypergraphs. Comput Optim Appl, 2018, 69: 133–158 https://doi.org/10.1007/s10589-017-9938-1
7	H B Chen, Y J Wang. High-order copositive tensors and its applications. J Appl Anal Comput, 2018, 8(6): 1863–1885 https://doi.org/10.11948/2018.1863
8	J Cooper, A Dutle. Spectra of uniform hypergraphs. Linear Algebra Appl, 2012, 436: 3268–3292 https://doi.org/10.1016/j.laa.2011.11.018
9	W Y Ding, L Q Qi, Y M Wei.ℳ-tensors and nonsingular ℳ-tensors. Linear Algebra Appl, 2013, 439: 3264–3278 https://doi.org/10.1016/j.laa.2013.08.038
10	L Farina, S Rinaldi. Positive Linear Systems: Theory and Applications. New York: Wiley Interscience, 2000 https://doi.org/10.1002/9781118033029
11	J B Hiriart-Urruty, A Seeger. A variational approach to copositive matrices. SIAM Rev, 2010, 52(4): 593–629 https://doi.org/10.1137/090750391
12	S L Hu, L Q Qi, J Y Shao. Cored hypergraphs, power hypergraphs and their Laplacian H-eigenvalues. Linear Algebra Appl, 2013, 439: 2980–2998 https://doi.org/10.1016/j.laa.2013.08.028
13	K Kannike. Vacuum stability of a general scalar potential of a few fields. Eur Phys J C, 2016, 76: 324 https://doi.org/10.1140/epjc/s10052-016-4160-3
14	H H Li, J Y Shao, L Q Qi. The extremal spectral radii of k-uniform supertrees. J Comb Optim, 2016, 32: 741–764 https://doi.org/10.1007/s10878-015-9896-4
15	L Li, X Z Zhang, Z H Huang, L Q Qi. Test of copositive tensors. J Ind Manag Optim, 2019, 15(2): 881–891 https://doi.org/10.3934/jimo.2018075
16	L H Lim. Singular values and eigenvalues of tensors: a variational approach.In: Proc1st IEEE International Workshop on Computational Advances of Multisensor AdaptiveProcessing, Puerto Vallarta, 2005. 2005, 129–132
17	T S Motzkin. Copositive quadratic forms. National Bureau Standards Report, 1952, 1818: 11–12
18	J W Nie, Z Yang, X Z Zhang. A complete semidefinite algorithm for detecting copositive matrices and tensors. SIAM J Optim, 2018, 28: 2902–2921 https://doi.org/10.1137/17M115308X
19	J Pena, J C Vera, L F Zuluaga. Completely positive reformulations for polynomial optimization. Math Program, 2014, 151: 405. Complet431 https://doi.org/10.1007/s10107-014-0822-9
20	L Q Qi. Eigenvalues of a real supersymmetric tensor. J Symbolic Comput, 2005, 40: 1302–1324 https://doi.org/10.1016/j.jsc.2005.05.007
21	L Q Qi. Symmetric nonnegative tensors and copositive tensors. Linear Algebra Appl, 2013, 439: 228–238 https://doi.org/10.1016/j.laa.2013.03.015
22	L Q Qi, Z Y Luo. Tensor Analysis: Spectral Theory and Special Tensors. Philadelphia: SIAM, 2017 https://doi.org/10.1137/1.9781611974751
23	N Shaked-Monderer. SPN graphs: When copositive= SPN. Linear Algebra Appl, 2016, 509: 82–113 https://doi.org/10.1016/j.laa.2016.07.018
24	J Y Shao, L Q Qi, S L Hu. Some new trace formulas of tensors with applications in spectral hypergraph theory. Linear Multilinear Algebra, 2015, 63: 971–992 https://doi.org/10.1080/03081087.2014.910208
25	Y S Song, L Q Qi. Necessary and sufficient conditions for copositive tensors. Linear Multilinear Algebra, 2015, 63: 120–131 https://doi.org/10.1080/03081087.2013.851198
26	C Y Wang, H B Chen, Y J Wang, G L Zhou. On copositiveness identification of partially symmetric rectangular tensors. J Comput Appl Math, 2020, 372: 112678 https://doi.org/10.1016/j.cam.2019.112678
27	L P Zhang, L Q Qi, G L Zhou. M-tensors and some applications. SIAM J Matrix Anal Appl, 2014, 35: 437–452 https://doi.org/10.1137/130915339

[1]	Wen-Huan WANG, Ling YUAN. Uniform supertrees with extremal spectral radii[J]. Front. Math. China, 2020, 15(6): 1211-1229.
[2]	Yizheng FAN, Zhu ZHU, Yi WANG. Least H-eigenvalue of adjacency tensor of hypergraphs with cut vertices[J]. Front. Math. China, 2020, 15(3): 451-465.
[3]	Jun HE, Yanmin LIU, Junkang TIAN, Xianghu LIU. Upper bounds for signless Laplacian Z-spectral radius of uniform hypergraphs[J]. Front. Math. China, 2019, 14(1): 17-24.
[4]	Dongmei CHEN, Zhibing CHEN, Xiao-Dong ZHANG. Spectral radius of uniform hypergraphs and degree sequences[J]. Front. Math. China, 2017, 12(6): 1279-1288.
[5]	Xiying YUAN, Xuelian SI, Li ZHANG. Ordering uniform supertrees by their spectral radii[J]. Front. Math. China, 2017, 12(6): 1393-1408.
[6]	Junjie YUE,Liping ZHANG,Mei LU. Largest adjacency, signless Laplacian, and Laplacian H-eigenvalues of loose paths[J]. Front. Math. China, 2016, 11(3): 623-645.
[7]	Changjiang BU,Yamin FAN,Jiang ZHOU. Laplacian and signless Laplacian Z-eigenvalues of uniform hypergraphs[J]. Front. Math. China, 2016, 11(3): 511-520.
[8]	Jinshan XIE, An CHANG. H-Eigenvalues of signless Laplacian tensor for an even uniform hypergraph[J]. Front Math Chin, 2013, 8(1): 107-127.

Viewed

Full text

Abstract

Cited

Shared

Discussed