TY - JOUR
T1 - Properties of Tensor Complementarity Problem and Some Classes of Structured Tensors
AU - Song , Yisheng
AU - Qi , Liqun
JO - Annals of Applied Mathematics
VL - 3
SP - 308
EP - 323
PY - 2022
DA - 2022/06
SN - 33
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/aam/20612.html
KW - $Q$-tensor, $R$-tensor, $R_0$-tensor, strictly semi-positive, tensor
complementarity problem.
AB - <p style="text-align: justify;">This paper deals with the class of $Q$-tensors, that is, a $Q$-tensor is a real
tensor $\mathcal{A}$ such that the tensor complementarity problem $(q, \mathcal{A}):$ finding an $x ∈\mathbb{R}^n$ such that $x ≥ 0,$ $q+\mathcal{A}x^{m−1} ≥ 0,$ and $x^⊤(q+\mathcal{A}x^{m−1}) = 0,$ has a solution for each vector $q ∈ \mathbb{R}^n.$ Several subclasses of $Q$-tensors are
given: $P$-tensors, $R$-tensors, strictly semi-positive tensors and semi-positive $R_0$-tensors. We prove that a nonnegative tensor is a $Q$-tensor if and only if all
of its principal diagonal entries are positive, and so the equivalence of $Q$-tensor, $R$-tensors, strictly semi-positive tensors was showed if they are nonnegative
tensors. We also show that a tensor is an $R_0$-tensor if and only if the tensor
complementarity problem $(0, \mathcal{A})$ has no non-zero vector solution, and a tensor
is a $R$-tensor if and only if it is an $R_0$-tensor and the tensor complementarity
problem $(e, A)$ has no non-zero vector solution, where $e = (1, 1 · · · , 1)^⊤.$</p>