Properties of Tensor Complementarity Problem and Some Classes of Structured Tensors

Abstract

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 $e = (1, 1 · · · , 1)^⊤.$