Properties of Tensor Complementarity Problem and Some Classes of Structured Tensors

This paper deals with the class of Q-tensors, that is, a Q-tensor is a real tensor A such that the tensor complementarity problem (q,A): finding x ∈ R such that x ≥ 0,q+Axm−1 ≥ 0, and x⊤(q+Axm−1) = 0, has a solution for each vector q ∈ Rn. Several subclasses of Q-tensors are given: P-tensors, R-tensors, strictly semi-positive tensors and semi-positive R0-tensors. We prove that a nonnegative tensor is a Q-tensor if and only if all of its diagonal entries are positive, and a symmetric tensor is a Q-tensor if and only if it is strictly copositive. We also show that the zero vector is the unique feasible solution of the tensor complementarity problem (q,A) for q ≥ 0 if A is a nonnegative Q-tensor.