This paper deals with the class of Q-tensors, that is, a Q-tensor is a real tensor ,4 such that the tensor complementarity problem （q, A）： finding an x ∈R^n such that x ≥ 0, q＋Axm-1 ≥ 0, and xT（q＋Ax^m-1） = 0...This paper deals with the class of Q-tensors, that is, a Q-tensor is a real tensor ,4 such that the tensor complementarity problem （q, A）： finding an x ∈R^n such that x ≥ 0, q＋Axm-1 ≥ 0, and xT（q＋Ax^m-1） = 0, has a solution for each vector q ∈R^n. Several subclasses of Q-tensors are given： F-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 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 R0-tensor if and only if the tensor complementarity problem （0, A） has no non-zero vector solution, and a tensor is a R-tensor if and only if it is an R0-tensor and the tensor complementarity problem （e,A） has no non-zero vector solution, where e = （1, 1…. , 1）T展开更多
基金supported by the National Natural Science Foundation of China(Grant No.11571095,11601134)the Hong Kong Research Grant Council(Grant No.PolyU502111,501212,501913 and 15302114)
文摘This paper deals with the class of Q-tensors, that is, a Q-tensor is a real tensor ,4 such that the tensor complementarity problem （q, A）： finding an x ∈R^n such that x ≥ 0, q＋Axm-1 ≥ 0, and xT（q＋Ax^m-1） = 0, has a solution for each vector q ∈R^n. Several subclasses of Q-tensors are given： F-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 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 R0-tensor if and only if the tensor complementarity problem （0, A） has no non-zero vector solution, and a tensor is a R-tensor if and only if it is an R0-tensor and the tensor complementarity problem （e,A） has no non-zero vector solution, where e = （1, 1…. , 1）T
