Exceptionally regular tensors and tensor complementarity problems

Recently, many structured tensors are defined and their properties are discussed in the literature. In this paper, we introduce a new class of structured tensors, called exceptionally regular (ER) tensor, which is relevant to the tensor complementarity problem (TCP). We show that this class of tensors is a wide class of tensors which includes many important structured tensors as its special cases. By constructing two examples, we demonstrate that an ER-tensor can be, but not always, an R-tensor. We also show that within the class of the semi-positive tensors, the class of ER-tensors coincides with the class of R-tensors. In particular, we consider the TCP with an ER-tensor and show that its solution set is nonempty and compact. In addition, we also obtain that the solution sets of the TCP with an R-tensor or a P-0 + R-0-tensor are nonempty and compact.