Research On Copositive Plus Tensors And Tensor Complementarity Problems

In recent years,researches on tensor(hypermatrices),a high-level extension of the matrix,have received more and more attention.The tensor Complementarity problem(TCP),correspondingly,has also been extensively studied as a subclass of nonlinear complementary problems.There are many literatures on the theoretical properties of TCP solution sets now,including the existence of solutions,the global uniqueness of solutions,the boundedness of solution sets,and so on.In addition,the application of tensor complementarity problem has also been discussed by many scholars.Structure tensor also plays an important role in research on TCP.In the past studies on structure tensors,most of them extended the concept,related theories and methods of structure matrix to tensor conditions.Since the tensor case is much more complex than matrix,there are many related matrix theories that cannot be extended to the tensor case.In this paper,we extend the concept of Copositive Plus matrix to the case of tensor.We define a class of structured tensors,called Copositive Plus tensor,and discuss the properties of the Copositive Plus tensor.Some properties of the Copositive Plus matrices are extended to the tensor case,and we also have some counter-examples to show that the properties of some Copositive Plus matrices properties cannot be extended to tensor conditions.Finally,We prove that the solution set of the tensor complementarity problem is nonempty and bounded if the involved tensor is a Q tensor and a Copositive Plus tensor.The results obtained in this paper provide a theoretical basis for further study of Copositive Plus tensor and related problems.