The Existence Of Solutions To Tensor Complementarity Problems

The complementarity problem(CP)is a classic problem in the field of mathematical programming,it has many applications in engineering and economics.Tensor complementarity problem(TCP)is a class of complementarity problems(CP),the newly proposed Song and Qi.TCP is a generalization of linear complementarity problems and a subclass of nonlinear complementarity problems.The purpose of this paper is to study the existence of solutions to tensor complementarity problems by discussing the properties of a special kind of structure tensor-Q tensors.Q tensor is a kind of structure tensor that introduced by Song and Qi.Q tensor is a generalization of the Q matrix in the linear complementarity problems.The property of Q matrices is a central problem in the existence of solutions to the linear complementarity problem,there are a lot of literatures on the properties of Q matrix.This paper shows that within the class of strong P0 tensors,four classes of tensors,i.e.,R0 tensors,R tensors,ER tensors and Q tensors,are all equivalent.Meanwhile,by constructing some counterexamples,it is proved that several famous conclusions about Q matrices in linear complementarity problems can not be extended to tensor space,and one of counterexamples answers the question raised by Song and Qi: Whether or not a nonzero solution of the tensor complementarity problem contains at least two nonzero components if the involved tensor is a semi-positive Q tensor.The work in this paper lays a foundation for the theoretical study about the algorithm which can solve TCP effectively.