Theoretical Analysis Of Tensor Complementarity Problems

Tensor complementarity problem is the generalization of linear complementarity problem,also is the special case of nonlinear complementarity problem.Most properties of linear complementarity problem can not be extended directly to tensor complementarity problem due to the complicated structure of tensors.On the other hand,despite many results of nonlinear complementarity problem are certainly applicable to tensor complementarity problem,which,however,do not often embody the structure of tensors.Therefore,how to get the specialized properties of tensor complementarity problem under the premise of fully considering the tensor structure is a focus of current research work.At present,most paper about the tensor complementarity problem is dedicated to showing square tensor complementarity problem over the nonnegative cone,and requires some relatively stronger conditions,e.g.,strictly semi-positive,strictly copositive.However,the tensor complementary problem is not confined to nonnegative cone in practice,and there may have more general rectangular tensors.In addition,few people study the topological properties of the solution set and the stability of the square tensor complementarity problem.Aiming at the insufficient of previous research work,in this paper,we study the related properties of tensor complementarity problem.The main research work is as follows:(1)For square tensor complementarity problem over nonnegative cone,we present the existence of solutions of square tensor complementarity problem with copositive tensor.To my knowledge,copositivity is the weakest requirement for the solution set nonempty of current tensor complementarity problem.(2)For square tensor complementarity problem over general cone,we study the topological properties of the solution set and stability of the square tensor complementarity problem at a given solution,the above result further enrich the theory of tensor complementarity problem.(3)For the given rectangular tensors,define a continuous positively homogeneous operator and a constant,and prove that the considered rectangular tensor is strictly semipositive if and only if this constant is positive.Then,present some upper and lower bounds of solutions for the complementarity problem with strictly semi-positive rectangular tensor.The structure of this paper is organized as follows: firstly,we review the development of the linear complementarity problem and tensor complementarity problem.Secondly,we present the existence of a solution of square tensor complementarity problem under copositivity.Thirdly,we show the topological properties of the solution set and stability of square tensor complementarity problem at a given solution.Finally,we present some upper and lower bounds of solutions for the complementarity problem with strictly semipositive rectangular tensor.