Theoretical Analysis Of Complementarity Problems With Some Structured Tensors

Tensor complementarity problem and polynomial complementarity problem are not only a natural generalization of linear complementarity problem,but also a special case of nonlinear complementarity problem.On the one hand,due to the complexity of tensor structure,many properties of linear complementarity problems can not be directly extended to tensor complementarity problems and polynomial complementarity problems.On the other hand,although the existing results of nonlinear complementarity problems can be directly used to analyze the properties of tensor complementarity problems,this direct application often ignores the special structure of tensors.Therefore,further study on the special properties of tensor complementarity problem is still one of the focuses of current research work under the condition of fully considering the structure of tensor.At present,most of the literatures about tensor complementarity problems mainly focus on the tensor complementarity problems with non-negative cones,and require strong conditions for the existence of solutions of tensor complementarity problems,such as strict copositive tensors,strict semi-positive tensors and so on.However,the models of tensor complementarity problem established in practical application are not limited to non-negative cones,and there may appear tensors with more general structure.In order to overcome the shortcomings of the existing research work,the related properties of tensor complementarity problems are theoretically analyzed in Chapter3.The main research work is as follows: the structural tensors on non-negative cones are extended to general closed convex cones,and the concept of K-ER tensor is given,and the existence of solutions and the compactness of the solution set for K-ER tensor complementarity problems are proved.At the same time,the conclusion is extended to the polynomial complementarity problems.Tensor variational inequality problem is a special kind of polynomial optimization problem and variational inequality problem,and polynomial tensor variational inequality problem is a natural generalization of tensor variational inequality problem and tensor polynomial complementarity problem.In Chapter 4,the properties of polynomial tensor variational inequalities are studied where the functions involvedare determined by multiple tensors and arbitrary vectors.Firstly,the existence of solutions and the compactness of solution sets for tensor variational inequalities are proved with symmetric strict P-tensor.Secondly,the existence and uniqueness theorem of solutions for tensor variational inequalities are generalized to polynomial tensor variational inequalities under the conditions of(strictly)positive semi-definite tensors and(strictly)positive definite tensors.In this dissertation,firstly,the development of linear complementarity problem,variational inequality problem,tensor complementarity problem and tensor variational inequality problem are reviewed.Secondly,for a class of tensor complementarity problems with K-ER tensors,the existence of solutions and the compactness of solution set for the tensor complementarity problem are proved,and the conclusion is extended to the polynomial complementarity problem.Finally,the existence of solutions and the compactness of solution sets for tensor variational inequalities are proved with symmetric strict P-tensor,and the properties and results of solution set of tensor variational inequality problems are extended to polynomial variational inequality problems.