Studies On Theory And Algorithm Of Tensor Complementarity Problems

The thesis studies the theory and algorithm of tensor complementarity problems.The complementary problem is an interdisciplinary research field between operational re-search and computational mathematics and has been widely used in scientific research,engineering technology and many other fields.As a generalization of the linear comple-mentarity problem and a subclass of the nonlinear complementarity problem,the tensor complementarity problem,since it was proposed in 2015,attracts a lot of attention of do-mestic and international optimization,numerical algebra and other fields,and it has been developed rapidly.In this paper,the discussion of the tensor complementarity problem can be divided into two parts:in terms of theory,we study the uniqueness and stabil-ity of solution,the non-emptiness and compactness of solution sets and the continuity of solution maps of the tensor complementarity problem;in terms of algorithm,for a special kind of tensor complementarity problems,an effective algorithm is designed and numerical experiments are carried out.The content is summarized as follows:First,the uniqueness of solution and the non-emptiness and compactness of solution sets are studied.On the one hand,based on an important conclusion about the global uniqueness and solvability of the linear complementarity problem,Song and Qi proposed a guess about generalizing this result to the tensor complementarity problem.By con-structing two counter examples,we give a negative answer to this guess.Then,a new class of structured tensors is defined,and the result that tensor complementarity problems involving this kind of tensors possess global uniqueness and solvability is obtained.On the other hand,by using the property of the tensor complementarity problem,a new wide kind of structured tensors is given,and the relationships between this class of structured tensors and other classes of structured tensors are discussed,and it is proved that the solu-tion sets of tensor complementarity problems are nonempty and compact when involving such kind of tensors.Next,the stability of solutions and the continuity of solution maps are established.First,the definitions of stability and continuity are given.Second,with the help of the tensor variational inequality and good properties of some special structured tensors,the conditions under which the solutions of tensor complementarity problems possess the stability are given.Third,by using properties of some structured tensors,the continuity of solution maps of tensor complementarity problems is obtained,and the relationship between the uniqueness of solution and the continuity of solution maps of tensor comple-mentarity problems is proved.Last,we discuss a special class of tensor complementarity problems and present that although the tensor complementarity problem is a subclass of the nonlinear complemen-tarity problem,the general algorithms for solving nonlinear complementarity problems cannot be directly applied to this case.Therefore,in this paper,by excavating and us-ing the good properties of the relevant structured tensors,we propose an index detect-ing partial Newton-type algorithm by introducing a pivot minimum function,which is well-defined in the sense that it generates a nonnegative nonincreasing sequence globally converging to a solution of the problem under consideration.Finally,numerical results further support the efficiency and reliability of the proposed algorithm.