Some Theory Analysis And Algorithem Design For Tensor Complementarity And Tensor Equations

Tensor is a high-dimensional array,which has broad application prospects in practical fields such as medical engineering,literature analysis,high-order network link analysis and partial differential equations.This paper mainly focuses on theoretical analysis and algorithm design for the tensor quadratic eigenvalue complementarity problem on the second-order cone,the tensor absolute value equations and the generalized tensor equations.This thesis is divided into six chapters:In Chapter 1,we mainly introduce the research background and research status of tensor quadratic eigenvalue complementarity problem on second-order cone,tensor absolute value equations and generalized tensor equations problem.In Chapter 2,we first give some basic symbols,then review the related concepts of structural tensors,exception cluster,local error bound,and introduce two new types of structural tensors to make necessary preparations for subsequent chapters.In Chapter 3,for the tensor quadratic eigenvalue complementarity problem on the second-order cone,the nonlinear programming transformation forms are given and the relationships of their solutions are proved.The content of this chapter can be regarded as a generalization of the quadratic eigenvalue complementarity problem of matrices.In Chapter 4,we mainly studies the tensor absolute value equations.This chapter is divided into three sections.In the first section,the existence of the solution of the tensor absolute value equations is studied.It is proved that when the tensor A is a H+-tensor,the solution set of the tensor absolute value equations is a non-empty compact set,etc.In the second section,under the appropriate conditions,the upper and lower bounds of the solution of the tensor absolute value equations are discussed.In the third section,we use the generalized Newton method to solve the tensor absolute value equations and report some numerical results through experiments.In Chapter 5,we mainly studies the generalized tensor equations.This chapter is divided into three sections.In the first section,the degree theory is used to study the existence of solutions of generalized tensor equations.In the second section,under the appropriate conditions,the local error bound conditions of the function are given.In the third section,the corresponding algorithm is designed for solving the general tensorequations,and some numerical results are given.In Chapter 6,we summarize the full text and indicate the further research direction.