Theoretical Analysis Of Eigenvalue Complementarity Problems With Structured Tensors

Tensor is an important research subject in multiple linear algebra,which is closely related to polynomial optimization and has many applications in quantum mechanics,medical imaging and signal processing.This thesis focuses on the theoretical analysis of tensor(eigenvalue)complementarity problems,including the strictly semi-positive tensor eigenvalue complementarity problems,polynomial complementarity problem with structured tensors(PCPs),and generalized polynomial complementarity problem with structured tensors(GPCPs).This thesis is divided into six chapters:In Chapter 1,we introduce the research background and research status of complementary problems.Particularly,tensor eigenvalue complementary problems,polynomial complementary problems and generalized polynomial complementary problems are introduced respectively.In Chapter 2,we introduce the preliminaries.Firstly,we give some basic symbols which will be used in this article.And then reviews the concepts of structure tensors,exceptional family of elements,residual function,operator norm,which is in connection with tensor(eigenvalues)complementary problem with structure tensor.This provides the necessary preparation for subsequent chapters.In Chapter 3,we mainly consider the Pareto-spectrum estimations of eigenvalue complementarity problem with strictly semi-positive tensors.Firstly,we discuss symbolic features of Pareto-eigenvalue of eigenvalue complementarity problem with strictly semipositive tensors.Then,we further study the estimation properties of Pareto-spectrum for eigenvalue complementarity problem with strictly semi-positive tensors by using the properties of the operator norm of strictly semi-positive tensors.In Chapter 4,we mainly consider the polynomial complementary problems with structured tensors.In the first section,We show that the solution existence of PCPs.Then,we prove the solution set of PCPs with a leading ER-tensor is nonempty and compact.In the second section,we analyze lower bounds of solutions of PCPs under the existence of solutions.In the third section,we analyze the error bounds of solutions of PCPs under the m-uniformly P-function.In Chapter 5,we mainly consider the generalized polynomial complementarity problems with structured tensors.In the first section,we consider the solution existence of PCPs with closed convex cone.And we show that,when a pair of leading tensors is cone ER,the solution set of GPCPs is nonempty and compact.In the second section,we study some basic topological properties of the solution set of GPCPs under the condition that the leading tensor pair is cone R~0,including closed,locally bounded,compactness,and upper semicontinuous.In the third section,it is proved that the natural residual function on GPCPs is a global Lipschitzian error bound of the solution set of GPCPs under the appropriate assumptions and conditions.In Chapter 6,we summarize the full text and indicate the further research direction.