The Solution Set Analysis Of Polynomial Complementarity Problems And Tensor Variational Inequalities

Tensors are higher-order generalizations of vectors and matrices,and tensors variational inequalities and tensors complementary problems have been widely used in stochastic processes,data processing,Markov chains and so on.It is associated with polynomial optimization,fixed point problem,the generalized equations and game theory are also closely related.The tensor complementarity problem is a special case of the tensor variational inequality,and the tensor variational inequality is an important nonlinear problem.The polynomial variational inequality problem is a natural extension of the polynomial complementary problem and the tensor variational inequality.The study of polynomial variational inequalities and complementarity problems is helpful to the theoretical analysis and solution of a class of classical nonlinear optimizations.In this thesis,the properties of solution sets for a class of polynomial variational inequalities and complementary problems are analyzed theoretically by combining tensor structure with polynomial optimization techniques.The thesis is split into six parts:In Chapter 1,introduction.The research background and current status of variational inequality,polynomial variational inequality,generalized tensor variational inequality and polynomial complementary problem are introduced respectively.In Chapter 2,preparatory knowledge.Firstly,some basic symbols used in this thesis are given,and then some concepts related to polynomial variational inequalities and complementary problems such as exceptionally family of elements and structural tensors are reviewed,which makes necessary preparation for the following content.In Chapter 3,the properties of the solution set of a class of polynomial variational inequalities are discussed.This chapter is divided into four sections,the first section introduces the related definitions and propositions.In the second section,we study the existence of solutions for structural tensor polynomial variational inequalities,and prove that the solution set is non-empty compact when the first-term coefficient tensor is positive definite.In Section 3,it is known that the solution is unique under the condition that the polynomial function is highly pseudo-monotone.Basis this,the bounds of the solutions of the polynomial variational inequalities are given.The fourth section concludes this chapter.In Chapter 4,in this thesis,generalized tensor variational inequalities are studied.This chapter is divided into three sections.The first section describes in detail a generalized tensor variational inequality problem model on the nonempty closed convex set K in the space Tm,n.Then,the properties of the solution sets and related theorems are reviewed.Function is given in the second quarter is m order K-pseudo monotone and m order K-strong new definition of pseudo monotone function,and to get an F(x)is m order K-strong pseudo monotone,GTVI(F,K)have unique solution.On this basis,the error bounds of generalized tensor variational inequalities are discussed.The third section is the summary of this chapter.In Chapter 5,we study the complementarity of generalized polynomials.This chapter is divided into three sections.The first section reviews the solution set properties and definitions related to polynomial complementarity problems.In the second section,the bounds of solutions are given via Frobenius norm.The third section is the summary of this chapter.In Chapter 6,summary and prospect.The main contents,contributions and shortcomings of this thesis are summarized,and the future research directions of polynomial variational and complementary problems are prospected.