Properties Of Solution Sets For Mixed Tensor Variational Inequalities

The theory of variational inequalities is one of the hot topics in optimization theory.Since the tensor variational inequality was proposed in 2018,it has attracted wide attention.In this paper,we study the existence of solutions and stability of solution sets for mixed tensor variational inequalities.This paper is divided into three chapters,and the main contents are as follows:In chapter 1,we introduce the background and research status of mixed variational inequalities,tensor complementarity problems and tensor variational inequalities,as well as the research methods.Some common symbols and basic concepts to be used later are given.In chapter 2,by employing the notion of exceptional family of elements,we investigate the existence of solutions for mixed tensor variational inequalities.First of all,we prove that the nonexistence of an exceptional family of elements is a sufficient condition for the solvability of mixed tensor variational inequality.For positive semidefinite mixed tensor variational inequalities,the nonexistence of an exceptional family of elements is proved to be an equivalent characterization of the nonemptiness of the solution sets.Secondly,we derive several sufficient conditions of the nonemptiness and compactness of the solution sets for the mixed tensor variational inequalities with some special structured tensors.By extending the concept of the kernel of matrix,a necessary condition of the nonemptiness and compactness of the solution set is given.As a consequence,a characterization of the nonemptiness and compactness of the solution set is given.Finally,we show that the mixed tensor variational inequality problems can be defined as a class of convex optimization problems,and a class of multi-person noncooperative games can be reformulated as a mixed tensor variational inequality.In chapter 3,by employing degree theory,we investigate the stability of solution sets for mixed tensor variational inequalities.First of all,we study the lower semicontinuity of the solution mappings of the mixed tensor variational inequalities by employing the homotopy invariance of Brouwer degree.Secondly,We study the upper semicontinuity of the solution mappings of the mixed tensor variational inequalities with the help of some structured tensors.Finally,We show that the mixed tensor variational inequality has a stable solution with the involved tensor being strictly positive definite...