Theoretical Analysis Of Tensor Variational Inequalities Problems

This paper mainly proposes a class of rectangular tensor variational inequality problems that are closely related to complementary problems and polynomial optimization problems.Using the concept of the exceptionally family of elements and correlation topological degree theory,the nonemptiness and compactness of the solution set for rectangular tensor variational inequalities is obtained under the condition that the related rectangular tensor is positive semi-definite in its feasible domian.In addition,some new properties and conclusions of tensor orthogonal similarity are given.The variational inequalities problem is an important research topic in the field of nonlinear analysis and optimization.It is widely used in many fields.For example,the optimal control in the problem of non-uniform heterogeneous plate balance and the economic equilibrium model can be transformed into the variational inequalities problem.When the feasible domain of the variational inequalities is a point closed convex cone,the variational inequalities problem is a complementary problem that plays an important role in nonlinear programming.In 2017,Wang Y.et al.first proposed the tensor variational inequalities problem,which is a special kind of nonlinear variational inequalities problem.For the square tensor that is positive definite on the nonempty closed convex set,Wang Y.et al.give the existence theorem of the tensor variational inequalities problem.Like a rectangular matrix,a rectangular tensor is a more general tensor than a square tensor,which is more widely used.For example,the standard biquadratic optimization model extracted in the classification portfolio investment model is a special form of the rectangular tensor variational inequalities problem.Due to the existence of two independent decision vectors in the rectangular tensor variational inequality problem,the existence of the solution is more difficult.At present,there are few related literatures on the problem of rectangular tensor variational inequality.In view of the blank points existing in the above existing research work,this paper theoretically analyzes the related properties of a class of rectangular tensor variational inequalities.The structure of this paper is as follows: Firstly,the research overview of the variational inequalities problem and the tensor variational inequality problem is reviewed.Secondly,the related knowledge of the exceptionally family of elements is introduced,and the related concepts and properties of the eigenvalue and eigenvector of the tensor are summarized respectively.Thirdly,the topological properties of the solution of square tensor variational inequalities are introduced.The non-empty compactness of the solution set of a class of rectangular tensor variation is given.This is the main content of this paper.Finally,a new definition of orthogonal matrix is given,and the correlation properties of the orthogonal similarity of tensors are obtained.