Connectedness Of The Solution Set Of The Tensor Complementarity Problem

In today's big data era,the data of many problems is stored as tensor,so tensor and related problems have become a hot research topic both at home and abroad.In recent years,a class of complementarity problem defined by tensor,called the tensor complementarity problem,has gained a lot of research.Tensor complementarity problem is a subclass of general nonlinear complementarity problem,and it is also a generalization of linear complementarity problem.The connectedness of the solution set is an important research direction for linear complementary problem and nonlinear complementary problem.Although the tensor complementarity problem has obtained many results in theory,the connectedness of its solution set has not been studied.This paper aims to study the connectedness of the solution set of tensor complementarity problem.This paper first proves that the solution set of the tensor complementarity problem is a semi-algebraic set,and then the connectedness of its solution set is equivalent to the path connectedness of its solution set.With the help of this equivalence,we use the analytical technique of path connectedness to study the connectedness of the solution set of the tensor complementarity problem.Using the structure of the tensor and the characteristics of the polynomial,we first obtain a necessary condition that: if the solution set of the tensor complementarity problem is connected,the underlying tensor must be a semi-positive tensor;after that we propose a sufficient condition for the connectedness of the solution set of the tensor complementarity problem,in which we make certain restrictions on the sub-tensors of the tensor involved in the tensor complementarity problem.In addition,we also verify all conclusions through some specific examples.The results obtained in this paper are the extension of the corresponding conclusions in the linear complementarity problem and the improvement of the corresponding conclusions in the nonlinear complementarity problem.The research work done in this paper enriches the theoretical research system of the tensor complementarity problem.