Research On The Structure Tensor And The Tensor Complementarity Problem

With the deepening of study, tensor theory system is increasingly perfect, more and more structure tensor was found and put forward. Some special properties and useful conclusions of the structure tensor are gradually being established. They also have a wide range of application in stochastic processes, data processing and Markov chains. There are close relations between the tensor complementarity problem and the mathematical programming, the variational inequality, the fixed point problem,the generalized equation and the strategy theory. Many theories of nonlinear analysis and topology are used in its study. After the tensor complementarity problem is proposed, it is widely used in mechanics, engineering, economy, transportation, and many other practical departments. This makes the tensor complementarity problem become a very popular research topic in mathematical programming.This paper is divided into four chapters. The first chapter, we introduce some basic knowledge about the tensor, tensor complementarity problems and some basic concepts in the paper.In second chapter, we mainly study the positive definiteness, positive semidefiniteness and copositivity of the generalized Cauchy tensor, and discuss the relationship between generalized Cauchy tensor and doubly nonnegative tensor , S tensor and S0 tensor, including positive and negative conditions of generalized Cauchy tensor's H-eigenvalue and Z-eigenvalue. Finally, the application of the generalized Cauchy tensor in the tensor complementarity problem is studied.In third chapter, we mainly study the tensor stochastic complementarity problems. Tensor stochastic complementarity problems is put forward based on the matrix stochastic complementarity problems. Under certain assumptions, the tensor stochastic complementarity problems of uncertain factors is transformed into the determine tensor complementarity problems model, and then we put forward smoothing Newton algorithm for solving the tensor stochastic complementarity problems. The feasibility of the algorithm is verified by numerical experiment.In forth chapter, we mainly study a kind of special tensor complementarity problem, namely tensor eigenvalue complementarity problem. By introducing a smoothing function, tensor eigenvalue complementarity problem is transformed into a set of nonlinear equation system, and then we put forward the smoothing Newton algorithm for solving the tensor eigenvalue complementarity problems. Finally, we proved the feasibility of the smoothing Newton algorithm,and effectiveness of the proposed algorithm is verified through numerical examples.Finally, the work of this paper is summarized and prospected.