Studies On Tensor Problems

The tensor problem is widely used in many fields of scientific research and engineering technology.Based on the existing research results,this thesis carries on the detailed theoretical analysis and the algorithm research to the tensor absolute value equation and the tensor splitting problem.The main contents of this paper are as follows:In Chapter 1,we mainly introduce the basic knowledge of tensor absolute value equa-tion and tensor splitting problem,and the research situation of tensor problem at home and abroad.In addition,it also introduces some basic knowledge required in this paper.This paper starts from the perspective of numerical calculation,combining numerical algebra and optimization methods,mainly the Levenberg-Marquardt(LM)method,to study the tensor absolute value equation and tensor splitting problem and the conver-gence of the designed algorithm is also proved.Experiments verify the effectiveness of the proposed algorithm.In Chapter 2,firstly,tensor absolute value equation is reconstructed into tensor complementarity problem by introducing complementary function,and its key properties are analyzed.For the constructed tensor complementary problem,an adaptive inexact Levenberg-Marquardt(LM)algorithm is designed to prove that convergence of the algo-rithm.In addition,a continuous-time neural network algorithm(CTNN)is proposed to solve a class of tensor complementarity problems(A-I is a strong M tensor).Some subsequent numerical experiments are performed and reported.Finally,the experimental results show that the CTNN algorithm is more effective in solving a class of tensor absolute value equation.Some subsequent numerical experiments are performed and reported.In Chapter 3,based on the theoretical research in Chapter 2,the tensor absolute value equation is reconstructed as a tensor complementarity problem.And then,solving the system of equations H(x)=0 is actually a nonlinear equation.Based on the inexact LM algorithm,an accelerated two-step inexact LM algorithm is proposed.And an appropriate convex combination of function values in the previous iteration and the current iteration is proposed to obtain the inexact two-step LM algorithm to solve the tensor absolute value equation.From the numerical experiments,we can see that the proposed algorithm in this chapter needs more iterations,but the time complexity of its single iteration is low,so the efficiency of the accelerated two-step LM algorithm is improved in general.In Chapter 4,firstly,the preliminary knowledge and properties of projection are introduced.The tensor splitting problem(TSFP)is transformed into a constrained op-timization problem by introducing a projection function,and the key properties are ana-lyzed.Based on these analysis,we design a class Levenberg-Marquardt(LM)algorithm to solve the tensor splitting problem.Then,we further propose the modified trust region LM method(MLM)and the convergence of the algorithm is proved.Finally,some numerical experiments are performed.We can see that the algorithm MLM experimental results are better.In Chapter 5,the research work of this thesis is summarized,and put forward the idea of future research work and the problem to be solved.