Estimations Of Pareto-eigenvalues For Higher-order Tensors

This thesis focuses on the tensor eigenvalue complementarity problems' estimations and theoretical analysis.Based on the equivalent transfer between tensor eigenvalue complementary problems and tensor eigenvalue problems,the author proposed some estimations and theoretical analysis of Pareto-eigenvalue for high order tensors.Complementarity problems are important problems which have close connection with optimization problems.Eigenvalue complementarity problems of matrix are special formation of complementarity problems.A class of differential inclusion problems determined by the linear complementary problem can be converted into eigenvalue complementarity problems of matrix to study and solve.Eigenvalue complementarity problems of tensors which are more general than eigenvalue problems have closely connection with a class of nonlinear differential inclusion problems.Since the homogeneity and nonlinear,the solving of largest Pareto-eigenvalue of tensors is an NP-hard problem in general,i.e.,there are no polynomial-time for solving the largest Pareto-eigenvalue of TGEiCP.In this way,there is a need for further study of methods for these problems.One method is to convert them into eigenvalue problems,then we can use estimating methods to solve these problems caused by transfer.Aiming at some special tensors,such as M-tensor,Ztensor,nonnegative irreducible tensor and other special tensors,the author proposed some approximate estimations of the largest Pareto-eigenvalues of TGEiCP,and also proposed some properties analysis of Pareto-eigenvalues for special tensors.The main contents are as follows: First,review the development situations of complementarity problems,eigenvalue complementarity problems of matrix and eigenvalue complementarity problems of tensor.Secondly,the analysis from the eigenvalue complementarity problems of high-order tensor to eigenvalue problems,and introduces the related symbols and basic concepts.Again,this thesis presents the important elements:estimations of Pareto-eigenvalue for some special tensors.Finally,analyze the properties of Pareto-eigenvalue for some special tensors,such as M-tensor,Z-tensor,monotone tensor.