Two Optimization Methods For Z-eigenvalues Of Symmetric Tensors

In recent years,tensor analysis has become the popular research direction in mathematical field.Tensor analysis can be regarded as Matrix extension in mathematical,including tensor decomposition and the theory and methods of tensor eigenvalues.In 1927,Hitchcock first proposed the concept of tensor decomposition,the most representative of which is CP decomposition and Tucker decomposition.Around 2005,eigenvalues of tensors and some properties of eigenvalues are given by Qi and Lim independently.There are varieties of forms,such as Z-eigenvalues,H-eigenvalues,E-eigenvalues and so on.In this paper,we mainly research the calculation of the Z-eigenvalues of symmetric tensors.On the one hand,the Z-eigenvalue problems of tensors can be converted to optimization problems.Subsequently,we present a projected gradient method to calculate the Z-eigenvalues of symmetric tensors.On the other hand,the Z-eigenvalues problems of tensors can be also converted to systems of nonlinear equations,and a Gauss-Newton algorithm is given to solve the Z-eigenvalues of symmetric tensors.Consequently,the thesis consists of three chapters of which the second and third chapters are the main parts.The rest of the paper is organized as follows:In the first chapter,the part of introduction mainly introduces the research background and research status of tensor eigenvalues,and gives the basic knowledge to be used in this paper.In the second chapter,based on the convex set theory,we present nonmonotone projected gradient method for solving Z-eigenvalues of symmetric tensors.Moreover,we utilize the nonmonotone linear search technology to speed up convergence,and the global convergence of the algorithm is established.In numerical experiments,comparing figures between visual images and numerical results,the numerical results show that the new algorithm is efficient both in low dimension and higherdimensional problems.In the third chapter,we transform Z-eigenvalues of symmetric tensors into system of equations.Based on merit function and direction of conjugate gradient,we come up with Gauss-Newton method for solving symmetric tensors.Moreover,the global convergence of the algorithm is established.The numerical results verified the responsibility and efficiency of the algorithm.