The Research Of Optimization Algorithms On Eigenvalues Of Symmetric Tensors

Tensors are widely used in high-order mathematical statistics,mathematical finance,biological computing,medical imaging,signal processing,nuclear magnetic resonance imaging and elasticity,etc.Many scholars have done a lot of meaningful work in tensor calculation,among which tensor eigenvalues calculation is an important research direction in this field.In this dissertation,we mainly study the computation of several kinds of largest(smallest)eigenvalues of tensors.On the basis of transforming these problems into optimization problems or nonlinear equations equivalently,considering the structural characteristics of each kind of specific problems,we propose adaptive trust region algorithm for solving B-eigenvalues of symmetric tensors;subspace trust region method and accelerated Levenberg-Marquardt method for solving generalized eigenvalue problems of symmetric tensors,respectively;and the accelerated spectral conjugate gradient method for Z-eigenvalue problems of symmetric tensors.The concrete contents and innovative achievements are as follows:Firstly,we transform the B-eigenvalue problem of symmetric tensors into a homogeneous polynomial optimization problem on unit hyper-sphere.Using projection idea and combining with adaptive technology,an adaptive trust region method(SATR)is proposed for finding the largest(smallest)B-eigenvalue of symmetric tensors.Global convergence of the proposed algorithm and second-order necessary conditions of the optimal solutions are established,respectively.Numerical experiments are listed to illustrate the efficiency of the proposed method.When the B-eigenvalue problem degenerates to Z-eigenvalue problem,the numerical comparison with the existing results shows that the SATR algorithm is more effective.Secondly,the generalized eigenvalue problem of symmetric tensors is transformed into a least squares problem,and a subspace trust region method(SSTR)is proposed.Its basic idea is to construct a low-dimensional subspace at each iteration,and to construct an approximate subproblem of the least squares problem in this low-dimensional subspace.Combining with the modified BFGS formula,a concise method for updating the subspace is proposed which can save a lot of computation and storage.The global convergence of the algorithm is proved.The numerical experiments show the effective of this algorithm.Thirdly,a new Levenberg-Marquardt(LM)method is proposed by using the special structure of nonlinear equations which transformed from the generalized eigenvalue problem of symmetric tensors.Using a non-monotonic technique to relax the LM parameters,the proposed algorithm is a non-monotonic accelerated LM algorithm.This algorithm has global convergence and local third-order convergence rate.The numerical results show that the algorithm is effective.Fourthly,based on the variational principle of Z-eigenvalue of symmetric tensors,the Z-eigenvalue problem of symmetric tensors is transformed into an unconstrained optimization problem.Based on conjugate gradient direction and Newton direction,an accelerated spectral conjugate gradient method for solving Z-eigenvalue problem of symmetric tensor is proposed.The global convergence of this algorithm is proved.In numerical experiments,the new conjugate gradient methods proposed in this chapter are compared with classical conjugate gradient methods.The results show that the algorithm in this chapter is more effective.