An accelerated conjugate gradient method for the Z-eigenvalues of symmetric tensors

We transform the Z-eigenvalues of symmetric tensors into unconstrained optimization problems with a shifted parameter. An accelerated conjugate gradient method is proposed for solving these unconstrained optimization problems. If solving problem results in a nonzero critical point, then it is a Z-eigenvector corresponding to the Z-eigenvalue. Otherwise, we solve the shifted problem to find a Z-eigenvalue. In our method, the new conjugate gradient parameter is a modified CD conjugate gradient parameter, and an accelerated parameter is presented by using the quasi-Newton direction. The global convergence of new method is proved. Numerical experiments are listed to illustrate the efficiency of the proposed method.