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.

This paper presents a novel nonsmooth objective penalty function for inequality constrained optimization problems. A modified flattened aggregate function, which is a smooth approximation of the max-value function, is discussed. Then, the smooth objective penalty function that contains the flattened aggregate function is proposed, and the exactness of the new function is studied. Based on this, an objective penalty function algorithm is proposed and its convergence is proven under mild conditions. Because of the flattened aggregate function, the gradient computation can usually be greatly reduced for problems with many constraints. Numerical experiments are included to illustrate the efficiency of the new algorithm through a series of numerical examples, especially for solving problems with many constraints.