A Local Minimax-Newton Method for Finding Multiple Saddle Points with Symmetries

In this paper, a local minimax-Newton method is developed to solve for multiple saddle points. The local minimax method [15] is used to locate an initial guess and a version of the generalized Newton method is used to speed up convergence. When a problem possesses a symmetry, the local minimax method is invariant to the symmetry. Thus the symmetry can be used to greatly enhance the efficiency and stability of the local minimax method. But such an invariance is sensitive to numerical error and the Haar projection has been used to enforce the symmetry [27]. In this paper, we prove that the Newton method is invariant to symmetries and that such an invariance is insensitive to numerical error. When a symmetric degeneracy takes place, it is proved that the Newton direction can be easily solved in an invariant subspace. Thus the Newton method can be used not only to speed up convergence but also to avoid using the Haar projection if the symmetric degeneracy is removable by a discretization. Finally, numerical examples are presented to illustrate the theory.