A Second-Order Method for Locating Critical Points of Prescribed Index on Energy Landscapes

RESUMO

Exploring high dimensional energy landscape is a challenging problem in many applications where both ground states and transition states offer important information about the underlying physical systems. To facilitate the computational exploration, we propose a cubic-regularized second-order method for locating critical points of prescribed index, including both energy minima (with an index $0$) and transition states (saddle points with positive index). At each step of the iteration, the proposed method involves several ingredients, such as updating the eigen-directions of the Hessian according to the prescribed index, reflecting the gradient and the Hessian along selected unstable eigen-directions, if any, and constructing the next step by solving a cubic-regularized subproblem. Under suitable assumptions, we show that, when the eigenspace and the cubic subproblem are solved exactly, the iteration converges locally and quadratically to a critical point of the prescribed index. We further establish convergence results in the inexact setting, where both the eigenspace computation and the solution of the cubic subproblem are performed approximately. We show that the resulting iteration retains local linear convergence, with the convergence rate depending on the accuracy of the inexact computations. Moreover, we prove that the fixed points of the algorithm are precisely critical points of the prescribed index. In addition, we introduce an adaptive strategy for updating the cubic regularization parameter to improve robustness and computational efficiency of the proposed algorithm. Numerical experiments confirm the predicted convergence behavior, demonstrate the effectiveness of the adaptive strategy, and illustrate the capability of the proposed method for locating critical points and constructing solution landscapes.