GENERALIZED GENTLEST ASCENT DYNAMICS METHODS FOR HIGH-INDEX SADDLE POINTS

A geometric perspective on the gentlest ascent dynamics is presented, revealing that the dynamics is utilizing the Householder reflector___constructed via the continuous power method-to adapt the negative gradient and identify index-1 saddle points. While the adaptation appears intuitive, it is governed by a precise criterion. Building on this geometric insight, three generalized dynamical systems are introduced for locating high-index saddle points, each centered on estimating directions for constructing generalized reflectors. The first approach employs the Oja flow to evolve eigenspaces, encompassing the continuous power method as a special case. The second approach formulates a matrix Riccati differential equation for the projector operator on the Grassmann manifold, which is shown to be equivalent to a double bracket flow with inherent sorting properties. The third approach is a hybrid method based on conventional subspace iteration, incorporating QR factorization for normalization. The equilibrium points of all three systems are classified, and convergence analyses are provided. These dynamical systems are readily solvable by using high-precision numerical ODE integrators. Numerical experiments confirm the theoretical results.