Hybrid Newton-BFGS high-index saddle dynamics for accelerating saddle-point searches

Abstract

We propose a hybrid Newton-BFGS high-index saddle dynamics (HiSD) method for accelerating saddle-point searches. The method constructs a Newton correction on the unstable subspace from the available negative eigenpairs, while BFGS updates based on projected secant information approximate the inverse Hessian on the stable subspace. By combining Newton and quasi-Newton corrections, the method incorporates curvature information in both subspaces while preserving the ascent-descent structure of HiSD. Accounting for the evolving stable subspaces, we establish local $Q$-superlinear convergence of the full-memory version near nondegenerate saddle points under suitable assumptions. We further develop a limited-memory version for large-scale problems and provide fixed-step and line search strategies. Numerical experiments show local convergence behavior consistent with the theory and demonstrate acceleration on ill-conditioned problems and degenerate saddle manifolds in neural-network loss landscapes. The method also supports efficient upward and downward saddle searches for a liquid crystal model, further illustrating its broad applicability to saddle-point computation.