A Derivative-Free Saddle-Search Algorithm with Linear Convergence Rate

We propose a derivative-free saddle-search algorithm designed to locate transition states using only function evaluations. The algorithm employs a nested architecture consisting of an inner eigenvector search and an outer saddle-point search. Through rigorous numerical analysis, we prove the almost sure convergence of the inner step under suitable assumptions. Furthermore, we establish the convergence of the outer search using a decaying step size while demonstrating linear convergence under constant step size and boundedness conditions. Numerical experiments are provided to validate our theoretical results and demonstrate the algorithm’s practical applicability. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://github.com/BaomingShi/Derivative-free-saddle-search-algorithm and in the supplementary materials ( Derivative-free-saddle-search-algorithm-main.zip [6.05KB]). [Formula: see text]