A Derivative-Free Saddle-search Algorithm With Linear Convergence Rate

TL;DR

This paper introduces a derivative-free saddle-search algorithm that guarantees linear convergence to transition states using only function evaluations, validated by numerical experiments.

Contribution

It presents a novel nested saddle-search algorithm with proven convergence properties, including linear convergence under specific conditions.

Findings

Proven almost sure convergence of the inner eigenvector search.

Established convergence of the outer saddle-point search with decaying step size.

Demonstrated linear convergence with constant step size through numerical experiments.

Abstract

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.