First Order Methods for Geometric Optimization of Crystal Structures

TL;DR

This paper evaluates first order optimization methods for crystal structure relaxation, introducing dynamic step size rules, and providing algorithmic recipes based on experimental trade-offs, supported by open-source software.

Contribution

It systematically compares popular first order methods with static and dynamic step sizes in crystal structure optimization, offering new insights and practical algorithms.

Findings

Dynamic step sizes improve convergence performance.

Trade-offs exist between convergence speed and success probability.

Open-source Python software facilitates further research.

Abstract

The geometric optimization of crystal structures is a procedure widely used in Chemistry that changes the geometrical placement of the particles inside a structure. It is called structural relaxation and constitutes a local minimization problem with a non-convex objective function whose domain complexity increases according to the number of particles involved. In this work we study the performance of the two most popular first order optimization methods in structural relaxation. Although frequently employed, there is a lack of their study in this context from an algorithmic point of view. We run each algorithm in combination with a constant step size, which provides a benchmark for the methods' analysis and direct comparison. We also design dynamic step size rules and study how these improve the two algorithms' performance. Our results show that there is a trade-off between convergence rate and the possibility of an experiment to succeed, hence we construct a function to assign utility to each method based on our respective preference. The function is built according to a recently introduced model of preference indication concerning algorithms with deadline and their run time. Finally, building on all our insights from the experimental results, we provide algorithmic recipes that best correspond to each of the presented preferences and select one recipe as the optimal for equally weighted preferences. Alongside our results we present our open source Python software veltiCRYS, which was used to perform the geometric optimization experiments. Our implementation, can be easily edited to accommodate other energy functions and is especially targeted for testing different methods in structural relaxation.