First Order Methods for Geometric Optimization of Crystals: Experimental Analysis

A Tsili and MS Dyer and VV Gusev and P Krysta and R Savani, ADVANCED THEORY AND SIMULATIONS, 7 (2024).

DOI: 10.1002/adts.202400124

The geometric optimization of crystal structures is a procedure widely used in computational 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. This work studies the performance of the two most popular gradient methods in structural relaxation, Steepest Descent and Conjugate Gradient. Although frequently employed, there is a lack of their study in this context from an algorithmic point of view. The algorithms are initially benchmarked on the basis of a constant step size. Three concepts for designing dynamic step size rules are then examined in detail and analyzed. Results show that there is a trade-off between convergence rate and the possibility of an experiment to succeed. In order to address this, a function is proposed as a formal means for assigning utility to each method based on preference. The function is built according to a recently introduced model of preference indication concerning algorithms with deadline and their run time. It introduces the quantification of the optimization algorithms' performance according to convergence speed and success rate, thus enabling the appointment of a specific algorithmic recipe as the best choice for balanced preferences. This article benchmarks and compares Steepest Descent and Conjugate Gradient in the context of the geometric optimization of crystals. Step size scheduling rules are designed and tested experimentally using a custom implementation. Extensive analysis of the experiments is provided and each method is evaluated using a proposed metric that trades off optimization success rate and convergence time. image

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