The geometric optimisation 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 along with the number of particles involved. In this work we study the performance of the two most popular first order optimisation methods, Gradient Descent and Conjugate Gradient, in structural relaxation. The respective pseudocodes can be found in Section 6. Although frequently employed, there is a lack of their study in this context from an algorithmic point of view. In order to accurately define the problem, we provide a thorough derivation of all necessary formulae related to the crystal structure energy function and the function's differentiation. 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.

翻译：晶体结构的几何优化是化学领域广泛使用的一种方法，用于改变结构内粒子的几何排布。该过程称为结构弛豫，是一个具有非凸目标函数的局部最小化问题，其定义域的复杂性随涉及粒子数量的增加而增加。本研究评估了两种最流行的一阶优化方法——梯度下降法和共轭梯度法——在结构弛豫中的性能表现，相应伪代码见第6节。尽管这些方法被频繁使用，但缺乏从算法角度对其在该语境下的系统研究。为精确定义该问题，我们详细推导了与晶体结构能量函数及其微分相关的所有必要公式。我们采用恒定步长运行每种算法，为方法分析与直接比较建立基准。此外，我们设计了动态步长规则，并研究其如何提升两种算法的性能。结果表明收敛速度与实验成功可能性之间存在权衡，因此我们构建了一个基于各自偏好的效用函数来评估每种方法。该函数依据最近提出的关于带截止时间算法及其运行时间的偏好指示模型构建。最后，基于实验成果的洞察，我们提供了与每种呈现偏好最佳匹配的算法方案，并为等权重偏好选择最优方案。