The problem of packing equal circles in a circle is a classic and famous packing problem, which is well-studied in academia and has a variety of applications in industry. This problem is computationally challenging, and researchers mainly focus on small-scale instances with the number of circular items n less than 320 in the literature. In this work, we aim to solve this problem on large scale. Specifically, we propose a novel geometric batch optimization method that not only can significantly speed up the convergence process of continuous optimization but also reduce the memory requirement during the program's runtime. Then we propose a heuristic search method, called solution-space exploring and descent, that can discover a feasible solution efficiently on large scale. Besides, we propose an adaptive neighbor object maintenance method to maintain the neighbor structure applied in the continuous optimization process. In this way, we can find high-quality solutions on large scale instances within reasonable computational times. Extensive experiments on the benchmark instances sampled from n = 300 to 1,000 show that our proposed algorithm outperforms the state-of-the-art algorithms and performs excellently on large scale instances. In particular, our algorithm found 10 improved solutions out of the 21 well-studied moderate scale instances and 95 improved solutions out of the 101 sampled large scale instances. Furthermore, our geometric batch optimization, heuristic search, and adaptive maintenance methods are general and can be adapted to other packing and continuous optimization problems.

翻译：等圆填充问题是一个经典且著名的几何填充问题，在学术界已有深入研究，并在工业领域具有广泛应用。该问题在计算上极具挑战性，现有文献主要关注圆形物品数量n小于320的小规模实例。本研究旨在求解大规模等圆填充问题。具体而言，我们提出了一种新颖的几何批处理优化方法，该方法不仅能显著加速连续优化过程的收敛速度，还能降低程序运行时的内存需求。随后，我们提出了一种称为解空间探索与下降的启发式搜索方法，能够在大规模问题中高效发现可行解。此外，我们提出了一种自适应邻域对象维护方法，用于维护连续优化过程中应用的邻域结构。通过这种方式，我们可以在合理的计算时间内为大规模实例找到高质量解。在n=300至1,000的基准实例上开展的广泛实验表明，我们提出的算法优于现有最优算法，并在大规模实例上表现出色。特别地，该算法在21个经典中等规模实例中发现了10个改进解，在101个采样大规模实例中发现了95个改进解。此外，我们的几何批处理优化、启发式搜索和自适应维护方法具有通用性，可适用于其他填充问题和连续优化问题。