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：等圆填充问题是经典且著名的填充问题，在学术界被广泛研究，并在工业领域具有多种应用。该问题计算难度大，现有文献主要聚焦于圆形物品数量n小于320的小规模实例。本文旨在解决大规模实例上的该问题。具体而言，我们提出了一种新颖的几何批处理方法，不仅能显著加速连续优化过程的收敛速度，还能减少程序运行时的内存需求。随后，我们提出了一种启发式搜索方法——解空间探索与下降法，该方法能在大规模实例上高效发现可行解。此外，我们还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，我们能够在合理时间内找到大规模实例的高质量解。在n=300至1000的基准实例上进行的大量实验表明，我们的算法优于现有最优算法，并在大规模实例上表现卓越。特别地，在21个经过充分研究的中等规模实例中，我们的算法找到了10个改进解；在101个采样的大规模实例中，找到了95个改进解。此外，我们所提出的几何批处理优化、启发式搜索及自适应维护方法具有通用性，可适用于其他填充及连续优化问题。