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:在圆圈中包装平均圆圈的问题是一个经典和著名的包装问题, 这个问题在学术界研究过, 并具有各种工业应用。 这个问题在计算上具有挑战性, 研究人员主要关注小事件, 循环项目的数量在文献中小于320个。 在这项工作中, 我们的目标是大规模解决这个问题。 具体地说, 我们提出一种新的几何批次优化方法, 不仅可以大大加快连续优化的趋同过程, 还可以减少程序运行期间的记忆要求。 然后我们提出一种超常搜索方法, 叫做解决方案空间探索和下降, 可以大规模地找到可行的解决方案。 此外, 我们提出一个适应性邻居对象维护方法, 以维持在连续优化过程中应用的邻居结构。 这样, 我们就可以在合理的计算时间内大规模找到高质量的解决方案。 从 n= 300到 1 000 的抽样实验表明, 我们提议的算法超越了最先进的标准算法, 并在大范围内进行出色的测试。 特别是, 我们的算法在不断改进的地理优化后, 在21个深度的深度的深度的精确的精确度上, 找到了10个, 的精确的精确的精确的精确的精确的精确的调整方法 。</s>