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The problem of packing equal circles in a circle is a classic and famouspacking problem, which is well-studied in academia and has a variety ofapplications in industry. This problem is computationally challenging, andresearchers mainly focus on small-scale instances with the number of circularitems n less than 320 in the literature. In this work, we aim to solve thisproblem on large scale. Specifically, we propose a novel geometric batchoptimization method that not only can significantly speed up the convergenceprocess of continuous optimization but also reduce the memory requirementduring the program's runtime. Then we propose a heuristic search method, calledsolution-space exploring and descent, that can discover a feasible solutionefficiently on large scale. Besides, we propose an adaptive neighbor objectmaintenance method to maintain the neighbor structure applied in the continuousoptimization process. In this way, we can find high-quality solutions on largescale instances within reasonable computational times. Extensive experiments onthe benchmark instances sampled from n = 300 to 1,000 show that our proposedalgorithm outperforms the state-of-the-art algorithms and performs excellentlyon large scale instances. In particular, our algorithm found 10 improvedsolutions out of the 21 well-studied moderate scale instances and 95 improvedsolutions out of the 101 sampled large scale instances. Furthermore, ourgeometric batch optimization, heuristic search, and adaptive maintenancemethods are general and can be adapted to other packing and continuousoptimization problems.

Geometric Batch Optimization for the Packing Equal Circles in a Circle Problem on Large Scale