A heuristic algorithm with multi-scale perturbations for point arrangement and equal circle packing in a convex container

The point arrangement and equal circle packing problems are a category of classic max–min constrained optimization problems with many important applications. Being computationally very challenging to solve, they have been widely studied in operations research and mathematics. We propose a heuristic algorithm for the point arrangement and equal circle packing problems in various convex containers. The algorithm relies on several complementary search components, including an unconstrained optimization procedure that ensures diversified and intensified searches, an optima exploitation based adjustment method for the radius of circles, and a monotonic basin-hopping method with multi-scale perturbations. Computational results on numerous benchmark instances show that the proposed algorithm significantly outperforms the existing state-of-the-art algorithms, especially for hard instances or large-scale instances. For the well-known equal circle packing problem in a circular container, it improves the best-known result for 69 out of the 96 hardest instances widely used in the literature. For the majority of the remaining instances tested, the algorithm improves or matches the best-known results with a high success rate, despite of the fact that these instances have been tested by many existing algorithms. Experimental analysis shows that the optima exploitation based adjustment method for the radius of circles plays a crucial role for the high performance of the algorithm and that the multi-scale perturbations are able to significantly enhance the search ability and robustness of the algorithm. Given the general feature of the proposed framework, it can be applied to other related max–min constrained optimization problems.