An improved column generation algorithm for minimum sum-of-squares clustering

Given a set of entities associated with points in Euclidean space, minimum sum-of-squares clustering (MSSC) consists in partitioning this set into clusters such that the sum of squared distances from each point to the centroid of its cluster is minimized. A column generation algorithm for MSSC was given by du Merle et al. in SIAM Journal Scientific Computing 21:1485-1505. The bottleneck of that algorithm is the resolution of the auxiliary problem of finding a column with negative reduced cost. We propose a new way to solve this auxiliary problem based on geometric arguments. This greatly improves the efficiency of the whole algorithm and leads to exact solution of instances with over 2,300 entities, i.e., more than 10 times as much as previously done.

给定一组与欧几里得空间中的点相关联的实体，最小平方和聚类（MSSC）在于将该集合划分为多个聚类，使得从每个点到其所属聚类的质心的平方距离之和最小化。杜·梅尔勒等人在《工业与应用数学学会科学计算杂志》21卷1485 - 1505页给出了一种用于MSSC的列生成算法。该算法的瓶颈在于解决寻找具有负简约成本的列这一辅助问题。我们基于几何原理提出了一种解决此辅助问题的新方法。这极大地提高了整个算法的效率，并能够精确求解具有2300多个实体的实例，即比之前所能处理的数量多10倍以上。