Fundamental qualitative properties of the minimum sum-of-squares clustering problem are established in this paper. We prove that the problem always has a global solution and, under a mild condition, the global solution set is finite. Moreover, the components of each global solution can be computed by an explicit formula. Based on a new concept of non-trivial local solution, we get necessary conditions for a system of centroids to be such a local solution. Interestingly, these necessary conditions are also sufficient ones. Finally, it is proved that the optimal value function is locally Lipschitz, the global solution map is locally upper Lipschitz, and the local solution map has the Aubin property, provided that the original data points are distinct. The obtained complete characterizations of the non-trivial local solutions allow one to understand better the performance of not only the k-means algorithm, but also of other solution methods for the problem in question.

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最小平方和聚类问题的定性性质

本文建立了最小平方和聚类问题的基本定性性质。我们证明问题总是有全局解，并且在温和条件下，全局解集是有限的。此外，每个全局解的分量可以通过显式公式计算。基于非平凡局部解的新概念，我们得到了质心系统成为这种局部解的必要条件。有趣的是，这些必要条件也是充分条件。最后证明最优值函数为局部Lipschitz，全局解图为局部上Lipschitz，局部解图具有Aubin性质，前提是原始数据点不同。