论文标题
通过半决赛编程的全局优化基数受限的最低平方总和聚类
Global Optimization for Cardinality-constrained Minimum Sum-of-Squares Clustering via Semidefinite Programming
论文作者
论文摘要
最近已经扩展了最小平方群集（MSSC）或K-均值类型聚类，以利用有关每个群集的基数的先验知识。这种知识用于提高性能以及解决方案质量。在本文中，我们提出了一种基于分支和切割技术的全局优化方法，以解决基数受限的MSSC。对于下边界的例程，我们使用Rujeerapaiboon等人最近提出的半决赛编程（SDP）放松。 [Siam J. Optim。 29（2），1211-1239，（2019）]。但是，这种放松只能用于小型实例中的分支和切割方法。因此，我们得出了一种新的SDP松弛，该松弛随着实例大小和簇的数量而缩放得更好。在这两种情况下，我们都通过添加多面体切割来增强结合。从量身定制的分支策略中受益，该策略可以实施成对的约束，我们减少了儿童节点中出现的问题的复杂性。相反，对于上限，我们提出了一个本地搜索过程，该过程利用了在每个节点上解决的SDP松弛解决方案。计算结果表明，所提出的算法在全球范围内首次求解了大小的实际实例，其大小比通过最新精确方法求解的算法大10倍。
The minimum sum-of-squares clustering (MSSC), or k-means type clustering, has been recently extended to exploit prior knowledge on the cardinality of each cluster. Such knowledge is used to increase performance as well as solution quality. In this paper, we propose a global optimization approach based on the branch-and-cut technique to solve the cardinality-constrained MSSC. For the lower bound routine, we use the semidefinite programming (SDP) relaxation recently proposed by Rujeerapaiboon et al. [SIAM J. Optim. 29(2), 1211-1239, (2019)]. However, this relaxation can be used in a branch-and-cut method only for small-size instances. Therefore, we derive a new SDP relaxation that scales better with the instance size and the number of clusters. In both cases, we strengthen the bound by adding polyhedral cuts. Benefiting from a tailored branching strategy which enforces pairwise constraints, we reduce the complexity of the problems arising in the children nodes. For the upper bound, instead, we present a local search procedure that exploits the solution of the SDP relaxation solved at each node. Computational results show that the proposed algorithm globally solves, for the first time, real-world instances of size 10 times larger than those solved by state-of-the-art exact methods.
