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 an exact 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.
翻译:最小平方组合(MSSC)或k-bours 类型组合(K-bours tym)最近得到扩展,以利用关于每个组群基本内容的先前知识。这种知识用于提高绩效和解决方案质量。在本文件中,我们建议了一种基于分支和切割技术的精确方法来解决受限制的基本组合(MSSC)。对于较低的约束常规,我们使用鲁杰拉帕伊博翁等人最近提议的半无限制编程(SDP)放松[SIAM J.optim. 29(2), 1211-1239, (2019 )]。然而,这种放松只能在小的案例中使用分支和切割方法。因此,我们提出了一个新的SDP宽松方法,根据实例大小和组群数量来更好地制定尺度。在这两种情况下,我们通过增加多面削减来强化约束。我们从执行双向约束的定制分支战略中受益,我们减少了儿童节点中出现的问题的复杂性。相反,我们提出了一种本地搜索程序,即利用全球10度稳定方程的解决方案的解决方案的更大程度,通过每10度来展示真正的解算。