  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.
