  The input of most clustering algorithms is a symmetric matrix quantifying
similarity within data pairs. Such a matrix is here turned into a quadratic set
function measuring cluster score or similarity within data subsets larger than
pairs. In general, any set function reasonably assigning a cluster score to
data subsets gives rise to an objective function-based clustering problem. When
considered in pseudo-Boolean form, cluster score enables to evaluate fuzzy
clusters through multilinear extension MLE, while the global score of fuzzy
clusterings simply is the sum over constituents fuzzy clusters of their MLE
score. This is shown to be no greater than the global score of hard clusterings
or partitions of the data set, thereby expanding a known result on extremizers
of pseudo-Boolean functions. Yet, a multilinear objective function allows to
search for optimality in the interior of the hypercube. The proposed method
only requires a fuzzy clustering as initial candidate solution, for the
appropriate number of clusters is implicitly extracted from the given data set.
