  We introduce and prove convergence of a damped Newton algorithm to
approximate solutions of the semi-discrete optimal transport problem with
storage fees, corresponding to a problem with hard capacity constraints. This
is a variant of the optimal transport problem arising in queue penalization
problems, and has applications to data clustering. Our result is novel as it is
the first numerical method with proven convergence for this variant problem;
additionally the algorithm applies to the classical semi-discrete optimal
transport problem but does not require any connectedness assumptions on the
support of the source measure, in contrast with existing results. Furthermore
we find some stability results of the associated Laguerre cells. All of our
results come with quantitative rates. We also present some numerical examples.
