  Capacity constrained optimal transport is a variant of optimal transport,
which adds extra constraints on the set of feasible couplings in the original
optimal transport problem to limit the mass transported between each pair of
source and sink. Based on this setting, constrained optimal transport has
numerous applications, e.g., finance, network flow. However, due to the large
number of constraints in this problem, existing algorithms are both
time-consuming and space-consuming. In this paper, inspired by entropic
regularization for the classical optimal transport problem, we introduce a
novel regularization term for capacity constrained optimal transport. The
regularized problem naturally satisfies the capacity constraints and
consequently makes it possible to analyze the duality. Unlike the matrix-vector
multiplication in the alternate iteration scheme for solving classical optimal
transport, in our algorithm, each alternate iteration step is to solve several
single-variable equations. Fortunately, we find that each of these equations
corresponds to a single-variable monotonic function, and we convert solving
these equations into finding the unique zero point of each single-variable
monotonic function with Newton's method. Extensive numerical experiments
demonstrate that our proposed method has a significant advantage in terms of
accuracy, efficiency, and memory consumption compared with existing methods.
