  Optimal mass transport, also known as the earth mover's problem, is an
optimization problem with important applications in various disciplines,
including economics, probability theory, fluid dynamics, cosmology and
geophysics to cite a few. Optimal transport has also found successful
applications in image registration, content-based image retrieval, and more
generally in pattern recognition and machine learning as a way to measure
dissimilarity among data. This paper introduces the problem of constrained
optimal transport. The time-dependent formulation, more precisely, the fluid
dynamics approach is used as a starting point from which the constrained
problem is defined by imposing a soft constraint on the density and momentum
fields or restricting them to a subset of curves that satisfy some prescribed
conditions. A family of algorithms is introduced to solve a class of
constrained saddle point problems, which has convexly constrained optimal
transport on closed convex subsets of the Euclidean space as a special case.
Convergence proofs and numerical results are presented.
