  We investigate the problem of optimal transport in the so-called Kantorovich
form, i.e. given two Radon measures on two compact sets, we seek an optimal
transport plan which is another Radon measure on the product of the sets that
has these two measures as marginals and minimizes a certain cost function. We
consider quadratic regularization of the problem, which forces the optimal
transport plan to be a square integrable function rather than a Radon measure.
We derive the dual problem and show strong duality and existence of primal and
dual solutions to the regularized problem. Then we derive two algorithms to
solve the dual problem of the regularized problem: A Gauss-Seidel method and a
semismooth quasi-Newton method and investigate both methods numerically. Our
experiments show that the methods perform well even for small regularization
parameters. Quadratic regularization is of interest since the resulting optimal
transport plans are sparse, i.e. they have a small support (which is not the
case for the often used entropic regularization where the optimal transport
plan always has full measure).

Optimization and Control;Numerical Analysis;Numerical Analysis