  We investigate the continuous optimal transport problem 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 regularization of the problem with so-called Young's functions,
which forces the optimal transport plan to be a function in the corresponding
Orlicz space rather than a Radon measure. We derive the predual problem and
show strong duality and existence of primal solutions to the regularized
problem. Existence of (pre-)dual solutions will be shown for the special case
of $L^p$ regularization for $p\geq2$. Then we derive four algorithms to solve
the dual problem of the quadratically regularized problem: A cyclic projection
method, a dual gradient decent, a simple fixed point method, and Nesterov's
accelerated gradient, all of which have a very low cost per iteration.
