The classical duality theory of Kantorovich and Kellerer for the classical optimal transport is generalized to an abstract framework and a characterization of the dual elements is provided. This abstract generalization is set in a Banach lattice %DIFDELCMD < \cal %%% X with a unit order . The primal problem is given as the supremum over a convex subset of the positive unit sphere of the topological dual of %DIFDELCMD < \cal %%% X and the dual problem is defined on the bidual of%DIFDELCMD < \cal %%% X . These results are then applied to several extensions of the classical optimal transport . In particular, an alternate proof of Kellerer's result is given without using the Choquet Theorem .
The classical duality theory of Kantorovich and Kellerer for the classical optimal transport is generalized to an abstract framework and a characterization of the dual elements is provided. This abstract generalization is set in a Banach lattice %DIFDELCMD < \cal %%% with a order unit . The primal problem is given as the supremum over a convex subset of the positive unit sphere of the topological dual of %DIFDELCMD < \cal %%% and the dual problem is defined on the %DIFDELCMD < \cal %%% bi-dual of . These results are then applied to several extensions of the classical optimal transport .
R
X with a unit order
with a order unit
D
X
D
bidual of
R
X
bi-dual of
D
. In particular, an alternate proof of Kellerer's result is given without using the Choquet Theorem