  We present a general method, based on conjugate duality, for solving a convex
minimization problem without assuming unnecessary topological restrictions on
the constraint set. It leads to dual equalities and characterizations of the
minimizers without constraint qualification. As an example of application, the
Monge-Kantorovich optimal transport problem is solved in great detail. In
particular, the optimal transport plans are characterized without restriction.
This characterization improves the already existing literature on the subject.
