Strong duality of the Monge-Kantorovich mass transfer problem in metric spaces

Abstract

This paper studies the Monge-Kantorovich mass transfer (MT) problem on metric spaces and with an unbounded cost function. Conditions are given under which the strong duality condition holds; that is, MT and its dual MT * are both solvable and their optimal values coincide. Classification (1991): 90C08, 90C48 Mathematics Subject

Optimal transportation problem as stochastic mechanics

“…Duality Theorem was proved by H.G. Kellerer [47], using a functional analytic method, in a more general setting (the proof by the minmax principle is given in [81], see also [38] for the recent development). When c(x, y) → ∞ (|y − x| → ∞), Selection Lemma which is useful in the stochastic optimal control theory gives a simple proof of the Duality Theorem (see [59]).…”