Transportation Theory

Problem Formation

wiki Transportation Theory (Monge–Kantorovich transportation problem) - the study of optimal transportation and allocation of resources.

Problem Formation - mines-factories example: There are mines and factories form two disjoint subsets M and F of the Euclidean plane $\mathbb{R}^{2}$. Suppose also that we have a cost function c : $\mathbb{R}^{2} \times \mathbb{R}^{2} \to [0, \infty)$, so that c(x, y) is the cost of transporting one shipment of iron from x to y.

• ignore the time taken to do the transporting.
• objective function : $c(T) = \sum_{m\in {M}} c(m, T(m))$
• a transport plan is a bijection: if each mine can supply only one factory.

Monge and Kantorovich formulations: let X Y be two separable metric spaces such that any probability measure on X or Y is a Radon measure ¹ (i.e. they are Radon spaces). Let $c : X \times Y \to [0, \infty]$ be a Borel-measurable function ² Given probability measures $\mu$ on X and $\nu$ on Y Monge’s formulation of the optimal transportation problem is to find a transport map $T : X \to Y$ that realizes the infimum :

\[\inf \{\int_{X} c(x, T(x)) d\mu (x) | T_{*}(\mu) \} = \nu\]

where $T_{*}(\mu)$ denotes the push forward ³ of $\mu$ by T. A map T that attains this infimum (i.e. makes it a minimum instead of an infimum) is called an "optimal transport map".

• can be ill-posed (no T satisfying $T_{*}(\mu) = \nu$).
• Kantorovich's formulation - find a probability measure $\gamma$ on $X\times Y$ that attains the infimum (where $\Gamma(\mu, \nu)$ denotes the collection of all probability measures on $X\times Y$ with marginals $\mu$ on X and $\nu$ on Y) :

\[\inf \{ \int_{X\times Y} c(x, y) d\gamma (x, y) | \gamma \in \Gamma(\mu, \nu) \}\]

• Duality formula :

\[\sup \{ \int_{X}\phi(x)d\mu (x) + \int_{Y} \psi(y) d\nu (y) : \phi(x) + \psi(y) \le c(x, y) \}\]

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