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 R2. Suppose also that we have a cost function c : R2×R2→[0,∞), 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)=∑m∈Mc(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×Y→[0,∞] be a Borel-measurable function 2 Given probability measures μ on X and ν on Y Monge’s formulation of the optimal transportation problem is to find a transport map T:X→Y that realizes the infimum :
inf{∫Xc(x,T(x))dμ(x)|T∗(μ)}=νwhere T∗(μ) denotes the push forward 3 of μ 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∗(μ)=ν).
- Kantorovich's formulation - find a probability measure γ on X×Y that attains the infimum (where Γ(μ,ν) denotes the collection of all probability measures on X×Y with marginals μ on X and ν on Y) :
- Duality formula :
FootNotes :
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Radon measure (wiki) : a measure on the σ-algebra of Borel sets 4 of a Hausdorff topological space X that is finite on all compact sets, outer regular on all Borel sets, and inner regular on open sets. ↩
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a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. ↩
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a pushforward measure is obtained by transferring (“pushing forward”) a measure from one measurable space to another using a measurable function. ↩
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Borel Set : any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. ↩