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)=mMc(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:XY that realizes the infimum :

inf{Xc(x,T(x))dμ(x)|T(μ)}=ν

where 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) :
inf{X×Yc(x,y)dγ(x,y)|γΓ(μ,ν)}
  • Duality formula :
sup{Xϕ(x)dμ(x)+Yψ(y)dν(y):ϕ(x)+ψ(y)c(x,y)}

FootNotes :

  1. 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. 

  2. 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. 

  3. a pushforward measure is obtained by transferring (“pushing forward”) a measure from one measurable space to another using a measurable function. 

  4. 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.