mass transport theory and its applicationshomepages.wmich.edu/~ledyaev/zhu-talk-sp2016.pdfthe...

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The problemKantorovich Duality

Applications

Mass Transport Theory and its ApplicationsAn introduction

Qiji Zhu

Analysis Seminar

Feburary 5, 2016

Qiji Zhu Mass Transport Theory and its Applications

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Applications

Monge problemKantorovich problemRelationshipProbabilistic formulationsHistory

Monge problem

Monge optimal transport problem (1781)

Given two probability measures µ on X, ν on Y , and a costfunction c : X × Y , can we find a map T : X → Y such thatT#µ = ν and that minimizes∫

Xc(x, T (x))dµ(x)

Here T#µ = ν means ν(B) = µ(T−1(B)).


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Applications


Kantorovich problem

Kantorovich problem (1942)

Given two probability measures µ on X, ν on Y , and a costfunction c : X × Y , can we find a probability measure π ∈ Π(µ, ν)that minimizes ∫

Xc(x, y)dπ(x, y).

Here Π(µ, ν) is the set of all probability measures on X × Y withmarginals µ and ν on X and Y , respectively.


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Applications


Relationship

Restrictingdπ(x, y) = dµ(x)δ(y = T (x))

or equivalentlyπ = (id, T )#µ,

the Kantorovich problem becomes a Monge problem.

We can view the Kantorovich problem as a convexification of theMonge problem.


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Applications


Probabilistic formulations

A probabilistic formulation of the Kantorovich problem: Givensome economical output Φ(x, y)(= −c(x, y)) and random variablesX ∼ µ and Y ∼ ν,

supπ∈Π(µ,ν)

Eπ[Φ(X,Y )]

The Monge problem can be written as

sup{Eµ[Φ(X,T (x))] : T#µ = ν}.


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Applications


History

• Monge proposed the problem in 1781 to model the followingsituation:

• Excavating and transport material for for construction.

• Scientists in the formal USSR started research on suchproblem motivated by applicaitons in transportationscheduling.

• Kantorovich gets interested in the late 1930s to 1940s.

• Linear programming duality was derived in 1938.

• Kantorovich duality theorem was pubished in 1942.


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Applications


History

• Kantorovich also applied the mass transport theory toeconomy by was criticised as deviate from Max’s politicaleconomy.

• As a result Kantorovich maintained low key on these resultsuntil Nikita Khrushchev’s “shaw” in 1956.

• Kantorovich eventually awarded the Nobel Econ Price for hiswork in 1975 (jointly with Koopmans) which stimulatedrenewed interest in the mass transport theory.

• More applications since the 1980s lead to further research inthis subject.

• Two books from the fields medelist C. Villani on this subjectin 2003 and 2006 add new interest.


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Applications

Kantorovich DualityHeuristicsSolution to the dualSolution to the primal

Kantorovich Duality

Kantorovich Duality

Under a certain constraint qualification conditions

infπ∈Π(µ,ν)

∫X×Y

c(x, y)dπ(x, y)

= supφ(x)+ψ(y)≤c(x,y)

{∫Xφ(x)dµ(x) +

∫Yψ(y)dν(y)

}.


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Applications


Kantorovich Duality

Heuristic explanation:

• c(x, y) is the cost of moving x to y.

• the primal problem is to find a transport plan that minimizethe cost.

• A contractor come a propose that he would charge φ(x) tomove x out and ψ(y) to move y in.

• Moreover, he ensures that you get a good deal thatφ(x) + ψ(y) ≤ c(x, y).

• The dual problem is the contractor’s problem.

• Duality theorem tells us that if the contractor is smart, he canget you pay the same price.


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Applications


Lagrangian and minimax

Define a Lagrangian by

L(π, (φ,ψ)) =

∫X×Y

c(x, y)dπ(x, y)

+

∫Xφ(x)dµ(x) +

∫Yψ(y)dν(y)−

∫X×Y

[φ(x) + ψ(y)]dπ(x, y).

We have

supφ,ψ

{∫Xφ(x)dµ(x) +

∫Yψ(y)dν(y)−

∫X×Y

[φ(x) + ψ(y)]dπ(x, y)

}=

{0 π ∈ Π(µ, ν)

+∞ π ∈ Π(µ, ν).


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Applications



Thus, denoting (M) the set of probability measures, we can writethe Kantorovich transport problem as a minimax problem

infπ∈M

supφ,ψ

L(π, (φ,ψ)).

The corresponding maximin problem is exactly the Kantorovichdual problem.

supφ,ψ

infπ∈M

L(π, (φ,ψ))

= supφ(x)+ψ(y)≤c(x,y)

{∫Xφ(x)dµ(x) +

∫Yψ(y)dν(y)

}.


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Applications



This is because

infπ∈M

{∫X×Y

[c(x, y)− φ(x)− ψ(y)]dπ(x, y)

}=

{0 c(x, y) ≥ φ(x) + ψ(y)

−∞ otherwise.


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Applications


Kantorovich Duality: max form

Kantorovich Duality

Under a certain constraint qualification conditions

supπ∈Π(µ,ν)

∫X×Y

c(x, y)dπ(x, y)

= infφ(x)+ψ(y)≥c(x,y)

{∫Xφ(x)dµ(x) +

∫Yψ(y)dν(y)

}.


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Applications


Generalized Fenchel conjugate

φc(y) = sup[c(x, y)− φ(x)],

φc′(x) = sup[c(x, y)− φ(y)],

andφcc

′(x) = φ(x),

if φ is nice enough.


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Applications


Generalized Fenchel conjugate

A solution pair to the dual always have the form

(φ,φc).

This is because for any pair (φ,ψ) satisfying constraint

φ(x) + ψ(y) ≥ c(x, y)

one hasψ(y) ≥ φc(y) = sup[c(x, y)− φ(x)].


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Applications


Solution to the primal

Knott and Smith 1984

Suppose that (φ,φc) is a solution pair to the Kantorovich dualproblem. Let π∗ be a solution to the Kantorovich primal problem.Then

supp π∗ ⊂ Graph ∂cφ a.e.

Here (x, y) ∈ Graph ∂cφ iff (x, y) satisfies the generalized Fenchelequality

φ(x) + φc(y) = c(x, y).


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Applications


Solution to the primal: special cases

When c(x, y) = ⟨x, y⟩, φc = φ∗ is the usual Fenchel conjugate.Furthermore, if φ is differentiable the solution reduces to one forthe Monge problem:

π∗ = (id,∇φ)#µ, ν = ∇φ#µ.

When c(x, y) = c(x− y) whith c convex and both c, φdifferentiable, φ(x) + φc(y) = c(x− y) implieas∇φ(x) = ∇c(x− y) since ∇c∗ is the inverse of ∇c we have

π∗ = (id, T )#µ, T (x) = x−∇c∗(∇φ(x)) ν = T#µ.


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Applications


Sketch of the proof

∫X×Y

c(x, y)dπ∗(x, y) =

∫Xφ(x)dµ(x) +

∫Yφc(y)dν(y),

implies that∫X×Y

[φ(x) + φc(y)− c(x, y)]dπ∗(x, y) = 0

and the integrand is non-negative by the generalized Fenchelinequality, which follows directly from the definition of φc.


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Applications

Finite measuresOption price boundKantorovich-Rubinstein Theorem

Finite measures

Let X = {x1, . . . , xN}, Y = {y1, . . . , yM}. Then the primalproblem becomes the following linear programming problem:

P min∑n,m

c(xn, ym)π(xn, ym)

s.t.∑m

π(xn, ym) = µ(xn), n = 1, . . . , N∑n

π(xn, ym) = ν(ym),m = 1, . . . ,M

π(xn, ym) ≥ 0.


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Applications


Finite measures

The dual problem is:

D max∑n

φ(xn)µ(xn) +∑m

ψ(yn)ν(yn)

s.t. φ(xn) + ψ(ym) ≤ c(xn, ym),

n = 1, . . . , N,m = 1, . . . ,M.

We see that in this case the Kantorovich duality theorem reducesto linear programming duality for this special linear programmingproblem.


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Applications


The problem

Find the bound for no arbitrage pricing of a contingent claimwhose payoff is the function c of two underlying assets with pricerepresented by random variables ξ and η on probability spaces(X,µ) and (Y, ν), respectively. Here µ and ν are assumed to bemartingale measures for the related financial markets.

Concrete example: currency swaps.


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Applications


Mass transport model

Then the problem of finding the upper bound for no arbitrage priceof c(ξ, η) is simply:

supπ∈Π(µ,ν)

Eπ[c(ξ, η)].


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Applications


The dual problem and its financial meaning

Kantorovich duality tells us that

supπ∈Π(µ,ν)

Eπ[c(ξ, η)]

= infφ(x)+ψ(y)≥c(x,y)

{Eµ[φ(ξ)] + Eν [ψ(η)]} .

Financial meaning: there exists contingent claims φ,ψ on ξ, ηseparately providing a tight upper bound for the no arbitrage priceof c.


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Applications


Kantorovich-Rubinstein Theorem

Kantorovich-Rubinstein Theorem

T∥·∥(µ, ν) = infπ∈Π(µ,ν)

∫X×X

∥x− y∥dπ(x, y)

= sup

{∫Xφd(µ− ν) : ∥φ∥Lip ≤ 1

}.

Here

∥φ∥Lip = supx=y

|φ(y)− φ(x)|∥x− y∥

.

This provides a definition for the distance between two measures.


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Applications


Proof

The solution pair of the dual problem has the form (φ,φc) where

φc(y) = inf[∥x− y∥ − φ(y)], c(x, y) = ∥x− y∥

is Lipschitz with constant 1. So we have

−φc(x) ≤ infy[∥x− y∥ − φc(y)] ≤ −φc(x).

It follows that φ(x) = φcc(x) = −φc(x).


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Applications


A comment and a quesiton

Comment: Kantorovich duality is essentially a linear programmingduality under a function-measure space pairing. Dealing withfunctional spaces it naturally emcompasses generalized Fenchelconjugate – a nonlinear object.

Question: Is the mass transport theory equivalent to the theory ofgeneralize Fenchel conjugate and duality?


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Applications


References

G. Monge, Memoire sur la theo rie des deeblais et des remblais, InHistorie de l’Academie Royale des Sciences de Paris (1781) 666-704.A. Galichon, Optimal Transport Methods in Economics, preprint 2015.L. V. Kantorovhich, Mathematical methods in the organization andplanning of production. Leningrad Univ. 1939.L. V. Kantorovhich, On the translocation of masses. Dokl. Akad. Nauk.USSR 37 (1942) 199-201.C. Villani, Optimal Transport, Old and New, Springer 2006

C. Villani, Topics in Optimal Transportation, AMS 2003.


mass transport theory and its applicationshomepages.wmich.edu/~ledyaev/zhu-talk-sp2016.pdfthe...

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