Transportation theory (mathematics) explained

In mathematics and economics, transportation theory or transport theory is a name given to the study of optimal transportation and allocation of resources. The problem was formalized by the French mathematician Gaspard Monge in 1781.^[1]

In the 1920s A.N. Tolstoi was one of the first to study the transportation problem mathematically. In 1930, in the collection Transportation Planning Volume I for the National Commissariat of Transportation of the Soviet Union, he published a paper "Methods of Finding the Minimal Kilometrage in Cargo-transportation in space".^{[2] [3]}

Major advances were made in the field during World War II by the Soviet mathematician and economist Leonid Kantorovich.^[4] Consequently, the problem as it is stated is sometimes known as the Monge–Kantorovich transportation problem.^[5] The linear programming formulation of the transportation problem is also known as the Hitchcock–Koopmans transportation problem.^[6]

Motivation

Mines and factories

Suppose that we have a collection of

m mines mining iron ore, and a collection of n factories which use the iron ore that the mines produce. Suppose for the sake of argument that these mines and factories form two disjoint subsets M and F of the Euclidean plane R ^{2} . Suppose also that we have a cost function c : R ^{2} x R ^{2} \to [ 0 , inf ty) , so that c(x, y) is the cost of transporting one shipment of iron from x to y . For simplicity, we ignore the time taken to do the transporting. We also assume that each mine can supply only one factory (no splitting of shipments) and that each factory requires precisely one shipment to be in operation (factories cannot work at half- or double-capacity). Having made the above assumptions, a transport plan is a bijection T: M \to F .In other words, each mine m \in M supplies precisely one target factory T(m) \in F and each factory is supplied by precisely one mine. We wish to find the optimal transport plan, the plan T whose total costc(T) : = \sum _{m} c(m, T(m))

is the least of all possible transport plans from

M to F . This motivating special case of the transportation problem is an instance of the assignment problem.More specifically, it is equivalent to finding a minimum weight matching in a bipartite graph.

Moving books: the importance of the cost function

The following simple example illustrates the importance of the cost function in determining the optimal transport plan. Suppose that we have

n books of equal width on a shelf (the real line), arranged in a single contiguous block. We wish to rearrange them into another contiguous block, but shifted one book-width to the right. Two obvious candidates for the optimal transport plan present themselves:

1. move all

n books one book-width to the right ("many small moves");

1. move the left-most book

n book-widths to the right and leave all other books fixed ("one big move").If the cost function is proportional to Euclidean distance (c(x, y) = \alpha \|x - y\| for some \alpha > 0 ) then these two candidates are both optimal. If, on the other hand, we choose the strictly convex cost function proportional to the square of Euclidean distance (c(x, y) = \alpha \|x - y\| ^{2} for some \alpha > 0 ), then the "many small moves" option becomes the unique minimizer.

Note that the above cost functions consider only the horizontal distance traveled by the books, not the horizontal distance traveled by a device used to pick each book up and move the book into position. If the latter is considered instead, then, of the two transport plans, the second is always optimal for the Euclidean distance, while, provided there are at least 3 books, the first transport plan is optimal for the squared Euclidean distance.

Hitchcock problem

The following transportation problem formulation is credited to F. L. Hitchcock:^[7]

Suppose there are

m sources x _{1,} \ldots, x _{m} for a commodity, with a(x _{i)} units of supply at x _{i} and n sinks y _{1,} \ldots, y _{n} for the commodity, with the demand b(y _{j)} at y _{j} . If a(x _{i, y} _{j)} is the unit cost of shipment from x _{i} to y _{j} , find a flow that satisfies demand from supplies and minimizes the flow cost. This challenge in logistics was taken up by D. R. Fulkerson^[8] and in the book Flows in Networks (1962) written with L. R. Ford Jr.^[9]

Tjalling Koopmans is also credited with formulations of transport economics and allocation of resources.

Abstract formulation of the problem

Monge and Kantorovich formulations

The transportation problem as it is stated in modern or more technical literature looks somewhat different because of the development of Riemannian geometry and measure theory. The mines-factories example, simple as it is, is a useful reference point when thinking of the abstract case. In this setting, we allow the possibility that we may not wish to keep all mines and factories open for business, and allow mines to supply more than one factory, and factories to accept iron from more than one mine.

Let

X and Y be two separable metric spaces such that any probability measure on X (or Y ) is a Radon measure (i.e. they are Radon spaces). Let c : X x Y \to [ 0 , inf ty] be a Borel-measurable function. 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 infimuminf \left\{ \left. \int _{X} c(x, T(x)) d \mu (x) \right| T _{*} (\mu) = \nu \right\},

where

T _{*(\mu)} denotes the push forward 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".

Monge's formulation of the optimal transportation problem can be ill-posed, because sometimes there is no

T satisfying T _{*(\mu)} = \nu : this happens, for example, when \mu is a Dirac measure but \nu is not.

We can improve on this by adopting Kantorovich's formulation of the optimal transportation problem, which is to find a probability measure

\gamma on X x Y that attains the infimuminf \left\{ \left. \int _{X} c(x, y) d \gamma (x, y) \right| \gamma \in \Gamma (\mu, \nu) \right\},

where

\Gamma (\mu, \nu) denotes the collection of all probability measures on X x Y with marginals \mu on X and \nu on Y . It can be shown^[10] that a minimizer for this problem always exists when the cost function c is lower semi-continuous and \Gamma(\mu, \nu) is a tight collection of measures (which is guaranteed for Radon spaces X and Y ). (Compare this formulation with the definition of the Wasserstein metric W _{p} on the space of probability measures.) A gradient descent formulation for the solution of the Monge–Kantorovich problem was given by Sigurd Angenent, Steven Haker, and Allen Tannenbaum.^[11]

Duality formula

The minimum of the Kantorovich problem is equal to

\sup \left( \int _{X} \varphi (x) d \mu (x) + \int _{Y} \psi (y) d \nu (y) \right),

where the supremum runs over all pairs of bounded and continuous functions

\varphi : X → R and \psi : Y → R such that\varphi (x) + \psi (y) \leq c(x, y).

Economic interpretation

The economic interpretation is clearer if signs are flipped. Let $x\; \backslash in\; X$ stand for the vector of characteristics of a worker, $y\; \backslash in\; Y$ for the vector of characteristics of a firm, and $\backslash Phi(x,y)\; =-c(x,y)$ for the economic output generated by worker $x$ matched with firm $y$. Setting $u(x)\; =\; -\backslash varphi(x)$ and $v(y)\; =-\backslash psi(y)$, the Monge–Kantorovich problemrewrites:$$\backslash sup\; \backslash left\backslash$$which has dual :$$\backslash inf\; \backslash left\backslash$$where the infimum runs over bounded and continuous function $u:X\backslash rightarrow\; \backslash mathbb$ and $v:Y\backslash rightarrow\; \backslash mathbb$. If the dual problem has a solution, one can see that:$$v(y)\; =\backslash sup\_x\; \backslash left\backslash$$so that $u(x)$ interprets as the equilibrium wage of a worker of type $x$, and $v(y)$ interprets as the equilibrium profit of a firm of type $y$.

Solution of the problem

Optimal transportation on the real line

For

1 \leq p < inf ty , let l{P} _{p(R)} denote the collection of probability measures on R that have finite p -th moment. Let \mu, \nu \in l{P} _{p(R)} and let c(x, y) = h(x - y) , where h: R → [ 0 , inf ty) is a convex function.

1. If

\mu has no atom, i.e., if the cumulative distribution function F _{\mu} : R → [ 0 , 1 ] of \mu is a continuous function, then F ^{-1} _{\nu} \circ F _{\mu} : R \to R is an optimal transport map. It is the unique optimal transport map if h is strictly convex.

1. We have

min _{\gamma} \int _{R ^{2}} c(x, y) d \gamma (x, y) = \int ^{1} _{0} c \left( F ^{-1} _{\mu} (s), F ^{-1} _{\nu} (s) \right) d s.

The proof of this solution appears in Rachev & Rüschendorf (1998).^[12]

Discrete version and linear programming formulation

In the case where the margins $\backslash mu$ and $\backslash nu$ are discrete, let $\backslash mu\_x$and $\backslash nu\_y$ be the probability masses respectively assigned to $x\backslash in\; \backslash mathbf$ and $y\backslash in\; \backslash mathbf$, and let $\backslash gamma\; \_$ be the probability of an $xy$ assignment. The objective function in the primal Kantorovich problem is then

\sum _{x\in} \gamma _{xy} c _{xy}

and the constraint $\backslash gamma\; \backslash in\; \backslash Gamma\; \backslash left(\backslash mu,\backslash nu\; \backslash right)$ expresses as

\sum _{y\in} \gamma _{xy} = \mu _{x,\forall} x\in X and\sum _{x\in} \gamma _{xy} = \nu _{y,\forall} y\in Y .

In order to input this in a linear programming problem, we need to vectorize the matrix $\backslash gamma\_$ by either stacking its columns or its rows, we call $\backslash operatorname$ this operation. In the column-major order, the constraints above rewrite as

\left( 1 _{1 x} ⊗ I _{\left\vert} \right) \operatorname{vec}\left( \gamma \right) = \mu and \left( I _{\left\vert} ⊗ 1 _{1 x} \right) \operatorname{vec}\left( \gamma \right) = \nu

where $\backslash otimes$ is the Kronecker product, $1\_$ is a matrix of size $n\backslash times\; m$ with all entries of ones, and $I\_$ is the identity matrix of size $n$. As a result, setting $z=\backslash operatorname\backslash left(\backslash gamma\; \backslash right)$, the linear programming formulation of the problem is

\begin{align} & Minimize &&\operatorname{vec}(c) ^{\top} z \\[ 4 pt] & subjectto: && z \ge 0 , \\[ 4 pt] & && \begin{pmatrix} 1 _{1 x} ⊗ I _{\left\vert} \\ I _{\left\vert} ⊗ 1 _{1 x} \end{pmatrix} z = \binom{\mu }{\nu } \end{align}

which can be readily inputted in a large-scale linear programming solver (see chapter 3.4 of Galichon (2016)^[13]).

Semi-discrete case

In the semi-discrete case, $X=Y=\backslash mathbb^d$ and $\backslash mu$ is a continuous distribution over $\backslash mathbb^d$, while $\backslash nu\; =\backslash sum\_^\backslash nu\; \_\backslash delta\_$ is a discrete distribution which assigns probability mass $\backslash nu\; \_$ to site $y\_j\; \backslash in\; \backslash mathbb^d$. In this case, we can see^[14] that the primal and dual Kantorovich problems respectively boil down to:$$\backslash inf\; \backslash left\backslash$$for the primal, where $\backslash gamma\; \backslash in\; \backslash Gamma\; \backslash left(\backslash mu,\backslash nu\; \backslash right)$ means that $\backslash int\_\; d\backslash gamma\; \_\backslash left(x\backslash right)\; =\backslash nu\; \_$ and $\backslash sum\_d\backslash gamma\_\backslash left(x\backslash right)\; =d\backslash mu\; \backslash left(x\backslash right)$, and:$$\backslash sup\; \backslash left\backslash$$for the dual, which can be rewritten as:$$\backslash sup\_\backslash left\backslash$$which is a finite-dimensional convex optimization problem that can be solved by standard techniques, such as gradient descent.

In the case when $c\backslash left(x,y\backslash right)\; =\backslash left\backslash vert\; x-y\backslash right\backslash vert\; ^/2$, one can show that the set of $x\backslash in\; \backslash mathbf$ assigned to a particular site $j$ is a convex polyhedron. The resulting configuration is called a power diagram.^[15]

Quadratic normal case

Assume the particular case $\backslash mu\; =\backslash mathcal\backslash left(0,\backslash Sigma\_X\backslash right)$, $\backslash nu\; =\backslash mathcal\; \backslash left(0,\backslash Sigma\; \_\backslash right)$, and $c(x,y)\; =\backslash left\backslash vert\; y-Ax\backslash right\backslash vert^2/2$ where $A$ is invertible. One then has

\varphi(x) =- x ^{\top} \Sigma ^{-1 / 2} _{X} \left( \Sigma ^{1 / 2} _{X} A ^{\top} \Sigma _{Y} A\Sigma ^{1 / 2} _{X} \right) ^{1/2} \Sigma ^{-1 / 2} _{X} x/ 2 \psi(y) =- y ^{\top} A\Sigma ^{1 / 2} _{X} \left( \Sigma ^{1 / 2} _{X} A ^{\top} \Sigma _{Y} A\Sigma ^{1 / 2} _{X} \right) ^{-1/2} \Sigma ^{1 / 2} _{X} Ay/ 2 T(x) = (A ^{\top)} ^{-1} \Sigma ^{-1 / 2} _{X} \left(\Sigma ^{1 / 2} _{X} A ^{\top} \Sigma _{Y} A\Sigma ^{1 / 2} _{X} \right) ^{1/2} \Sigma ^{-1 / 2} _{X} x

The proof of this solution appears in Galichon (2016).

Separable Hilbert spaces

Let

X be a separable Hilbert space. Let l{P} _{p(X)} denote the collection of probability measures on X that have finite p -th moment; let l{P} ^{r(X)} _{p} denote those elements \mu \in l{P} _{p(X)} that are Gaussian regular: if g is any strictly positive Gaussian measure on X and g(N) = 0 , then \mu(N) = 0 also.

Let

\mu \in l{P} ^{r} _{p} (X) , \nu \in l{P} _{p(X)} , c (x, y) = | x - y | ^{p/p} for p\in( 1 , inf ty), p ^{-1} + q ^{-1} = 1 . Then the Kantorovich problem has a unique solution \kappa , and this solution is induced by an optimal transport map: i.e., there exists a Borel map r\in L ^{p(X,} \mu; X) such that\kappa = ( id _{X} x r) _{*} (\mu) \in \Gamma (\mu, \nu).

Moreover, if

\nu has bounded support, thenr(x) = x - | \nabla \varphi (x) | ^{q} \nabla \varphi (x)

for

\mu -almost all x\in X for some locally Lipschitz, c -concave and maximal Kantorovich potential \varphi . (Here \nabla \varphi denotes the Gateaux derivative of \varphi .)

Entropic regularization

Consider a variant of the discrete problem above, where we have added an entropic regularization term to the objective function of the primal problem

\begin{align} & Minimize \sum _{x\in} \gamma _{xy} c _{xy} + \varepsilon \gamma _{xy} ln \gamma _{xy} \\[ 4 pt] & subjectto: \\[ 4 pt] & \gamma\ge 0 \\[ 4 pt] & \sum _{y\in} \gamma _{xy} = \mu _{x} ,\forall x\in X \\[ 4 pt] & \sum _{x\in} \gamma _{xy} = \nu _{y,} \forall y\in Y \end{align}

One can show that the dual regularized problem is

max _{\varphi} \sum _{x\in} \varphi _{x} \mu _{x} + \sum _{y\in} \psi _{y} v _{y} - \varepsilon \sum _{x\in} \exp \left( (\varphi _{x} + \psi _{y} - c _{xy})/(\varepsilon) \right)

where, compared with the unregularized version, the "hard" constraint in the former dual ($\backslash varphi\_x\; +\; \backslash psi\_y\; -\; c\_\backslash geq\; 0$) has been replaced by a "soft" penalization of that constraint (the sum of the $\backslash varepsilon\; \backslash exp\; \backslash left((\backslash varphi\; \_x\; +\; \backslash psi\_y\; -\; c\_)/\backslash varepsilon\; \backslash right)$ terms). The optimality conditions in the dual problem can be expressed as

\mu _{x} = \sum _{y\in} \exp \left( (\varphi _{x} + \psi _{y} - c _{xy})/(\varepsilon) \right) ~\forall x\in X \nu _{y} = \sum _{x\in} \exp \left( (\varphi _{x} + \psi _{y} - c _{xy})/(\varepsilon) \right) ~\forall y\in Y

Denoting $A$ as the $\backslash left\backslash vert\; \backslash mathbf\backslash right\backslash vert\; \backslash times\; \backslash left\backslash vert\; \backslash mathbf\backslash right\backslash vert$ matrix of term $A\_=\backslash exp\; \backslash left(-c\_\; /\; \backslash varepsilon\; \backslash right)$, solving the dual is therefore equivalent to looking for two diagonal positive matrices $D\_$ and $D\_$ of respective sizes $\backslash left\backslash vert\; \backslash mathbf\backslash right\backslash vert$ and $\backslash left\backslash vert\; \backslash mathbf\backslash right\backslash vert$, such that $D\_AD\_1\_=\backslash mu$ and $\backslash left(D\_AD\_\backslash right)\; ^1\_=\backslash nu$. The existence of such matrices generalizes Sinkhorn's theorem and the matrices can be computed using the Sinkhorn–Knopp algorithm,^[16] which simply consists of iteratively looking for $\backslash varphi\; \_$ to solve, and $\backslash psi\; \_$ to solve . Sinkhorn–Knopp's algorithm is therefore a coordinate descent algorithm on the dual regularized problem.

Applications

The Monge–Kantorovich optimal transport has found applications in wide range in different fields. Among them are:

• Image registration and warping^[17]
• Reflector design^[18]
• Retrieving information from shadowgraphy and proton radiography^[19]
• Seismic tomography and reflection seismology^[20]
• The broad class of economic modelling that involves gross substitutes property (among others, models of matching and discrete choice).

See also

• Wasserstein metric
• Transport function
• Hungarian algorithm
• Transportation planning
• Earth mover's distance
• Monge–Ampère equation

Further reading

• Book: Brualdi, Richard A. . Combinatorial matrix classes . Encyclopedia of Mathematics and Its Applications . 108 . Cambridge . . 2006 . 978-0-521-86565-4 . 1106.05001 . registration .

Notes and References

1. G. Monge. Mémoire sur la théorie des déblais et des remblais. Histoire de l’Académie Royale des Sciences de Paris, avec les Mémoires de Mathématique et de Physique pour la même année, pages 666–704, 1781.
2. [Alexander Schrijver|Schrijver, Alexander]
3. Ivor Grattan-Guinness, Ivor, Companion encyclopedia of the history and philosophy of the mathematical sciences, Volume 1, JHU Press, 2003. Cf. p.831
4. L. Kantorovich. On the translocation of masses. C.R. (Doklady) Acad. Sci. URSS (N.S.), 37:199–201, 1942.
5. Book: Cédric Villani. Topics in Optimal Transportation. 2003. American Mathematical Soc.. 978-0-8218-3312-4. 66.
6. Book: Singiresu S. Rao. Engineering Optimization: Theory and Practice. 2009. John Wiley & Sons. 978-0-470-18352-6. 221. 4th.
7. Frank L. Hitchcock (1941) "The distribution of a product from several sources to numerous localities", MIT Journal of Mathematics and Physics 20:224–230 .
8. D. R. Fulkerson (1956) Hitchcock Transportation Problem, RAND corporation.
9. [L. R. Ford Jr.]
10. L. Ambrosio, N. Gigli & G. Savaré. Gradient Flows in Metric Spaces and in the Space of Probability Measures. Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel. (2005)
11. S. . Angenent . S. . Haker . A. . Tannenbaum . Minimizing flows for the Monge–Kantorovich problem . SIAM J. Math. Anal. . 35 . 1 . 61–97 . 2003 . 10.1137/S0036141002410927 . 10.1.1.424.1064 .
12. Rachev, Svetlozar T., and Ludger Rüschendorf. Mass Transportation Problems: Volume I: Theory. Vol. 1. Springer, 1998.
13. [Alfred Galichon|Galichon, Alfred]
14. Santambrogio, Filippo. Optimal Transport for Applied Mathematicians. Birkhäuser Basel, 2016. In particular chapter 6, section 4.2.
15. .
16. [Gabriel Peyré|Peyré, Gabriel]
17. Haker. Steven. Zhu. Lei. Tannenbaum. Allen. Angenent. Sigurd. 1 December 2004. Optimal Mass Transport for Registration and Warping. International Journal of Computer Vision. en. 60. 3. 225–240. 10.1023/B:VISI.0000036836.66311.97. 0920-5691. 10.1.1.59.4082. 13261370.
18. Glimm. T.. Oliker. V.. 1 September 2003. Optical Design of Single Reflector Systems and the Monge–Kantorovich Mass Transfer Problem. Journal of Mathematical Sciences. en. 117. 3. 4096–4108. 10.1023/A:1024856201493. 8301248. 1072-3374.
19. Kasim. Muhammad Firmansyah. Ceurvorst. Luke. Ratan. Naren. Sadler. James. Chen. Nicholas. Sävert. Alexander. Trines. Raoul. Bingham. Robert. Burrows. Philip N.. 16 February 2017. Quantitative shadowgraphy and proton radiography for large intensity modulations. Physical Review E. 95. 2. 023306. 10.1103/PhysRevE.95.023306. 28297858. 1607.04179. 2017PhRvE..95b3306K. 13326345.
20. Metivier . Ludovic . Measuring the misfit between seismograms using an optimal transport distance: application to full waveform inversion . Geophysical Journal International . 24 February 2016 . 205 . 1 . 345–377 . 10.1093/gji/ggw014 . 2016GeoJI.205..345M . free .