THE MONGE OPTIMAL TRANSPORTATION PROBLEM
Xu-Jia Wang
Centre for Mathematics and Its Applications
The Australian National University
Canberra, ACT 0200
Australia
§1. Introduction
We report here two recent developments in the theory of optimal transportation. The
first is a proof for the existence of optimal mappings to the Monge mass transportation
problem [16]. The other is an application of optimal transportation in geometric optics
[19].
The optimal transportation problem, in general, deals with the redistribution of materials in the most economical way. The original Monge problem [12] can be formulated
as follows. For two given bounded open sets U and V in the Euclidean n-space, Rn , together with corresponding mass distributions with equal total mass, namely non-negative
measures µ and ν on U and V satisfying the mass balance condition
µ(U ) = ν(V ) < ∞,
whether there exists a measure preserving map s0 : U → V which minimizes the Monge
cost functional
Z
(1.1)
C(s) =
|s(x) − x|dµ
U
among all measure preserving maps from U to V . A map s is called measure preserving if
it is measurable and satisfies
Z
Z
(1.2)
h(s(x))dµ =
h(y)dν
U
V
for any continuous function h.
The optimal transportation problem has also been studied for more general cost functionals
Z
(1.3)
C(s) =
c(x, s(x))dµ,
U
Supported by Australian Research Council
1
where c : Rn × Rn → R is the cost function, s ∈ S = S(f, g), the set of measure preserving
maps from (U, dµ) to (V, dν).
We need to suppose that the measures µ and ν are absolutely continuous, namely there
exist measurable functions f ∈ L1 (U ) and g ∈ L1 (V ) such that µ = f dx and ν = gdx, for
otherwise there is no optimal mappings in general, as is easily seen by putting µ = δo , the
Dirac measure at the origin, and ν = 21 (δo + δx ) for some point x 6= o. When µ = f dx,
ν = gdx, the mass balance condition takes the form
Z
Z
(1.4)
f=
g.
U
V
The first major contribution to the optimization problem is by Monge himself [6]. By
a heuristic geometric argument, Monge observed that an optimal mapping s should be in
part determined by a potential function ϕ, that is,
s(x) − x
= −Dϕ
|s(x) − x|
(1.5)
Monge’s discovery was deeply connected with the theory of developable surfaces, lines of
curvature, etc. See [14].
This optimization problem, and its variants and extensions, has been intensively studied
for over two hundred years. In 1885 the French academy offered a prize for a solution of
Monge’s problem [8]. A major breakthrough was made by Kantorovich [9,10], whose related
work won him the Nobel economy prize. He introduced a relaxed linear minimization
problem, that is
(1.6)
J(p) = inf J(q)
q∈P
for the linear functional
Z
(1.7)
Z
J(p) =
|x − y|dp,
Rn
Rn
where P is the set of Borel measures q defined on Rn × Rn satisfying
q(E × Rn ) = µ(E),
q(Rn × E) = ν(E)
for any Borel sets E ⊂ Rn .
Problem (1.6) and (1.7) has a dual functional (also introduced by Kantorovich),
Z
Z
(1.8)
I(u, v) =
udµ +
vdν,
U
V
where (u, v) ∈ K,
(1.9)
K = {(u, v) | u(x) + v(x) ≤ c(x, y) ∀ x ∈ U, y ∈ V }.
2
Then one has
(1.10)
inf C(s) = inf J(p)
s∈S
p∈P
= sup I(u, v),
(u,v)∈K
where S is the set of all measure preserving mappings. (1.10) shows the relation between
the cost functional (1.3) and Kantorovich’s functionals (1.7) and (1.8). As we will see below
that for many important cases, the optimal mappings for (1.3) can be determined by the
maximizers to the supremum in (1.10).
The existence of optimal mappings for the Monge mass transportation problem was
recently proved by Evans and Gangbo [6] under some regularity conditions, namely f
and g are Lipschitz continuous and supported on disjoint smooth domains. Their proof
uses both partial differential equations and ordinary differential equations, and is very
complicated.
Using Kantorovich’s dual functional, together with a recent result by Caffarelli [3],
Gangbo-McCann [7] on the existence of optimal mappings with strictly convex cost functions, we proved the existence of optimal mappings for the Monge mass transportation
problem, assuming only that µ and ν are absolutely continuous [16]. A similar proof was
simultaneously found in [4]. As we remarked above, the absolute continuity of µ and ν
cannot be relaxed.
We should also mention the work [15] in which Sudakov proved the existence of optimal
mappings for more general cost functionals. His proof uses the minimizer for (1.6) (1.7)
and a measure decomposition result developed by himself in [15]. However, a gap was
recently found [1], and his approach was only partially restored, see Remark 3.1.
The optimal transportation problem has found numerous applications [1,5,8,13]. Recently a new application in geometric optics was found in [19]. The design of reflector
antenna leads to a very complicated Monge-Ampere type equation subject to the second
boundary condition. Though the existence and uniqueness of solutions have been proved
in [18], an efficient algorithm for numerical solutions is still lacking, due to the strong nonlinearity of the equation. Also the equation may change type, namely it may change from
an elliptic equation to a hyperbolic equation if a convexity condition for the solution is not
preserved.
In [19] we proved that the reflector antenna design problem can indeed be reduced to
an optimal transportation problem for an appropriate cost function. Therefore to find
a numerical solution it suffices to find a maximizer of the dual functional. The linear
programming then provides a reliable approach to the numerical solutions of the problem.
In Section 2 we include a result from [3,7] on the existence of optimal mappings when
the cost function is strictly convex. This result is then used in Section 3 to prove the
existence of optimal mappings for the Monge problem. Our result on the reflector antenna
design problem will be presented in Section 4.
3
§2. The strictly convex case
We first consider the existence of optimal mappings for the optimal transportation problem with strictly convex cost function [3,7]. A cost function c(x, y) is called strictly convex
if
c(x, y) = e
c(x − y)
and e
c is a strictly convex function. We suppose that µ = f dx and ν = gdx, and (1.4) holds.
Lemma 2.1. Suppose c is Lipschitz continuous. Then there is a maximizer (ϕ, ψ) such
that
(2.1)
I(ϕ, ψ) = sup I(u, v),
(u,v)∈K
where I is the dual functional in (1.8).
Proof. Let {(uk , vk )} ⊂ K such that I(uk , vk ) → sup(u,v)∈K I(u, v). let
u∗k (x) = inf {c(x, y) − vk (y)},
(2.2)
y∈V
vk∗ (y)
= inf {c(x, y) − u∗k (x)}.
x∈U
Then u∗k ≥ uk , vk∗ ≥ vk , (u∗k , vk∗ ) ∈ K, and I(u∗k , vk∗ ) ≥ I(uk , vk ). Moreover, u∗k , vk∗ are
Lipschitz continuous, with their Lipschitz constants controlled by that of c.
By the mass balance condition (1.4), we have I(u, v) = I(u+C, v−C) for any constant C.
Hence we may suppose that u∗k , vk∗ are uniformly bounded in k, and (u∗k , vk∗ ) sub-converges
to (ϕ, ψ). Then (ϕ, ψ) ∈ K is a maximizer for (2.1), and is Lipschitz continuous. ¤
The function ϕ is called the Monge-Kantorovich potential of the optimal transportation
problem. For the Monge problem it is indeed the potential in (1.5). On the other hand,
the function ψ is indeed the potential for the inverse transportation problem.
Theorem 2.1. [3,7] Let c ∈ C 1 (Rn × Rn ) be strictly convex and C be the cost functional
given in (1.3). Then there exists an optimal mapping sopt ∈ S such that
(2.3)
C(sopt ) = inf C(s).
s∈S
Proof. From (2.2) we have
(2.4)
ϕ(x) = inf {c(x, y) − ψ(y)}
y∈V
It follows that for any x ∈ U , there is a point s(x) ∈ V such that
(2.5)
ϕ(x) = c(x, s(x)) − ψ(s(x)),
ϕ(x0 ) ≤ c(x0 , s(x)) − ψ(s(x))
4
∀ x0 ∈ U.
It follows Dϕ(x) = Dx c(x, y) with y = s(x), where ϕ is Lipschitz continuous and is
differentiable almost everywhere. Since c is strictly convex, Dx c(x, y) = De
c(x − y) is
invertible. We define
sopt (x) = x − (De
c)−1 (Dϕ(x)).
(2.6)
The mapping s = sopt is measure preserving. Indeed, for any continuous function h
and small constant ε, let ψε (y) = ψ(y) + εh(y), ϕε (x) = ϕ(x) − εh(s(x)) + δε , where the
constant δε can be chosen such that δε = o(ε) (as ε → 0) and (ϕε , ψε ) ∈ K. Then since
(ϕ, ψ) maximizes the functional I(·, ·), we have
1
0 = lim (I(ϕε , ψε ) − I(ϕ, ψ))
ε→0 ε
Z
Z
= − h(s(x))f (x) + h(y)g(y),
which gives (1.2).
Notice that for any s ∈ S, (u, v) ∈ K,
Z
Z
u(x)f (x) +
Z
v(y)g(y) =
(u(x)f (x) + v(s(x))f (x))
Z
≤
c(x, s(x))f (x).
Namely,
(2.7)
sup I(u, v) ≤ inf C(s),
s∈S
(u,v)∈K
and equality holds when s = sopt and (u, v) = (ϕ, ψ). Hence sopt is an optimal mapping.
¤
The optimal mapping sopt in Theorem 2.1 is unique in the sense that if se ∈ S(f, g) such
that C(e
s) = C(sopt ), then se = sopt for a.e. x ∈ supp f [7] . If the potential ϕ is smooth, it
satisfies a Monge-Ampere type equation. To see this, let us consider the special case when
c(x, y) = |x − y|2 . Then
(2.8)
sopt (x) = x − Dϕ(x).
To derive the equation we notice that the Jacobi determinant of the mapping is equal to
f /g. It follows
det(I − D2 ϕ) = f (x)/g(sopt (x)).
That is,
(2.9)
det(D2 u) = f /g,
5
where u = 12 |x|2 − ϕ. The boundary condition is
(2.10)
Du(U ) = V.
This is a Monge-Ampere equation subject to the second boundary condition. Regularity
for (2.9) and (2.10) was proven by Caffarelli [2] and Urbas [17]. The regularity of the
potential function for general strictly convex cost functions is still open.
§3. The Monge problem
For the Monge mass transportation problem we have
Theorem 3.1. [4,16] Suppose µ = f dx and ν = gdx for some non-negative integrable
functions f, g such that (1.4) holds. Then there exists an optimal mapping which minimizes
the Monge cost functional (1.1).
To prove Theorem 3.1 one would naturally like to prove the convergence of the optimal
mappings sp obtained in Theorem 2.1 with cost function c(x, y) = |x − y|p as p & 1.
s (x)−x
Although it is possible to prove the convergence of the directions |spp (x)−x| if |sp (x) − x| is
strictly positive, it has been unsuccessful to prove the convergence of |sp (x) − x| as p → 1.
Our proof [16] for the existence of optimal mappings is constructive (once an appropriate
covering of transfer rays is known). A similar proof was simultaneously found in [4]. The
following is the outline of our proof.
Proof. By extending the definition of f, g to the domain Ω = U ∪ V such that f, g vanish
outside U, V respectively, we may suppose U = V = Ω. Let (ϕp , ψp ) be the potentials
obtained in Lemma 2.1, and sp be the optimal mapping obtained in Theorem 2.1, with the
cost function c(x, y) = |x − y|p , 1 < p < ∞. From (2.5) we have
(3.1)
ϕp (x) + ψp (sp (x)) = |x − sp (x)|p .
Sending p & 1 we see that (ϕp , ψp ) sub-converges to (ϕ, ψ). By (2.4), the Lipschitz
constants of ϕp , ψp are controlled by supx,y∈Ω p|x − y|p−1 . It follows that
(3.2)
|ϕ(x) − ϕ(y)| ≤ |x − y|,
|ψ(x) − ψ(y)| ≤ |x − y|
for any x, y ∈ Ω. We may assume
(3.3)
ϕ = −ψ.
Indeed, since ϕ(x) + ψ(y) ≤ |x − y|, we have ϕ ≤ −ψ and by (3.2), ϕ = −ψ is admissible.
For any x ∈ Ω such that a subsequence sp (x) → y 6= x, we have
(3.4)
ϕ(y) − ϕ(x) = −(ψ(y) + ϕ(x))
= − lim {ψp (sp (x)) + ϕp (x)}
p→1
= − lim cp (x − sp (x)) = −|y − x|.
p→1
6
It follows that ϕ is linear on the line segment xy.
Therefore we can introduce the notions of transfer set and transfer ray. A transfer ray
is a maximal line segment such that |ϕ(x) − ϕ(y)| = |x − y| on the segment. The set of
all transfer rays will be denoted by L. The transfer set, denoted by T , is the collection of
points on transfer rays.
We have the following two fundamental properties, both of them follows from the existence of optimal mappings sp for p > 1. One is
Z
(3.5)
(f + g) = 0
Ω−T
That is f, g are supported on the transfer set. Hence we need only to consider the mass
transportation problem on the set T . The other one is, for any subset L1 ⊂ L such that
T1 , the set of all points on transfer rays in L1 , is measurable,
Z
Z
(3.6)
f=
g.
T1
T1
This property enables us to use decomposition for the transfer set T .
We can decompose L into the union of countably many disjoint subset {Lj }∞
j=1 such
that:
(a) For each j, the set Tj , that is the set of all points on transfer rays in Lj , is measurable.
(b) Tj is contained in a thin cylinder; that is after appropriate choice of coordinates,
Tj ⊂ {|x0 | < δj } × (−aj , aj ), where x0 = (x1 , · · · , xn−1 ), aj ≥ 16δj .
(c) The endpoints of any transfer ray in Lj are either contained in {aj − δj ≤ xn ≤ aj } or
in {−aj ≤ xn ≤ −aj + δj }.
We then introduce a transformation ηj on Tj which makes all transfer rays in Lj parallel
to the xn axis. On ηj (Tj ), we define the measure
Z
(3.7)
µj (E) =
ηj−1 (E)
f.
If µj is absolutely continuous, that is, if µj = fj dx for some integrable function fj , we
can use the Fubini theorem to reduce the Monge problem to the problem on each transfer
ray. For the one dimensional case, one can define a unique measure preserving mapping
satisfying the monotone condition
(3.8)
(s(y) − s(x)) · (y − x) ≥ 0.
Therefore a mapping is defined on ηj (Tj ), and so also on Tj . Since all the Tj ’s are disjoint,
we obtain a mapping on the transfer set T . This mapping is then proved to satisfy the
measure preserving condition and is optimal.
The absolute continuity of µj follows from the finite covering theorem together with the
following estimate: There is an absolute constant C such that for any two transfer rays
7
`1 , `2 ∈ Lj , where Lj is any set in the decomposition satisfying (a)-(c) above, if we denote
xi = `i ∩ {xn = 0} and xi,h = `i ∩ {x4 = h} (i = 1, 2), then
(3.9)
|x1,h − x2,h | ≤ C|x1 − x2 |.
This completes the proof. ¤
Remark 3.1. Sudakov claimed in his paper [15] the existence of optimal mappings for
Monge’s problem, and for more general cost functions. However his proof contains a gap,
pertaining to absence of proof for the absolute continuity. Indeed the absolute continuity
is false for arbitrarily given families of line segments, as shown by the example in [11]
which says that there exists a compact set of disjoint line segments in E 3 whose end set
has positive measure. Sudakov’s proof, which is contained in a paper of 180 pages, uses the
minimizer of the relaxed functional J (see (1.6)), for which he has to develop a measure
decomposition result. The approach in [15] can be partially restored by adding our proof of
the absolute continuity in [4,16], see [1]. The proof in [16], which uses the maximizer of the
dual functional, and the existence of optimal mappings for strictly convex cost functions,
and is much simpler in comparison.
An immediate question is whether there exist optimal mappings when the cost function
c(x, y) = |x − y| (the Euclidean norm) is replaced by arbitrary norm kx − yk. The proof in
[4,16] is valid if the norm is uniformly convex, namely if the level set {kxk = 1} is smooth
and uniformly convex. Recently we found that the proof is still valid if the Gauss curvature
of the level set {kxk = 1} is pinched in two positive constants.
§4. Reflector antenna design
The reflector antenna system we are concerned here consists of a point light source O, a
reflecting surface Γ, and a target area in the outer space, to be illuminated by the system.
The light source O is located at the origin of R3 . The reflecting surface Γ is a radial graph
of a function ρ over a domain Ω ⊂ S 2 , given by
(4.1)
Γ = {xρ(x); x ∈ Ω}
ρ > 0,
where S 2 is the unit sphere centered at O. The target area is identified with a domain
Ω∗ ⊂ S 2 in the way that a direction from O to the target area is regarded as a point in Ω∗ .
We want to construct the reflecting surface Γ such that the output domain Ω∗ is covered
by the reflected light, or more precisely by the direction of the reflected light. Let f be
the illumination on the input domain D, that is the distribution of the intensity of rays
from O, and let g be the illumination on the output domain Ω∗ . Suppose there is no loss
of energy in the reflection and the reflected ray does not run out of the domain Ω∗ . Then
we have, by the energy conservation,
Z
Z
(4.2)
f=
g.
Ω
8
Ω∗
For a ray from O to a point z = xρ(x) ∈ Γ (x ∈ Ω), the direction of the reflected ray is,
by the reflection law,
(4.3)
T (x) = Tρ (x) = x − 2hx, nin,
where n is the outward normal of Γ at z. By the energy conservation, T is a measure
preserving mapping, that is
Z
Z
(4.4)
f=
g ∀ Borel set E ⊂ Ω∗ .
T −1 (E)
E
In order that the reflected rays cover the domain Ω∗ and stay in Ω∗ , we must have
T (Ω) = Ω∗ .
(4.5)
From (4.4) we obtain a partial differential equation for the light reflection problem.
Indeed, if Γ is a smooth surface, then by (4.4), the Jocabi determinant of the mapping T
at x ∈ D is equal to f (x)/g(T (x)). Choose local orthonormal coordinates near x on S 2 .
Then we have the equation [18]
(4.6)
Lρ = η −2 det(−∇i ∇j ρ + 2ρ−1 ρi ρj + (ρ − η)δij ) = f (x)/g(T (x)),
where η = (|∇ρ|2 + ρ2 )/2ρ. This is an extremely complicated, fully nonlinear partial
differential equation of Monge-Ampere type.
Because of its importance in applications, it is highly desirable to find an efficient algorithm for the construction of the reflecting surface, which is still lacking due to the strong
nonlinearity of the equation. We have
Theorem 4.1. Suppose that Ω and Ω∗ are respectively strictly contained in the north
and south hemispheres. Suppose f and g are bounded positive functions.
(i) There exists a Lipschitz continuous maximizing pair (ϕ1 , ψ1 ) of the linear functional
Z
(4.7)
Z
I(u, v) =
f (x)u(x) +
g(y)v(y)
Ω∗
Ω
in the convex set
∗
K = {(u, v); u ∈ C(Ω), v ∈ C(Ω ), and
(4.8)
u(x) + v(y) ≤ c(x, y) ∀ x ∈ Ω, y ∈ Ω∗ },
where x · y is the inner product in R3 ,
(4.9)
c(x, y) = − log(1 − x · y),
such that ρ1 = eϕ1 is a solution of (4.6) (4.5) (in a generalized sense).
9
(ii) There exists a Lipschitz continuous minimizing pair (ϕ2 , ψ2 ) of the linear functional
(4.7) in the convex set
∗
K 0 = {(u, v); u ∈ C(Ω), v ∈ C(Ω ), and
(4.10)
u(x) + v(y) ≥ c(x, y) ∀ x ∈ Ω, y ∈ Ω∗ },
such that ρ2 = eϕ2 is a solution of (4.6) (4.5).
(iii) The solution of (4.6) (4.5) is unique, in the sense that if ρ is a smooth solution of
(4.6) (4.5), then either ρ = Cρ1 or ρ = Cρ2 for some positive constant C.
The proof of Theorem 4.1 follows similar lines to that of Theorem 2.1. For details we
refer to [19].
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