Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Interval analysis and Optimal Transport
Nicolas Delanoue - Mehdi Lhommeau - Philippe Lucidarme
LARIS - Universite d’Angers - France
Séminaire du LARIS
16th September 2014
Nicolas Delanoue
Interval analysis and Optimal Transport
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Outline
1
Introduction
Interval analysis
Introduction to Optimal Transport
Transportation
Optimal Transport
Some known results
2
A lower bound of the optimal value
Finite dimensional relaxation
3
An upper bound of the optimal value
Duality
Finite dimensional relaxation
4
Conclusion - Future work
Nicolas Delanoue
Interval analysis and Optimal Transport
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Interval analysis
Introduction to Optimal Transport
Definition
An interval is a compact subset of R of the following form :
[x] = [x, x] = {x ∈ R | x ≤ x ≤ x}.
Nicolas Delanoue
Interval analysis and Optimal Transport
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Interval analysis
Introduction to Optimal Transport
Definition
An interval is a compact subset of R of the following form :
[x] = [x, x] = {x ∈ R | x ≤ x ≤ x}.
Definition
Let f : R → R be a map, one says that [f ] : IR → IR is an
inclusion map of f if ∀[x] ∈ IR, f ([x]) ⊂ [f ]([x]).
Nicolas Delanoue
Interval analysis and Optimal Transport
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Interval analysis
Introduction to Optimal Transport
Definition
An interval is a compact subset of R of the following form :
[x] = [x, x] = {x ∈ R | x ≤ x ≤ x}.
Definition
Let f : R → R be a map, one says that [f ] : IR → IR is an
inclusion map of f if ∀[x] ∈ IR, f ([x]) ⊂ [f ]([x]).
Nicolas Delanoue
Interval analysis and Optimal Transport
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Interval analysis
Introduction to Optimal Transport
Definition
An interval is a compact subset of R of the following form :
[x] = [x, x] = {x ∈ R | x ≤ x ≤ x}.
Definition
Let f : R → R be a map, one says that [f ] : IR → IR is an
inclusion map of f if ∀[x] ∈ IR, f ([x]) ⊂ [f ]([x]).
Nicolas Delanoue
Interval analysis and Optimal Transport
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Interval analysis
Introduction to Optimal Transport
Definition
An interval is a compact subset of R of the following form :
[x] = [x, x] = {x ∈ R | x ≤ x ≤ x}.
Definition
Let f : R → R be a map, one says that [f ] : IR → IR is an
inclusion map of f if ∀[x] ∈ IR, f ([x]) ⊂ [f ]([x]).
Nicolas Delanoue
Interval analysis and Optimal Transport
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Interval analysis
Introduction to Optimal Transport
Interval Arithmetic
[x] + [y ] = [x + y , x + y ]
[x] − [y ] = [x − y , x − y ]
[x] × [y ] = [min{xy
h , xyi, xy , xy }, max{xy , xy , xy , xy }],
[x] ÷ [y ] = [x] ×
1 1
y, y
, if y y > 0.
Nicolas Delanoue
Interval analysis and Optimal Transport
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Interval analysis
Introduction to Optimal Transport
Interval Arithmetic
[x] + [y ] = [x + y , x + y ]
[x] − [y ] = [x − y , x − y ]
[x] × [y ] = [min{xy
h , xyi, xy , xy }, max{xy , xy , xy , xy }],
[x] ÷ [y ] = [x] ×
1 1
y, y
, if y y > 0.
Proposition
The four basic interval operations are inclusion maps of +, −, ×
and ÷ defined on reals.
Nicolas Delanoue
Interval analysis and Optimal Transport
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Interval analysis
Introduction to Optimal Transport
Proposition
If f and g are maps with inclusion maps [f ] and [g ],
then [f ] ◦ [g ] is an inclusion map of f ◦ g .
Nicolas Delanoue
Interval analysis and Optimal Transport
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Interval analysis
Introduction to Optimal Transport
Proposition
If f and g are maps with inclusion maps [f ] and [g ],
then [f ] ◦ [g ] is an inclusion map of f ◦ g .
Example
Let f : R → R be
f (x) = 1 − 2x + x 2 .
The map [f ] : IR → IR
[f ]([x, x]) = [1, 1] − [2, 2] × [x, x] + [x, x]2 ,
is an inclusion map for f .
Nicolas Delanoue
Interval analysis and Optimal Transport
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Interval analysis
Introduction to Optimal Transport
f (x) = 1 − 2x + x 2
Nicolas Delanoue
Interval analysis and Optimal Transport
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Interval analysis
Introduction to Optimal Transport
f (x) = 1 − 2x + x 2
x ∈ [2, 3]
Nicolas Delanoue
Interval analysis and Optimal Transport
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Interval analysis
Introduction to Optimal Transport
f (x) = 1 − 2x + x 2
x ∈ [2, 3]
⇒
1
∈
[
Nicolas Delanoue
1
,
1
],
Interval analysis and Optimal Transport
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Interval analysis
Introduction to Optimal Transport
f (x) = 1 − 2x + x 2
x ∈ [2, 3]
⇒
⇒
1
−2x
∈
∈
[
[
Nicolas Delanoue
1
−6
,
,
1
−4
],
],
Interval analysis and Optimal Transport
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Interval analysis
Introduction to Optimal Transport
f (x) = 1 − 2x + x 2
x ∈ [2, 3]
⇒
⇒
⇒
1
−2x
x2
∈
∈
∈
[
[
[
Nicolas Delanoue
1
−6
4
,
,
,
1
−4
9
],
],
],
Interval analysis and Optimal Transport
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Interval analysis
Introduction to Optimal Transport
f (x) = 1 − 2x + x 2
x ∈ [2, 3]
⇒
⇒
⇒
1
−2x
x2
∈
∈
∈
[
[
[
1
−6
4
,
,
,
1
−4
9
],
],
],
x ∈ [2, 3]
⇒
f (x)
∈
[
−1
,
6
].
Nicolas Delanoue
Interval analysis and Optimal Transport
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Interval analysis
Introduction to Optimal Transport
Main results based on interval analysis
Interval Arithmetic, Ramon E. Moore, 1966,
Nicolas Delanoue
Interval analysis and Optimal Transport
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Interval analysis
Introduction to Optimal Transport
Main results based on interval analysis
Interval Arithmetic, Ramon E. Moore, 1966,
Global optimization, R. Baker Kearfott, 90’s,
Nicolas Delanoue
Interval analysis and Optimal Transport
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Interval analysis
Introduction to Optimal Transport
Main results based on interval analysis
Interval Arithmetic, Ramon E. Moore, 1966,
Global optimization, R. Baker Kearfott, 90’s,
Solution set of systems of equations, Arnold Neumaier,
Nicolas Delanoue
Interval analysis and Optimal Transport
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Interval analysis
Introduction to Optimal Transport
Main results based on interval analysis
Interval Arithmetic, Ramon E. Moore, 1966,
Global optimization, R. Baker Kearfott, 90’s,
Solution set of systems of equations, Arnold Neumaier,
Reliable solutions to ordinary differential equations, Rudolf
Lohner 1988,
Nicolas Delanoue
Interval analysis and Optimal Transport
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Interval analysis
Introduction to Optimal Transport
Main results based on interval analysis
Interval Arithmetic, Ramon E. Moore, 1966,
Global optimization, R. Baker Kearfott, 90’s,
Solution set of systems of equations, Arnold Neumaier,
Reliable solutions to ordinary differential equations, Rudolf
Lohner 1988,
Applied Interval Analysis (to Robotics), Luc Jaulin, 2001,
Nicolas Delanoue
Interval analysis and Optimal Transport
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Interval analysis
Introduction to Optimal Transport
Main results based on interval analysis
Interval Arithmetic, Ramon E. Moore, 1966,
Global optimization, R. Baker Kearfott, 90’s,
Solution set of systems of equations, Arnold Neumaier,
Reliable solutions to ordinary differential equations, Rudolf
Lohner 1988,
Applied Interval Analysis (to Robotics), Luc Jaulin, 2001,
Kepler conjecture proved by T. Hales in 2003,
Nicolas Delanoue
Interval analysis and Optimal Transport
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Interval analysis
Introduction to Optimal Transport
Main results based on interval analysis
Interval Arithmetic, Ramon E. Moore, 1966,
Global optimization, R. Baker Kearfott, 90’s,
Solution set of systems of equations, Arnold Neumaier,
Reliable solutions to ordinary differential equations, Rudolf
Lohner 1988,
Applied Interval Analysis (to Robotics), Luc Jaulin, 2001,
Kepler conjecture proved by T. Hales in 2003,
The Lorentz equations support a strange attractor proved by
W. Tucker in 1998.
Nicolas Delanoue
Interval analysis and Optimal Transport
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Interval analysis
Introduction to Optimal Transport
Main results based on interval analysis
Interval Arithmetic, Ramon E. Moore, 1966,
Global optimization, R. Baker Kearfott, 90’s,
Solution set of systems of equations, Arnold Neumaier,
Reliable solutions to ordinary differential equations, Rudolf
Lohner 1988,
Applied Interval Analysis (to Robotics), Luc Jaulin, 2001,
Kepler conjecture proved by T. Hales in 2003,
The Lorentz equations support a strange attractor proved by
W. Tucker in 1998.
PDE, algebraic topology, . . .
Nicolas Delanoue
Interval analysis and Optimal Transport
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Interval analysis
Introduction to Optimal Transport
For the rest of the talk :
Interval arithmetic can generate bounds
for a given map over an interval.
Nicolas Delanoue
Interval analysis and Optimal Transport
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Interval analysis
Introduction to Optimal Transport
Example with books
Nicolas Delanoue
Interval analysis and Optimal Transport
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Interval analysis
Introduction to Optimal Transport
Example with books
Example in the discrete case
Nicolas Delanoue
Interval analysis and Optimal Transport
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Interval analysis
Introduction to Optimal Transport
Example in the discrete case
Nicolas Delanoue
Interval analysis and Optimal Transport
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Interval analysis
Introduction to Optimal Transport
Example in the discrete case
Nicolas Delanoue
Interval analysis and Optimal Transport
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Interval analysis
Introduction to Optimal Transport
Example in the discrete case
Nicolas Delanoue
Interval analysis and Optimal Transport
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Interval analysis
Introduction to Optimal Transport
Example in the discrete case
Nicolas Delanoue
Interval analysis and Optimal Transport
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Interval analysis
Introduction to Optimal Transport
Example in the discrete case
µ = (2, 1, 4)
ν = (4, 2, 1)
Transportation
4 2 1
2
1
4
Nicolas Delanoue
Interval analysis and Optimal Transport
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Interval analysis
Introduction to Optimal Transport
Example in the discrete case
µ = (2, 1, 4)
ν = (4, 2, 1)
Transportation
4 2 1
2
1
4
Nicolas Delanoue
Interval analysis and Optimal Transport
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Interval analysis
Introduction to Optimal Transport
Example in the discrete case
µ = (2, 1, 4)
ν = (4, 2, 1)
A plan transference π
4 2 1
2 2 0 0
,
1 1 0 0
4 1 2 1
Nicolas Delanoue


2 0 0
π= 1 0 0 
1 2 1
Interval analysis and Optimal Transport
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Interval analysis
Introduction to Optimal Transport
Plan transference problem
2
1
4
4 2 1
. . .
. . .
. . .
Nicolas Delanoue
Interval analysis and Optimal Transport
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Interval analysis
Introduction to Optimal Transport
Plan transference problem
2
1
4
4 2 1
. . .
. . .
. . .
Solutions
4
2 2
π=
1 1
4 1
2
0
0
2
1
0
,
0
1
Nicolas Delanoue
4
2 1
π̃ =
1 1
4 2
2
0
0
2
1
1
.
0
0
Interval analysis and Optimal Transport
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Interval analysis
Introduction to Optimal Transport
Definition - Transference plan
A transference plan (or a transportation) π is a measure on the
product space X × Y such that
π(A × Y ) = µ(A),
π(X × B) = ν(B).
all measurable subsets A of X and B of Y .
Nicolas Delanoue
Interval analysis and Optimal Transport
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Interval analysis
Introduction to Optimal Transport
Definition - Transference plan
A transference plan (or a transportation) π is a measure on the
product space X × Y such that
π(A × Y ) = µ(A),
π(X × B) = ν(B).
all measurable subsets A of X and B of Y .
In the discrete case
P
∀i, j πij = µi ,
P
∀j, i πij = νj .
Nicolas Delanoue
Interval analysis and Optimal Transport
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Interval analysis
Introduction to Optimal Transport
Comparing two plan transferences
Transportation π
Transportation π̃
I (π) = 0 + 0 + 1 + 2 + 1 + 1 + 0 = 5
I (π̃) = 0 + 2 + 1 + 2 + 2 + 1 + 1 = 9
Nicolas Delanoue
Interval analysis and Optimal Transport
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Interval analysis
Introduction to Optimal Transport
In the discrete case
X
min
n
π∈R ⊗Rm
cij πij
i,j
subject to ∀i,
X
πij = µi ,
(1)
j
∀j,
X
πij = νj .
j
where cij are non negative real numbers which tells how much it
costs to transport one unit of mass from location i to location j.
Nicolas Delanoue
Interval analysis and Optimal Transport
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Interval analysis
Introduction to Optimal Transport
Kantorovich formulation
The optimal transportation cost between µ and ν is the value :
Z
c(x, y )dπ(x, y )
Tc (µ, ν) = inf
π∈B(X ×Y )
X ×Y
subject to πX = µ,
(2)
πY = ν
The optimal π’s, i.e. those such that I (π) = Tc (µ, ν), if they exist,
will be called optimal transference plans.
Nicolas Delanoue
Interval analysis and Optimal Transport
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Interval analysis
Introduction to Optimal Transport
Remark
The optimal transportation problem is an infinite dimensional
linear programming problem.
i.e. I is a linear cost function, and constraints are linear.
Nicolas Delanoue
Interval analysis and Optimal Transport
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
1
Interval analysis
Introduction to Optimal Transport
c = kx − y kp , p > 1 , the strict convexity of c guarantees
that, if µ, ν are absolutely continuous with respect to
Lebesgue measure, then there is a unique solution to the
Kantorovich problem.
Nicolas Delanoue
Interval analysis and Optimal Transport
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
1
2
Interval analysis
Introduction to Optimal Transport
c = kx − y kp , p > 1 , the strict convexity of c guarantees
that, if µ, ν are absolutely continuous with respect to
Lebesgue measure, then there is a unique solution to the
Kantorovich problem.
c = kx − y k2 , optimal transference plans are the (restrictions
of) gradients of convex functions. (Legendre-Fenchel
transformation, inf convolution)
Nicolas Delanoue
Interval analysis and Optimal Transport
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
1
2
3
Interval analysis
Introduction to Optimal Transport
c = kx − y kp , p > 1 , the strict convexity of c guarantees
that, if µ, ν are absolutely continuous with respect to
Lebesgue measure, then there is a unique solution to the
Kantorovich problem.
c = kx − y k2 , optimal transference plans are the (restrictions
of) gradients of convex functions. (Legendre-Fenchel
transformation, inf convolution)
many others in
Topics in Optimal Transportation, Cédric Villani, AMS (2003)
Nicolas Delanoue
Interval analysis and Optimal Transport
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Finite dimensional relaxation
Proposition - Relaxation
Let µ and ν (with support X and Y ) be absolutely continuous measures
with respect to Lebesgue measure. If {Xi }i and {Yi }i be finite
λ−pavings of X and Y . Suppose that µ(Xi ) ∈ [µi , µi ], ν(Yj ) ∈ [ν j , ν j ],
and ∀x, y ∈ Xi × Yj , c ij ≤ c(x, y ),
Nicolas Delanoue
Interval analysis and Optimal Transport
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Finite dimensional relaxation
Proposition - Relaxation
Let µ and ν (with support X and Y ) be absolutely continuous measures
with respect to Lebesgue measure. If {Xi }i and {Yi }i be finite
λ−pavings of X and Y . Suppose that µ(Xi ) ∈ [µi , µi ], ν(Yj ) ∈ [ν j , ν j ],
and ∀x, y ∈ Xi × Yj , c ij ≤ c(x, y ),
T =
X
min
n
πij ∈R
⊗Rm
subject to
c ij πij
i,j
∀i, µi ≤
X
πij ≤ µi ,
j
∀j, ν j ≤
X
πij ≤ ν j ,
i
∀i, ∀j, πij ≥ 0.
Nicolas Delanoue
Interval analysis and Optimal Transport
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Finite dimensional relaxation
Proposition - Relaxation
Let µ and ν (with support X and Y ) be absolutely continuous measures
with respect to Lebesgue measure. If {Xi }i and {Yi }i be finite
λ−pavings of X and Y . Suppose that µ(Xi ) ∈ [µi , µi ], ν(Yj ) ∈ [ν j , ν j ],
and ∀x, y ∈ Xi × Yj , c ij ≤ c(x, y ),
T =
X
min
n
πij ∈R
⊗Rm
subject to
c ij πij
i,j
∀i, µi ≤
X
πij ≤ µi ,
j
∀j, ν j ≤
X
πij ≤ ν j ,
i
∀i, ∀j, πij ≥ 0.
then
Nicolas Delanoue
T ≤ Tc (µ, ν).
Interval analysis and Optimal Transport
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Finite dimensional relaxation
Spatial discretization
µ
ν
Nicolas Delanoue
Interval analysis and Optimal Transport
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Finite dimensional relaxation
Spatial discretization
Nicolas Delanoue
Interval analysis and Optimal Transport
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Finite dimensional relaxation
Spatial discretization
Nicolas Delanoue
Interval analysis and Optimal Transport
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Finite dimensional relaxation
Enclosing
If µ = f (x)dλ(x), and [f ] an inclusion function for f then
Z
X
f (x)dx ∈
[f ](Xi )λ(Xi )
X
i
µi
P
i
f (Xi )λ(Xi ) ≤
µ(X )
R
X
µi
f (x)dx
Nicolas Delanoue
≤
P
i
f (Xi )λ(Xi )
Interval analysis and Optimal Transport
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Finite dimensional relaxation
Proof
Let {Xi }, {Yj } be a λ−pavings, let πij = π(Xi × Yj ) then
∀π, ∃ξij ∈ Xi × Yj ,
Z
X
c(x, y )dπ(x, y )
c(ξij )πij =
i,j
X ×Y
Nicolas Delanoue
Interval analysis and Optimal Transport
(3)
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Finite dimensional relaxation
Proof
Let {Xi }, {Yj } be a λ−pavings, let πij = π(Xi × Yj ) then
∀π, ∃ξij ∈ Xi × Yj ,
Z
X
c(x, y )dπ(x, y )
c(ξij )πij =
i,j
Since c ij ≤ c(ξij ) and πij ≥ 0, then
∀π,
Z
X
c ij πij ≤
c(x, y )dπ(x, y )
i,j
(3)
X ×Y
X ×Y
Nicolas Delanoue
Interval analysis and Optimal Transport
(4)
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Finite dimensional relaxation
Proof
Let µ and ν (with support X and Y ) be absolutely continuous measures
with respect to Lebesgue measure. If {Xi }i and {Yi }i be finite
λ−pavings of X and Y . Suppose that µ(Xi ) ∈ [µi , µi ], ν(Yj ) ∈ [ν j , ν j ],
and ∀x, y ∈ Xi × Yj , c ij ≤ c(x, y ),
K=
X
min
n
πij ∈R
⊗Rm
subject to
c ij πij
i,j
∀i, µi =
X
πij = µi ,
j
∀j, νj =
X
πij = νj ,
i
∀i, ∀j, πij ≥ 0.
then
Nicolas Delanoue
K ≤ Tc (µ, ν).
Interval analysis and Optimal Transport
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Finite dimensional relaxation
Proof
Let µ and ν (with support X and Y ) be absolutely continuous measures
with respect to Lebesgue measure. If {Xi }i and {Yi }i be finite
λ−pavings of X and Y . Suppose that µ(Xi ) ∈ [µi , µi ], ν(Yj ) ∈ [ν j , ν j ],
and ∀x, y ∈ Xi × Yj , c ij ≤ c(x, y ),
T =
X
min
n
πij ∈R
⊗Rm
subject to
c ij πij
i,j
∀i, µi ≤
X
πij ≤ µi ,
j
∀j, ν j ≤
X
πij ≤ ν j ,
i
∀i, ∀j, πij ≥ 0.
then
Nicolas Delanoue
T ≤ Tc (µ, ν).
Interval analysis and Optimal Transport
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Finite dimensional relaxation
Example
X = Y = [0, 1]
1.6
1.6
1.4
1.4
1.2
1.2
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
ν = 1.5(1 − y 2 )dy
µ = 1dx
c(x, y ) = kx − y k2
Nicolas Delanoue
Interval analysis and Optimal Transport
1
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Finite dimensional relaxation
20
18
16
14
12
10
8
6
4
0
50
100
150
200
250
300
350
Figure : Guaranteed lower bounds of Tc (µ, ν) where n = Card{Xi }i
Nicolas Delanoue
Interval analysis and Optimal Transport
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Finite dimensional relaxation
Outline
1
Introduction
Interval analysis
Introduction to Optimal Transport
Transportation
Optimal Transport
Some known results
2
A lower bound of the optimal value
Finite dimensional relaxation
3
An upper bound of the optimal value
Duality
Finite dimensional relaxation
4
Conclusion - Future work
Nicolas Delanoue
Interval analysis and Optimal Transport
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Duality
Finite dimensional relaxation
Linear programming - Duality
Primal problem
min
x∈Rn
cT x
subject to Ax = b,
x ≥ 0.
Nicolas Delanoue
Interval analysis and Optimal Transport
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Duality
Finite dimensional relaxation
Linear programming - Duality
Primal problem
min
x∈Rn
cT x
subject to Ax = b,
x ≥ 0.
Dual problem
max
y ∈Rm
bT y
subject to yi ∈ R,
AT y ≤ c.
Nicolas Delanoue
Interval analysis and Optimal Transport
(5)
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Duality
Finite dimensional relaxation
Duality
Z
inf
c(x, y )dπ(x, y )
π∈B(X ×Y )
X ×Y
subject to
πX = µ,
πY = ν
Z
sup
Z
ϕ(x) dµ(x) +
φ,ψ∈Cb (X ,Y )
subject to
X
ψ(y ) dν(y )
Y
ϕ(x) + ψ(y ) ≤ c(x, y ).
Nicolas Delanoue
Interval analysis and Optimal Transport
(6)
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Duality
Finite dimensional relaxation
Duality
Z
inf
c(x, y )dπ(x, y )
π∈B(X ×Y )
X ×Y
subject to
πX = µ,
πY = ν
Z
sup
Z
ϕ(x) dµ(x) +
φ,ψ∈Cb (X ,Y )
subject to
X
ψ(y ) dν(y )
Y
(6)
ϕ(x) + ψ(y ) ≤ c(x, y ).
where Cb (X , Y ) denotes the set of all pairs of bounded and continuous
functions φ : X → R and ψ : Y → R.
Nicolas Delanoue
Interval analysis and Optimal Transport
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Duality
Finite dimensional relaxation
Duality
Z
inf
c(x, y )dπ(x, y )
π∈B(X ×Y )
X ×Y
subject to
πX = µ,
πY = ν
Z
sup
Z
ϕ(x) dµ(x) +
φ,ψ∈Cb (X ,Y )
subject to
X
ψ(y ) dν(y )
Y
(6)
ϕ(x) + ψ(y ) ≤ c(x, y ).
where Cb (X , Y ) denotes the set of all pairs of bounded and continuous
functions φ : X → R and ψ : Y → R.
If X is compact and Haussdorff, Cb (X )∗ = {Radon measure}
Nicolas Delanoue
Interval analysis and Optimal Transport
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Duality
Finite dimensional relaxation
Kantorovich Duality
The minimum of the Kantorovich problem is equal to
Z
Z
Tc (µ, ν) =
sup
ϕ(x) dµ(x) +
ψ(y ) dν(y )
φ,ψ∈Cb (X ,Y )
subject to
X
Y
ϕ(x) + ψ(y ) ≤ c(x, y ).
Nicolas Delanoue
Interval analysis and Optimal Transport
(7)
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Duality
Finite dimensional relaxation
Interpretation in the discrete case
X
X
sup
φ i µi +
ψj νi
(φi )∈Rn ,(ψj )∈Rm
subject to
Nicolas Delanoue
i
j
φi + ψj ≤ cij
Interval analysis and Optimal Transport
(8)
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Duality
Finite dimensional relaxation
Interpretation in the discrete case
X
X
sup
φ i µi +
ψj νi
(φi )∈Rn ,(ψj )∈Rm
subject to
Nicolas Delanoue
i
j
φi + ψj ≤ cij
Interval analysis and Optimal Transport
(9)
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Duality
Finite dimensional relaxation
Interpretation in the discrete case
X
X
sup
φ i µi +
ψj νi
(φi )∈Rn ,(ψj )∈Rm
subject to
Nicolas Delanoue
i
j
φi + ψj ≤ cij
Interval analysis and Optimal Transport
(10)
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Duality
Finite dimensional relaxation
Proposition - Relaxation
Let µ and ν (with support X and Y ) be absolutely continuous measures
with respect to Lebesgue measure. If {Xi }i and {Yi }i be finite
λ−pavings of X and Y . Suppose that µ(Xi ) ∈ [µi , µi ], ν(Yj ) ∈ [ν j , ν j ],
and ∀x, y ∈ Xi × Yj , c(x, y ) ≤ c ij ,
Nicolas Delanoue
Interval analysis and Optimal Transport
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Duality
Finite dimensional relaxation
Proposition - Relaxation
Let µ and ν (with support X and Y ) be absolutely continuous measures
with respect to Lebesgue measure. If {Xi }i and {Yi }i be finite
λ−pavings of X and Y . Suppose that µ(Xi ) ∈ [µi , µi ], ν(Yj ) ∈ [ν j , ν j ],
and ∀x, y ∈ Xi × Yj , c(x, y ) ≤ c ij ,
T =
sup
(φi )∈Rn ,(ψj )∈Rm
subject to
Nicolas Delanoue
X
φi µ i +
X
i
ψj ν i
j
φi + ψj ≤ c ij
Interval analysis and Optimal Transport
(11)
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Duality
Finite dimensional relaxation
Proposition - Relaxation
Let µ and ν (with support X and Y ) be absolutely continuous measures
with respect to Lebesgue measure. If {Xi }i and {Yi }i be finite
λ−pavings of X and Y . Suppose that µ(Xi ) ∈ [µi , µi ], ν(Yj ) ∈ [ν j , ν j ],
and ∀x, y ∈ Xi × Yj , c(x, y ) ≤ c ij ,
T =
sup
(φi )∈Rn ,(ψj )∈Rm
subject to
then
Nicolas Delanoue
X
φi µ i +
X
i
ψj ν i
j
φi + ψj ≤ c ij
Tc (µ, ν) ≤ T .
Interval analysis and Optimal Transport
(11)
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Duality
Finite dimensional relaxation
60
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Figure : Guaranteed upper bounds of Tc (µ, ν) where n = Card{Xi }i
Nicolas Delanoue
Interval analysis and Optimal Transport
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Software
filib - FI LIB - A fast interval library,
http://www2.math.uni-wuppertal.de/~xsc/software/filib.html
GLPK - GNU Linear Programming Kit (GLPK),
http://www.gnu.org/software/glpk/
GMP - GNU Multiple Precision Arithmetic Library,
https://gmplib.org/
Source code is available on my webpage.
Nicolas Delanoue
Interval analysis and Optimal Transport
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Future work
Compute guaranteed enclosures of the solution combining
linear programming and constraint propagation.
Nicolas Delanoue
Interval analysis and Optimal Transport
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Future work
Compute guaranteed enclosures of the solution combining
linear programming and constraint propagation.
Generalize this methodology to other problems (D. Henrion &
J.B. Lasserre):
Nicolas Delanoue
Interval analysis and Optimal Transport
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Future work
Compute guaranteed enclosures of the solution combining
linear programming and constraint propagation.
Generalize this methodology to other problems (D. Henrion &
J.B. Lasserre):
Probability and Markov Chains
Nicolas Delanoue
Interval analysis and Optimal Transport
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Future work
Compute guaranteed enclosures of the solution combining
linear programming and constraint propagation.
Generalize this methodology to other problems (D. Henrion &
J.B. Lasserre):
Probability and Markov Chains
Optimal Control with occupation measures (ODE),
Nicolas Delanoue
Interval analysis and Optimal Transport
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Future work
Compute guaranteed enclosures of the solution combining
linear programming and constraint propagation.
Generalize this methodology to other problems (D. Henrion &
J.B. Lasserre):
Probability and Markov Chains
Optimal Control with occupation measures (ODE),
Others as in Moments, Positive Polynomials and Their
Applications, J.B Lasserre, Imperial College Press Optimization
Series (2009)
Nicolas Delanoue
Interval analysis and Optimal Transport
Introduction
A lower bound of the optimal value
An upper bound of the optimal value
Conclusion - Future work
Future work
Compute guaranteed enclosures of the solution combining
linear programming and constraint propagation.
Generalize this methodology to other problems (D. Henrion &
J.B. Lasserre):
Probability and Markov Chains
Optimal Control with occupation measures (ODE),
Others as in Moments, Positive Polynomials and Their
Applications, J.B Lasserre, Imperial College Press Optimization
Series (2009)
Merci pour votre attention.
Nicolas Delanoue
Interval analysis and Optimal Transport