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Remarks on the Monge–Kantorovich problem in the

discrete setting

Haim Brezis

To cite this version:

Haim Brezis. Remarks on the Monge–Kantorovich problem in the discrete setting. Comptes

Ren-dus Mathématique, Elsevier Masson, 2018, 356 (2), pp.207-213. �10.1016/j.crma.2017.12.008�.

�hal-01727357�

Contents lists available atScienceDirect

C. R.

Acad.

Sci.

Paris,

Ser. I

www.sciencedirect.com

Functional analysis/Mathematical analysis

Remarks

on

the

Monge–Kantorovich

problem

in

the

discrete

setting

Remarques

sur

le

problème

de

Monge–Kantorovich

dans

le

cas

discret

Haïm Brezis

a,b,c

a_{DepartmentofMathematics,HillCenter,BuschCampus,RutgersUniversity,110FrelinghuysenRoad,Piscataway,NJ08854,USA} b_{DepartmentsofMathematicsandComputerScience,Technion,IsraelInstituteofTechnology,32000Haifa,Israel}

c_{LaboratoireJacques-Louis-Lions,UniversitéPierre-et-Marie-Curie,4,placeJussieu,75252Pariscedex05,France}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received19December2017 Accepted19December2017 Availableonline5January2018 PresentedbyHaïmBrezis

In Optimal Transport theory, three quantities play a central role: the minimal cost of transport,originallyintroducedbyMonge,itsrelaxedversionintroducedbyKantorovich, and adualformulation alsoduetoKantorovich. The goalofthisNote isto publicizea veryelementary,self-containedargument extractedfrom[9],whichshowsthatallthree quantitiescoincideinthediscretecase.

©2017Académiedessciences.PublishedbyElsevierMassonSAS.Thisisanopenaccess articleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

r

é

s

u

m

é

Enthéoriedutransportoptimal,troisquantités jouentunrôlecentral :lecoûtminimal de transport, introduit par Monge, sa version relaxée, introduitepar Kantorovich, et la formulationduale,dueaussiàKantorovich. L’objetdecettenoteest demettreen avant unedémonstrationtotalementélémentaire,extraitede[9],dufaitquecestroisquantités coïncidentdanslecasdiscret ;cettepreuvenerequiertaucuneconnaissancepréalable.

©2017Académiedessciences.PublishedbyElsevierMassonSAS.Thisisanopenaccess articleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

Considertwosets X

,

Y consistingofm points

(

Pi

)

and

(

Ni

),

1

≤

i

≤

m,i.e.

X

= {

Pi

,

P2

, . . . ,

Pm

}

and Y

= {

N1

,

N2

, . . . ,

Nm

} .

Letc

:

X

×

Y

→ R

beanyfunction(c standsfor“cost”).Weintroducethreequantities.Thefirstonedenoted M (forMonge) isdefinedby

E-mailaddress:brezis@math.rutgers.edu.

https://doi.org/10.1016/j.crma.2017.12.008

1631-073X/©2017Académiedessciences.PublishedbyElsevierMassonSAS.ThisisanopenaccessarticleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

208 H. Brezis / C. R. Acad. Sci. Paris, Ser. I 356 (2018) 207–213 M

:=

min σ∈Sm m



i=1 c

(

Pi

,

Nσ(i)

),

(1)

wheretheminimumistakenovertheset

S

m ofallpermutationsoftheintegers

{

1

,

2

,

. . . ,

m

}

.Thesecondone,denoted K

(forKantorovich),isdefinedby

K

:=

min A

⎧

⎨

⎩

m



i,j=1 ai jc

(

Pi

,

Nj

)

;

A

= (

ai j

)

is doubly stochastic

⎫

⎬

⎭

.

(2)

Recallthatamatrix A

= (

ai j

)

isdoublystochasticif

ai j

≥

0

∀

i

,

j

,

m



i=1 ai j

=

1

∀

j

,

and m



j=1 ai j

=

1

∀

i

.

(3)

Finallydefine D (forduality)by

D

:=

sup ψ:Y→R ϕ:X→R

_m



i=1

(

ϕ

(

Pi

)

− ψ(

Ni

))

;

ϕ

(

x

)

− ψ(

y

)

≤

c

(

x

,

y

),

∀

x

∈

X

,

∀

y

∈

Y

.

(4) Theorem1.1.Wehave M

=

K

=

D

.

(5)

Moreoverthe“sup”in(4)isachieved.

Equality K

=

D inTheorem 1.1isattheheartofKantorovich’spioneeringdiscoveryconcerningtheMongeproblem(see

[19] and[20]).Equality M

=

K makes totallytransparentthe connectionbetweenKantorovich’sformulationandMonge’s original goal (see item (3)in Section 4below). The purposeofthis note isto advertise the MK(

=

Monge–Kantorovich) theory initsmostelementary(butinitselfstrikinganduseful!) setting,asitappears,e.g.,inBrezis–Coron–Lieb [11] (see Section 3anditem (1) inSection 4 below). This“primitive” caseilluminates thefoundations ofthe MK sagawhich has “exploded” inrecent years;see, e.g.,the remarkableworksof [2], [3],[7],[15],[16],[17], [23], [24],[29],[34],[35], etc. I reproduceinSection2anelementaryself-containedproofofTheorem 1.1(accessibletofirst-yearstudents),extractedfrom apresentationof[11]thatIgavein1985(see[9]).

2. ProofofTheorem 1.1

ChoosingforA in(2)apermutationmatrixyields

K

≤

M

.

(6)

Ontheotherhand,assumethat

ϕ

and

ψ

areasin(4).LetA

= (

ai j

)

beadoublystochasticmatrix.Multiplyingthe

inequal-ities

ϕ

(

Pi

)

− ψ(

Nj

)

≤

c

(

Pi

,

Nj

)

byai j andsummingoveri

,

j yields m



i=1

(

ϕ

(

Pi

)

− ψ(

Ni

))

≤

m



i,j=1 ai jc

(

Pi

,

Nj

).

(7)

MinimizingoverA andmaximizingover

ϕ

,

ψ

gives

D

≤

K

.

(8)

Inviewof(6)and(8),itsufficestoestablishthat

M

≤

D

.

(9)

Proofof(9). Withoutlossofgeneralitywemayrelabelthepoints

(

Nj

)

sothat

M

=

m



i=1 c

(

Pi

,

Ni

)

≤

m



j=1 c

(

Pj

,

Nσ(j)

)

∀

σ

∈

S

m

.

(10)

m



i=1

(

ϕ

(

Pi

)

− ψ(

Ni

))

≥

M (11) and

ϕ

(

Pi

)

− ψ(

Nj

)

≤

c

(

Pi

,

Nj

)

∀

i

,

j

.

(12) Set di

=

c

(

Pi

,

Ni

)

∀

i (13) and bi j

=

c

(

Pi

,

Nj

)

−

di

=

c

(

Pi

,

Nj

)

−

c

(

Pi

,

Ni

)

∀

i

,

j

.

(14)

Considerthenumbers

λ

i

= ψ(

Ni

),

1

≤

i

≤

m,asbeingtheunknowns.Oncethe

λ

i’shavebeendeterminedset

ϕ

(

Pi

)

= ψ(

Ni

)

+

di

= λ

i

+

di

∀

i

.

(15)

(Thischoiceisdictatedby(10),(11),and(12)appliedwithi

=

j.)From(15),(13)and(10)weseethat(11)holds.Wenow rewrite(12)as

λ

i

− λ

j

≤

bi j

∀

i

,

j

.

(16)

Notethatby(13)and(14)

bii

=

0 1

≤

i

≤

m

,

(17)

andthatthehypothesis(10)reads

m



j=1

bjσ(j)

≥

0

∀

σ

∈

S

m

.

(18)

We complete the proof of (12) (and thus the existence of functions

ϕ

and

ψ

satisfying (11)–(12)) via the next lemma essentiallyduetoAfriat[1].

Lemma2.1.Assumethat

(

bi j

)

isageneralmatrixsatisfying(17)–(18).Thenthesystemofinequalities(16)admitsasolution.

Proof. (Copiedfrom[9],inspiredby[1]).Wefirstproposeanansatzforthe

λ

i’sandthenprovethatthisansatzhasallthe

requiredproperties.Achain K connectingi to j isafinitesequenceK

= (

i1

,

. . . ,

ik

)

suchthatk

≥

2

,

il

∈ {

1

,

. . . ,

m

} ∀

l,i1

=

i,

andik

=

j.(Wedonotassumethati1

,

. . . ,

ikaredistinct.)

GivenachainK connectingi

=

i1 to j

=

ik,set

SK

:=

bi1i2

+

bi2i3

+ · · · +

bik−1ik

.

(19)

Suppose nowthatasolution

(λ

i

)

to(16)existsandconsiderachain K connectingi to j.Wehave

λ

i1

− λ

i2

≤

bi1i2

,

λ

i2

− λ

i3

≤

bi2i3

,

. . .

λ

ik−1

− λ

ik

≤

bik−1ik

.

Addingtheseinequalitiesyields

λ

i

− λ

j

≤

SK

,

(20) andinparticular

λ

i

− λ

1

≤

inf K



SK

;

K is a chain connecting i to 1



.

(21)

Wenowturntotheexistence ofasolution

(λ

i

)

to(16).Sincethe

λ

i’saredefinedmoduloanadditiveconstantitistempting,

210 H. Brezis / C. R. Acad. Sci. Paris, Ser. I 356 (2018) 207–213

λ

i

:=

inf K



SK

;

K is a chain connecting i to 1



.

(22)

(A priori,it mayhappen that

λ

i

= −∞

,butthiswillbe excluded below.) Fix1

≤

i

,

j

≤

m andlet K

= (

i1

,

. . . ,

ik

)

beany

chainconnecting j to1,then K

˜

:= (

i

,

K

)

connectsi to1 andtherefore(by(22))

λ

i

≤

SK˜

=

bi j

+

SK

.

(23)

TakingtheinfoverK in(23)weobtain

λ

i

≤

bi j

+ λ

j

∀

1

≤

i

,

j

≤

m

.

(24)

This correspondsto thedesiredinequality (16)providedwe establishthat

λ

j

= −∞

∀

j; assumptions (17)and(18)enter

here.Wewillprovethat

λ

1

=

0

.

(25)

Then,combining(24)and(25)wededucethat

0

= λ

1

≤

b1 j

+ λ

j

∀

j

andthus

λ

j

= −∞

∀

j.Wenowturntotheproofof(25).First,wechoosethechain K

= (

1

,

1

)

in(22)andobtain

λ

1

≤

b11

=

0

.

(26)

Nextweestablishthat

λ

1

≥

0.Westartwithsometerminology.AchainK connectingi to j

=

i iscalledacycle (oraloop).

Acycleissimple ifi1

, . . . ,

ik−1aredistinct.Weclaimthat,foreverycycleK ,

SK

≥

0

.

(27)

Indeedwhen K isasimplecycle(27)followsfrom(18)(and(17)) appliedtothepermutation i1

→

i2

· · · →

ik (theother

integers are invariant).By decomposinga generalcycleinto simplecycleswe findthat(27) holdsforallcycles.Applying

(27)toanychainconnecting1 to1,wededucefrom(22)that

λ

1

≥

0.

2

Remark2.1. The aboveproof providesinfact anecessaryandsufficient conditionfortheexistence ofasolutionto (16).It readsasfollows.



j∈B

bjσ(j)

≥

0

,

∀

σ

∈

S

k

,

(28)

foreveryinteger1

≤

k

≤

m andforeverysubset B of

{

1

,

. . . ,

m

}

containingk distinctelements,wherethepermutations

σ

actonlyon B.Thisresultappearsalreadyin[1]asaconsequenceofTheorems3.1and7.2in[1].Unfortunately,theproofs in[1]areobscuredbyaflurryofdefinitions!

3. Whenthecostc isadistance

WenowpresentasimpleconsequenceofTheorem 1.1whenthecostc isadistance,whichcorrespondstothesettingof

[20].Letd

(

x

,

y

)

beapseudometric(i.e.thedistancebetweentwodistinctpointscanbezero)onasetZ .Let

(

Pi

),

(

Ni

),

1

≤

i

≤

m bepointsinZ suchthat Pi

=

Nj

∀

i

,

j (butitmayhappenthat Pi

=

PjorNi

=

Njforsome i

=

j).

Corollary3.1.Wehave DLip

:=

sup ζ

_m



i=1

(ζ (

Pi

)

− ζ(

Ni

))

; ζ :

Z

→ R, |ζ(

x

)

− ζ(

y

)

| ≤

d

(

x

,

y

)

∀

x

,

y

∈

Z

=

min σ∈Sm m



i=1 d

(

Pi

,

Nσ(i)

)

=

M

.

Proof. Clearly DLip

≤

M.Afterrelabelingthepoints

(

Nj

)

wemayassume,asin(10),that

M

=

m



i=1 d

(

Pi

,

Ni

)

≤

m



j=1 d

(

Pj

,

Nσ(j)

)

∀

σ

∈

S

m

.

(29)

ApplyingTheorem 1.1with X

= {

P1

,

P2

,

. . . ,

Pm

},

Y

= {

N1

,

N2

,

. . . ,

Nm

}

andc

(

x

,

y

)

=

d

(

x

,

y

)

weknowthat M

=

D;thuswe

obtainfunctions

ϕ

:

X

→ R

and

ψ

:

Y

→ R

suchthat

ϕ

(

Pi

)

− ψ(

Ni

)

=

d

(

Pi

,

Ni

)

∀

i

,

(30)

and

ϕ

(

Pi

)

− ψ(

Nj

)

≤

d

(

Pi

,

Nj

)

∀

i

,

j

.

(31)

Weclaimthat

|ψ(

Ni

)

− ψ(

Nj

)

| ≤

d

(

Ni

,

Nj

)

∀

i

,

j

.

(32)

Indeed,by(30),(31),andthetriangleinequalitywehave

ψ (

Ni

)

=

ϕ

(

Pi

)

−

d

(

Pi

,

Ni

)

≤ ψ(

Nj

)

+

d

(

Pi

,

Nj

)

−

d

(

Pi

,

Ni

)

≤ ψ(

Nj

)

+

d

(

Ni

,

Nj

),

whichimplies(32).Set,forz

∈

Z ,

ζ

0

(

z

)

=

inf j

ψ (

Nj

)

+

d

(

z

,

Nj

)



,

(33) sothat

|ζ

0

(

x

)

− ζ

0

(

y

)

| ≤

d

(

x

,

y

)

∀

x

,

y

∈

Z

.

From(32)weseethat

ζ

0

(

Ni

)

= ψ(

Ni

)

∀

i

.

(34)

Ontheotherhand,wehave,by(33)and(31),

ζ

0

(

Pi

)

≥

ϕ

(

Pi

)

∀

i,whiletaking j

=

i in(33),andusing(30),yields

ζ

0

(

Pi

)

≤

d

(

Pi

,

Ni

)

+ ψ(

Ni

)

=

ϕ

(

Pi

)

∀

i

.

Therefore,

ζ

0

(

Pi

)

=

ϕ

(

Pi

)

∀

i

.

(35)

Choosing

ζ

= ζ

0 inthedefinitionofDLipandapplying(30),(34),and(35)yields DLip

≥

M.

2

4. Finalcomments

(1) Corollary 3.1istakenfrom[11].Equality M

=

DLip plays animportantrole inprovingthat the“leastenergyrequired

toproduceprescribedsingularities”(inliquidcrystals)coincideswiththe“lengthofaminimalconnectionconnecting thesesingularities”(for subsequentdevelopmentssee,e.g., [5],[8], [12],[13] and[28]). Theproof ofCorollary 3.1 in

[11]takesafewlines,butitreliesheavilyonthreenontrivialtools.TheequalityK

=

DLipisderivedfromKantorovich’s

duality(seeitem(2)below).WhiletheequalityM

=

K reliesonBirkhoff’s theorem[6]ondoublystochastic matrices (also calledBirkhoff–von Neumann’s theorembecause von Neumann [36] rediscovered itindependently a few years later).Itassertsthattheextremepointsoftheconvexset A ofdoublystochasticmatricesarepreciselythepermutation matrices.Applying theKrein–Milman theorem,onededuces thatanymatrixin A is aconvexcombinationof permu-tationmatrices, andconsequently K

≥

M. (ForrecentdevelopmentsrelatedtoBirkhoff’stheorem,Ireferthereaderto

[22]and[14].) Bycontrast,theaboveproofofCorollary 3.1iselementaryandself-contained.Noprerequisiteisneeded andmoreoverityieldsthetwoequalitiesM

=

K andK

=

D inasingleshot!

(2) Equality K

=

D in Theorem 1.1 is at the heart of Kantorovich’sdiscovery (dating back to the late 1930s – see the referencesin[33])andgoesfarbeyond thediscretesettingconsideredhere.NotethatK andD involvetheminimization (resp.maximization)of linearfunctionals onconvexsets. Themostcommonwayto show that K

=

D isvia duality, either inthe sense of linearprogramming orin the sense ofconjugate convex functions(applying for examplethe theoremofFenchel–Rockafellar;see,e.g.,TheoremI.12in[10]).Ireferthereaderto[2],[3],[15],[17],[23],[24],[29],

[34],[35],etc.

(3) According to A. M. Vershik (personal communication), equality M

=

K was known to L. V. Kantorovich, based on Birkhoff’stheorem(seeitem(1)above),whichwas publishedaroundthesametimeas[19]-[20].ApparentlyBirkhoff’s ideas were intheairsince a precursorofBirkhoff’s theoremappeared alreadyin1931(see the historicalnote onp. 25of[14]).Surprisingly,Birkhoff’stheoremishardlyevermentionedinthevastMKliterature.Thereasonforitbeing thattheMKcommunityhasbeenmostlypreoccupiedwiththeequalityM

=

K inthenon-atomic case;inthissetting, theMongeformulationwasnotevenpreciselystateduntilthe1970swhenitwasposedexplicitlyinmoderntermsby A.M. Vershik[32] (seealso[7]). Intheir rushtothecontinuum case, theMK aficionadospaid littleattentiontothe discretecase—whichisinitselfstrikinganduseful!!

212 H. Brezis / C. R. Acad. Sci. Paris, Ser. I 356 (2018) 207–213

(4) Asalreadymentioned, ourelementaryproof of Theorem 1.1doesnot require anyofthetools described initems(1) and(2)above.Instead,itreliesontheconstruction(22)(copiedfrom[9])involving“chains”and“cycles”.Thisdeviceis reminiscentofRockafellar’scelebratedtheorem[26] oncyclicallymonotone operators.Thesameconstruction appears subsequently, atthesuggestion ofRockafellar,in[25] inthe contextof MathematicalEconomics,andthen in[31] in the MK context. In [31], Smith and Knott introduced the terminology “c-cyclical monotonicity”, which has become very fashionable inthe MK community,see [2], [3], [4], [15], [17], [21], [23],[24],[27],[30],[29],[34],[35], etc. In the literature, one can find two distinct definitions. The original definition says that if X

,

Y are arbitrary sets and c

:

X

×

Y

→ R

isanyfunction,thenaset



⊂

X

×

Y isc-cyclicallymonotoneifforeveryintegern,andforanyfinite sequence

(

xi

,

yi

)

,1

≤

i

≤

n,ofpointsin



(notnecessarilydistinct),onehas

n



i=1

{

c

(

xi

,

yi+1

)

−

c

(

xi

,

yi

)

} ≥

0

,

(36)

where yn+1

:=

y1.Inanotherdefinition,(36)isreplacedby

n



i=1

c

(

xi

,

yσ(i)

)

−

c

(

xi

,

yi

)



≥

0

∀

σ

∈

S

n

.

(37)

In fact,thetwodefinitions areequivalent. Clearly,(37)implies(36)(justchoose

σ

(

i

)

=

i

+

1 when1

≤

i

≤

n

−

1 and

σ

(

n

)

=

1).Forthereverseimplication,wereturntotheproof ofLemma 2.1withbi j

=

c

(

xi

,

yj

)

−

c

(

xi

,

yi

)

.Weclaimthat

foreverycycleK

= (

i1

,

i2

,

. . . ,

ik−1

,

i1

)

,onehasSK

≥

0 (sothattheconclusionofLemma 2.1holds,andclearlyimplies (37)).Applying (36)to

(

xi1

,

yi1

),

. . . ,

(

xik−1

,

yik−1

)

(instead of

(

xi

,

yi

)

) yields SK

≥

0.Ifwetake



= (

Pi

,

Ni

),

1

≤

i

≤

m,

asinthesettingofTheorem 1.1,assumption(37)seems(atleastformally)strongerthanassumption(18)inLemma 2.1

because (37)is assumedforall finite sequences

(

xi

,

yi

)

in



,andmoreover thesepoints are not necessarilydistinct

—buttheconclusionsarethe sameandthusthetwoassumptions areaposterioriequivalent!Finally,observethat if X

=

Y

=

H isaHilbertspaceandc

(

x

,

y

)

= |

x

−

y

|

2_{,then aset}

_

_⊂

_H

_×

_{H isc-cyclically monotoneifandonlyifitis}

cyclicallymonotoneintheusualsense(coinedbyRockafellar).

(5) E.Ghys[18]andA.Vershik[33] tellthefascinatingstoriesoftheMongeandKantorovichdiscoveries.Ihighly recom-mendthesepapers.

Acknowledgements

I thankP. Mironescu for catchinga smalloversight in [9]andL. Ambrosio forproviding usefulreferences.I am very gratefultoA.M.VershikforsharingwithmeinformationsconcerningthehistoryandlaterdevelopmentsoftheMKtheory, andtoL.C.Evansforhiswarmencouragements.

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