Damped Arrow-Hurwicz algorithm for sphere packing

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Pierre Degond, Marina Ferreira, Sébastien Motsch

To cite this version:

Pierre Degond, Marina Ferreira, Sébastien Motsch. Damped Arrow-Hurwicz algorithm for sphere packing. Journal of Computational Physics, Elsevier, 2017, 332, pp.47-65. �10.1016/j.jcp.2016.11.047�.

�hal-02499452�

Contents lists available atScienceDirect

Journal of Computational Physics

www.elsevier.com/locate/jcp

Damped Arrow–Hurwicz algorithm for sphere packing

Pierre Degond

^a

, Marina A. Ferreira

^a

, Sebastien Motsch

^b

aDepartmentofMathematics,SouthKensingtonCampus,ImperialCollegeLondon,SW72AZLondon,UnitedKingdom bSchoolofMathematical&StatisticalSciences,ArizonaStateUniversity,Tempe,AZ85287-1804,UnitedStates

a r t i c l e i n f o a b s t r a c t

Articlehistory:

Received16May2016

Receivedinrevisedform28October2016 Accepted30November2016

Availableonline1December2016 Keywords:

Non-convexminimizationproblem Spherepackingproblem Non-overlappingconstraints

We consider algorithms that, from an arbitrarily sampling of N spheres (possibly overlapping),findaclosepackedconfiguration withoutoverlapping. Theseproblemscan be formulatedas minimization problemswithnon-convex constraints.For suchpacking problems, we observe that the classical iterative Arrow–Hurwicz algorithm does not converge. Wederive anovel algorithmfromamulti-stepvariant oftheArrow–Hurwicz schemewithdamping.Wecomparethisalgorithmwithclassicalalgorithms belongingto the classof linearly constrained Lagrangian methods and show that it performs better.

Weprovideananalysisoftheconvergenceofthesealgorithmsinthesimplecaseoftwo spheresinone spatialdimension.Finally,weinvestigatethe behaviourofour algorithm whenthenumberofspheresislargeintwoandthreespatialdimensions.

©^{2016TheAuthors.PublishedbyElsevierInc.ThisisanopenaccessarticleundertheCC} BY-NC-NDlicense(http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

Particle packing problems can be encountered in many different systems, from the formation of planets or cells in livetissues tothedynamicsof crowdsofpeople.Theyhavebeenwidely investigatedinthestudyofgranular media [1], glasses[2]andliquids [3]. Morerecently, particlepackings haverevealed tobe importanttoolsin biology[4]andsocial sciences,inparticularincrowddynamics[5].

Packing problems give rise to NP-hard non-convex optimization problems [6] and the optimalsolution is in general notunique,sincepermutations,rotationsorreflectionsmaygenerateequivalentsolutions.Wereferthereaderto[6]fora reviewonpackingproblems.Intheliterature,onecanfindnumericalstudiesinvolvingparticleswithvariousshapessuchas ellipses[7]orevennon-convexparticles[8].However,inthepresentworkweassumethattheparticlesaresimplyidentical spheres withdiameterd in

R

^b,b

=

¹

,

2

,

3,butthemethodologyis generalandwill beextended toother casesin future work.Weconsideralgorithmsthat,givenaninitialconfigurationofN spheres(possiblyoverlapping),findanearbypacked configurationwithoutoverlapping. Indeed,inmanynaturalsystemsindividualsorparticles onlyseektoachieve alocally optimalsolution.Therefore,itismore likelythatthey reacha localconfigurationthat doesnot necessarilycorrespondto aglobaloptimum.Bycombiningourmethodwith, forexample,simulatedannealingtechniques[9],wecouldconvertour algorithmsintoglobalminimumsearchalgorithms.Itishowevernottheobjectivewepursuehere.

Classicalprocedures to solve non-convexminimization problemsinclude Uzawa–Arrow–Hurwicz type algorithms [10], augmentedLagrangian[11,12],linearlyconstrainedLagrangian(LCL),sequentialquadraticprogramming(SQP)[13],among others.The SQPandthe Uzawa–Arrow–Hurwiczalgorithms are widelyused.However they requiretheHessianmatrixof thefunctiontobeminimizedtobepositivedefinite,whichisnotalwaysthecaseinthistypeofproblems(seetheexample

E-mailaddress:pierre.degond@gmail.com(P. Degond).

http://dx.doi.org/10.1016/j.jcp.2016.11.047

0021-9991/©^{2016TheAuthors.PublishedbyElsevierInc.ThisisanopenaccessarticleundertheCCBY-NC-NDlicense} (http://creativecommons.org/licenses/by-nc-nd/4.0/).

presented in section 3). In general, all thesemethods perform well witha small numberof particles. However we are interestedinthecasewherethisnumberbecomeslarge.

In [14,15] the authors studythe shape of three dimensional clusters of atomsunder the effect of softpotentials by using moleculardynamics.Thisapproachdiffers fromourswithregard tothenon-overlapping constraints,whichare ap- proximatedbysoftpotentials,producingsoftdynamics.Althoughbeingmorecostlywhendealingwithalargenumberof particles,wehaveoptedbytheharddynamicsapproach,sinceitallowsforahigherprecisioninthetreatmentofthecon- straints.Thisproveseffectivewhendealingwithinteractionbetweenrigidbodies,wheretheeffectoftherigidboundaries playsanimportantrole.Thismotivatesthepresentwork.

We start insection 2.1bypresenting twoformulations oftheproblem. Thefirst oneis theclassical minimizationap- proach. The second one considers a constraineddynamical systemin the spirit of [16]. We also presenttwo equivalent types ofnon-overlapping constraints involvingsmooth ornon-smoothfunctionswhich are found inthe literature [6,17].

Tosolvethenon-convexminimization problemsarising intheseformulations,wefirstconsiderinsection2.2theclassical Arrow–Hurwiczalgorithm(AHA). Insection 2.3we introduceanovelmulti-stepschemebasedonasecond-orderODEin- terpretation oftheminimization problem:thedamped Arrow–Hurwiczalgorithm(DAHA). We testthe DAHAagainsttwo methodstakenfromthewidelyknownclassoflinearlyconstrainedLagrangianalgorithms[13,16].Thesealgorithmsconsist ofa sequenceofconvexminimization problems,forwe refertothem asnestedalgorithms(NA)andthey shallbereferred to astheNAPandNAV. Theconvergenceofthefour algorithms(the AHA,DAHA,NAPandNAV)is analyzedinsection 3 andAppendix Aforthecaseoftwospheresinonedimension.Insection4thealgorithmsarenumericallycomparedforthe casesofmanyspheresintwo dimensions.Abriefnumericalstudyofthepackingdensityintwoandthreedimensionsis alsopresented.Finally,conclusionsandfutureworksarepresentedinsection5.Wealsoreferto[18]foradetailedanalysis oftheminimizationproblem.InparticularweprovethatminimizersarenotsaddlepointsoftheLagrangian.Thisanalysis requiresdevelopmentthatarebeyondthescopeofthepaper.

2. ThedampedArrow–Hurwiczalgorithm(DAHA)

2.1. Minimizationproblemsforspherepacking

Wefirstrecalltwodifferentformulationsofgenericminimizationproblems.LetN andb betwogivenpositiveintegers.

WeconsiderfirsttheproblemoffindingaconfigurationX

¯

suchthat X

¯ ∈ ^argmin

φk(X)≤⁰,k,=¹,...,N,k<

W

(

X

), (2.1)

where W

: R

^bN

→ R

^{is a convex function (not} necessarily strictly convex). The functions

φ

k

: R

^bN

→ R

^{, k}

, =

¹

, ...,

N, k

<

are continuous butnot necessarilyconvex. We suppose that W has aminimum inthe set ofadmissiblesolutions

{

^X

∈ R

^bN

| φ

_k

(

X

) ≤

⁰

}

^{. Inthese}conditions,X

¯

exists butmaynot be unique.Wealsoassume that

φ

_k,k

, =

¹

, ...,

N,k

<

andW areC¹functionsintheneighbourhoodofX.

¯

Inwhatfollows,dwilldenotethediameterofasphere,N thenumber of spheres, b the spatial dimension, X the position of thecenter of the spheres and

φ

k the non-overlapping constraint functionsbetweenthekth and

th spheres.Thenon-overlappingconstraintsforasystemofidenticalspheresin

R

^{b canbe} expressedbymeansofasmoothoranon-smoothfunctionasspecifiedbellow.Althoughleading toequivalentconstraints, eachformhasanimpactontheconvergenceofthenumericalmethodtowardsalocalminimizer,aswewillseeinsections3 and4.

Definition2.1.Wecallnon-smoothformoftheconstraintfunctions(NS)thefollowingfunction

φ

_k

(

X

) =

^d

− |

^Xk

−

^X

| ,

k

, = ¹ , ...,

N

,

k

=

andsmoothformoftheconstraintfunctions(S)thefollowingfunction

φ

k

(

X

) =

d²

− |

Xk

−

X

|

²

,

k

, = 1 , ...,

N

,

k

= .

Anillustrationofthenon-overlappingconstraints,aswellas,apossiblesolutionforN

=

^{7 arepresentedinFig. 1.}

Wenowpresentasecondformulationconsistinginsolvingaminimizationproblemassociatedwithadiscretedynamical systemwhichhasX

¯

asafixedpoint.Let

| · |

^{denotetheEuclideannormon}

R

^{b.Theproblemisformulated}iteratively:given aninitialconfigurationX⁰

= {

^X0_i

}

i=¹,...,N,wepassfromiterateX^p toiterateX^p+1 asfollows

⎧ ⎪

⎪ ⎨

⎪ ⎪

⎩

X^p+1

=

^Xp

+ τ

V^p+1

(a)

V^p+1

∈ ^argmin

φk(X^p+τV)≤⁰,k,=¹,...,N,k<

1 2

N i=¹

|

^Vi

+ ∇

XiW

(

X^p

)|

²

, (b) (2.2)

where

τ >

0 isagivenparameterandV

= {

^Vi

}

i=¹,...,N.WedefineX

˜

asafixedpointofthisproblem.Consequently,X

˜

satisfies

Fig. 1.Representation of the non-overlapping constraints, a, and a possible optimal solution of(2.1)forN=7, b.

0 ∈ ^argmin

φk(˜X+τV)≤0,k,=1,...,N,k<

1 2

N i=¹

|

^Vi

+ ∇

X_iW

(

X

˜ ) |

²

. (2.3)

Notethattheminimaof(2.2)(b)exist,butmaynotbeunique.

Theminimizationproblem(2.1)canbeformulatedintermsoftheLagrangian

L : R

^bN

× ( R

⁺0

)

^N(N−1)/2

→ R

^definedby

L (

X

,λ) =

^W

(

X

) +

k,∈{¹,...,N},^k<

λ

_k

φ

_k

(

X

),

where

λ = {λ

k

}

k,=¹,...,N,k<representsthesetofLagrangemultipliers.IfX

¯

isasolutionoftheminimizationproblem(2.1), then,theAbadie constraintqualification(ACQ) [19]holds atX

¯

and, consequently, thereexists

λ ¯ ∈ ( R

⁺₀

)

^N(N−1)/2 such that

(

X

¯ , λ) ¯

isacritical-pointoftheLagrangian,namely,

(

X

¯ , λ) ¯

satisfiestheKKT-conditions[20,21]:

∇

Xi

L (

X

¯ , λ) ¯ = ⁰ ,

i

= ¹ , ...,

N

∇

λk

L (

X

¯ , λ) ¯ = ^{0 and} λ ¯

_k

≥ ^{0 or}

∇

λk

L (

X

¯ , λ) < ¯ 0 and λ ¯

_k

= ⁰ ,

k

, = ¹ , ...,

N

,

k

<

whichisequivalentto

⎧ ⎨

⎩

∇

XiW

(

X

¯ ) +

k,∈{¹,...,N},^k<

λ ¯

k

∇

Xi

φ

k

(

X

¯ ) = ⁰ ,

i

= ¹ , ...,

N

φ

k

(

X

¯ ) = ^{0 and} λ ¯

k

≥ ^{0 or}

φ

k

(

X

¯ ) < 0 and λ ¯

k

= ⁰ ,

k

, = ¹ , ...,

N

,

k

< .

(2.4)

Wehavereducedouroriginalproblem(2.1)toacritical-pointsystem,withapossibleenlargementofthesetofsolutions.

Contrarilytoconvexoptimization,inthecaseofpackingproblems,thesecritical-pointsmaynotbesaddle-points.Inrefer- ence[18]weprovideadetailedanalysisofthispoint,whichrequiresnewtechnicaldevelopmentsthatgobeyondthescope ofthepresentpaper.

Wealsoformulatetheminimizationproblem(2.2)(b)intermsofaLagrangian

L

^p,

L

^p

(

V

, μ ) = ¹

2

N i=¹

|

^Vi

+ ∇

X_iW

(

X^p

) |

²

+

k,∈{¹,...,N},^k<

μ

_k

φ

_k

(

X^p

+ τ

V

),

where

μ _{= {} μ

_k

_}

_{k,=1,...,N,k<} is the set of Lagrange multipliers associated to the constraints. The gradients of the La- grangianaregivenby

∇

Vi

L

^p

(

V

, μ ) =

^Vi

+ ∇

XiW

(

X^p

) + τ

k,∈{¹,...,N},^k<

μ

k

∇

Xi

φ

k

(

X^p

+ τ

V

),

i

= ¹ , ...,

N

∇

μ_k

L

^p

(

V

, μ ) = φ

k

(

X^p

+ τ

V

),

k

, = ¹ , ...,

N

.

Thedynamicalsystemiswritten:X

˜

^p+1

= ˜

^Xp

+ τ

V

^˜

^p+1suchthat

(

V

˜

^p+1

, μ ˜

^p+1

)

isasolutionofthecritical-pointproblem

⎧ ⎪

⎪ ⎪

⎪ ⎨

⎪ ⎪

⎪ ⎪

⎩

V

˜

_i^p+1

+ ∇

XiW

(

X

˜

^p

) + τ

k,∈{¹,...,N},^k<

˜

μ

_k^p+1

∇

Xi

φ

k

(

X

˜

^p

+ τ

V

^˜

^p+1

) = ⁰ ,

i

= ¹ , ...,

N

φ

k

(

_X

˜

^p

+ τ

V

^˜

^p+1

) = 0 and μ ˜

_k^p+1

≥ 0 or

φ

k

(

X

˜

^p

+ τ

V

^˜

^p+1

) < 0 and μ ˜

_k^p+1

= ⁰

,

k

, = ¹ , ...,

N

,

k

<

(2.5)

Likewise,thefixedpointX

˜

ofthedynamicalsystemisdefinedsuchthatthereexists

μ ˜

suchthat

⎧ ⎪

⎨

⎪ ⎩

∇

X_iW

(

_X

˜ ) + τ

k,∈{1,...,N},k<

˜

μ

k

∇

X_i

φ

k

(

_X

˜ ) = 0 ,

i

= 1 , ...,

N

φ

_k

(

X

˜ ) = ^{0 and} μ ˜

_k

_≥ 0

or

φ

_k

(

X

˜ ) < 0 and μ ˜

_k

₌ 0

,

k

, = ¹ , ...,

N

,

k

< .

(2.6)

Then, it is clearthat problems (2.4) and(2.6) are equivalent for all valuesof

τ >

0 by setting

λ ¯ = τ μ ˜

. However, the choiceof

τ

isimportanttoensureconvergenceofthedynamicalsystem(2.5)tothefixedpoint.

As itwillbe obviousbelow,allfunctionsW and

φ

k usedthroughoutthepaperwillsatisfytheconditionsconsidered inthissection. Thenonlinearsystems(2.4) or(2.5)will havetobesolved byan iterativealgorithm.We nowpresentthe algorithmsconsideredinthepaper.

2.2. TheArrow–Hurwiczalgorithm(AHA)

The classical Arrow–Hurwicziterativealgorithm[10] searches asaddle-pointof theLagrangian byalternating steps in the directionof

−∇

X

L

^and

+∇

λ

L

^{. Using thisidea, a} saddle-pointis thena steady-state solutionof the Arrow–Hurwicz systemofODE’s(AHS)whichisdefinednext.

Definition2.2.TheArrow–Hurwiczsystem(AHS)isdefinedby

⎧ ⎪

⎪ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎪

⎪ ⎩

X

˙

i

= − α

⎛

⎝ ∇

X_iW

(

X

) +

k,∈{1,...,N},k<

λ

k

∇

X_i

φ

k

(

X

)

⎞

⎠ ,

i

= 1 , ...,

N

(a) λ ˙

_k

=

0 , if λ

k

= ^{0 and} φ

k

(

X

) < 0

βφ

k

(

X

), otherwise ,

k

, = ¹ , ...,

N

,

k

< , (b)

(2.7)

where

α

and

β

arepositiveconstants.Consideringasmalltime-step

t,asemi-implicitEulerdiscretizationschemeofthe previoussystemleadstotheArrow–Hurwiczalgorithm(AHA),whichisdefinediterativelyby

⎧ ⎪

⎨

⎪ ⎩

Xⁿ_i⁺¹

=

X_iⁿ

− α

∇

X_iW

(

Xⁿ

) +

k,∈{1,...,N},k<

λ

ⁿ_k

∇

X_i

φ

k

(

Xⁿ

)

,

i

= 1 , ...,

N

λ

ⁿ_k⁺¹

= ^max { ⁰ , λ

ⁿ_k

+ βφ

k

(

Xⁿ⁺¹

)},

^k

, = ¹ , ...,

N

,

k

<

(2.8)

where

α

and

β

nowcorrespondto

α ˜ ₌ α

tand

β ˜ = β

tandthetildeshavebeendroppedforsimplicity.

The original AHAwas formulated usinga fullyexplicit Eulerscheme,butithas proved moreaccurate touse asemi- implicitscheme.Findingalocalsteady-satesolutionof(2.7)(a)–(2.7)(b)inthecaseofapackingproblemhasrevealednot to be always possible because it oftenhappens that no critical-point is a saddle-point[18]. Thismanifests itself by the existence ofperiodicsolutionsoftheAHSwhichdonotconvergetothecritical-point.Inordertoovercomethisdifficulty weproposethedampedArrow–Hurwiczalgorithmwhichispresentednext.Thismethodisbasedonamodificationofthe dynamicsofthe AHSthattransformsan unstablecritical-pointintoan asymptotically stableone. Theperformance ofour methodwillbetestedbycomparingwithpreviousapproaches[13,16],whicharebasedonamodificationoftheLagrangian bylinearlyapproximatingtheconstraints.Theseapproachesarepresentedinsection2.4.

2.3. ThedampedArrow–Hurwiczalgorithm

Inordertoavoidperiodicsolutionswewilladdadampingtermasdescribedbelow.Notethatwearenotinterestedon thetransientdynamicsofthesystem,butratheronitsasymptoticbehaviour.

Weproposethefollowingdefinition.

Definition2.3.WedefinethedampedArrow–Hurwiczsystem(DAHS)as

⎧ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎩

X

¨

_i

= − α

²

_[∇

_XiW

(

X

) +

k,∈{¹,...,N},^k<

λ

_k

∇

Xi

φ

_k

(

X

) ]

− α β

k,∈{1,...,N},k<

φ

k

(

X

)λ

k

∇

Xi

φ

k

(

X

) −

^cX

˙

_i

,

i

= ¹ , ...,

N

(a) λ ˙

k

=

0 , if λ

_k

= ^{0 and} φ

_k

(

X

) < 0

βφ

_k

(

X

), otherwise ,

k

, = ¹ , ...,

N

,

k

< (b)

(2.9)

where

α

,

β

andc arepositive constants andthedampedArrow–Hurwiczalgorithm(DAHA)asthe corresponding semi- implicitdiscretescheme:

⎧ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎨

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎩

Xⁿ_i⁺¹

= ¹ 1 +

^c

/ 2

2

X_iⁿ

− ( 1 −

^c

/ 2 )

Xⁿ_i⁻¹

− α

²

1 +

^c

/ 2 [∇

X_iW

(

Xⁿ

) +

k,∈{¹,...,N},k<

λ

ⁿ_k

∇

X_i

φ

_k

(

Xⁿ

) ]

− α β 1 +

^c

/ 2

k,∈{1,...,N},k<

φ

k

(

Xⁿ

)λ

ⁿ_k

∇

Xi

φ

k

(

Xⁿ

),

i

= ¹ , ...,

N

(a) λ

ⁿ_k⁺¹

= ^max { ⁰ , λ

ⁿ_k

+ βφ

k

(

Xⁿ⁺¹

)},

^k

, = ¹ , ...,

N

,

k

< , (b)

(2.10)

where

α

,

β

andc correspondnowtonumericalparameters.

NotethattheDAHAisamulti-stepscheme,sincenotonlyone,buttwopreviousconfigurations Xⁿ⁻¹ andXⁿ areused toobtainXⁿ⁺¹.Bysettingc

=

^{2,themethodisreducedtoaone-stepmethod.}

Asecond-orderODEsystemwithdampinghaspreviouslybeenproposedwithinthescopeofconvexprogramming[22, 23]. Besides comprisingthe non-convexcase, ourapproach differs fromthis withregard to the extra term

α β

in equa- tion(2.9)(a).

Inthefollowingwepresentthederivation oftheDAHS.Westartby consideringtheAHS(2.7)(a)–(2.7)(b)presentedin theprevioussection.Wethentakethesecond-orderversionof(2.7)(a).Foreachi

=

¹

, ...,

N wehave

X

¨

_i

= − α

N m=1

∇

X_m

∇

X_iW

(

X

) +

k,∈{¹,...,N},k<

λ

_k

∇

X_i

φ

_k

(

X

)

X

˙

m

− α

k,∈{¹,...,N},^k<

λ ˙

_k

∇

Xi

φ

_k

(

X

). (2.11)

Using(2.7)(b),wecanreplace

λ ˙

kin(2.11)by

βφ

k

(

X

)

H

(λ

k

)

,whereHistheHeavisidefunction.Moreover,inordertokeep thesamesteadystatesastheAHS,wereplaceH

(λ

k

)

by

λ

k,asatequilibrium

λ

k

φ

k

=

^{0.Notethatotherchoicescouldbe} made,suchasapowerof

λ

_kforinstance,whichwouldinfluencethespeedofconvergenceofthealgorithm.Howeverwe donotexplorethisaspectfurtherhere.Weget

X

¨

i

= − α

N m=1

∇

X_m

∇

X_iW

(

X

) +

k,∈{¹,...,N},k<

λ

_k

∇

X_i

φ

_k

(

X

)

X

˙

m

(2.12)

− α β

k,∈{¹,...,N},^k<

φ

_k

(

X

)λ

_k

∇

Xi

φ

_k

(

X

). (2.13)

Itturnsoutthatpassingtothesecond-orderintroducesexponentiallygrowingmodes(seeRemark 2.1).

Remark2.1.Consider the simple ODE ^u

˙ = − α

u whose solution is u

(

t

) =

^u0e⁻αt, where u₀ is the initial configuration.

Differentiatingbothsidesoftheequationandsubstitutingu

˙

by

− α

uyieldsu

¨ = α

²u,whosesolutionincludesnowanexpo- nentiallygrowingmode:u

(

t

) =

^c1e⁻αt

+

^c2eαt,wherec1andc2 arerealconstantsdeterminedbytheinitialconfigurations.

In orderto remove thesemodes, we replace thetermin (2.12) by a simplesecond-order dynamics inthe force field givenbytherighthandsideof(2.7)(a).Weget:

X

¨

_i

= − α

²

⎡

⎣ ∇

XiW

(

X

) +

k,∈{¹,...,N},^k<

λ

k

∇

Xi

φ

k

(

X

)

⎤

⎦

− α β

k,∈{¹,...,N},k<

φ

_k

(

X

)λ

_k

∇

X_i

φ

_k

(

X

). (2.14)

Now,we justadd avelocity dampingtermin theformof

−

^cX

˙

_i andwe finally obtain (2.9)(a).We endup withthesys- tem(2.9)(a)–(2.9)(b).

Remark2.2.Wecaninterpretthefirstterm,attherighthandsideof(2.14)asasecond-orderdynamicsversionof(2.7)(a).

Denotingby T1 andT2 thetermsin(2.14) whicharemultipliedby

− α

² and

− α β

,respectively,we recover(2.7)(a)inan over-dampedlimit

X

¨

_i

+ α β

T₂

= − α

²T₁

−

^cX

˙

_i,with

→

^{0 andc}

=

^1.

Proposition2.4.TheAHS(2.7)(a)–(2.7)(b)andtheDAHS(2.9)(a)–(2.9)(b)havethesameequilibriumsolutions.

Proof. If

(λ

^∗

,

X^∗

)

is a steadystate ofthe AHS, then either

φ

k

(

X^∗

) =

^{0 or}

λ

^∗_k

=

^0. Consequently,

λ

k

φ

k

(

X^∗

) =

^{0, which} impliesthat thesecond partofequation (2.9)(b)isnullandX

¨

^∗

=

^{0.Using asimilar argumentweconcludethat asteady} stateofDAHSisalsoasteadystateofAHS.

2

2.4. Previousapproaches

Acommonapproachtosolvethegenericminimizationproblems(2.1)and(2.2)(a)–(2.2)(b)isbasedonthelinearization oftheconstraintfunctions

φ

karoundacertainconfigurationX^p,whichwedenoteby

φ

_k^p

(

X

)

,i.e.,

φ

_k^p

(

X

) = φ

k

(

X^p

) + ∇

X

φ

k

(

X^p

) · (

X

−

^Xp

). (2.15)

The solution X^p+1 of theresulting linearlyconstrainedoptimization problemis usedto improvethe linearizationof the constraint functionsand this process is iterated until convergence. Note that this transformation turns the non-convex minimization problems (2.1) and (2.2)(a)–(2.2)(b) into a sequence of convex problems, for which there are many tools available[24].WehavechosentheArrow–Hurwiczalgorithm,however,anyothermethodforconvexoptimizationproblems wouldsuitourpurpose.

ThismethodbelongstotheclassoflinearlyconstrainedLagrangian(LCL)methods[13]whichhavebeenusedforlarge constrainedoptimizationproblems.

2.4.1. Thenestedalgorithmforthepositions(NAP)

Considerthesystem(2.4)withlinearizedconstraintfunctions.Weproposethefollowingdefinition.

Definition2.5(NestedAlgorithmforthePositions(NAP)).Let

(

X^p

, λ

^p

)

be given. Define X^p,0

=

^Xp,

λ

^p,0

= λ

^p and

φ

_k^p as in(2.15).Foragiven

(

X^p,n

, λ

^p,n

)

,letthestepoftheinner-loopbedefinedas

⎧ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎩

X_i^p,n+1

=

^X_i^p,n

− α

⎡

⎣ ∇

XiW

(

X^p,n

) +

k,∈{1,...,N},k<

λ

_k^p,n

∇

Xi

φ

_k^p

(

X^p,n

)

⎤

⎦ ,

i

= ¹ , ...,

N

(a) λ

_k^p,n+1

= ^max

0 , λ

_k^p,n

+ βφ

_k^p

(

X^p,n+1

)

,

k

, = ¹ , ...,

N

,

k

< , (b)

(2.16)

then

(

X^p+1

, λ

^p+1

) =

^limn→∞

(

X^p,n

, λ

^p,n

)

.

Ifweonlycomputeone stepoftheinner-loopperiterationoftheouter-loopwegetavariantoftheAHAformulation, where

φ

k

(

X^p+1

)

isreplacedby

φ

_k^p

(

X^p+1

)

in(2.16)(b).

2.4.2. Thenestedalgorithmforthevelocities(NAV)

Weconsidertheminimizationproblem(2.5)withlinearizedconstraintfunctions.

Definition2.6(NestedAlgorithmfortheVelocities(NAV)).Let

τ >

0 and

(

X^p

,

V^p

, μ

^p

)

begiven.DefineV^p,0

=

^Vp,

μ

^p,0

= μ

^p

and

φ

_k^pasin(2.15).Foragiven

(

V^p,n

, μ

^p,n

)

,letthestepoftheinner-loopbedefinedas

⎧ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎩

V_i^p,n+1

=

V_i^p,n

− α

⎛

⎝

V_i^p,n

+ ∇

X_iW

(

X^p

) + τ

k,∈{1,...,N},k<

μ

_k^p,n

∇

X_i

φ

_k^p

(

X^p

+ τ

V^p,n

)

⎞

⎠ ,

i

= 1 , ...,

N

(a)

μ

_k^p,n+1

= max

0 , μ

_k^p,n

+ βφ

_k^p

(

X^p

+ τ

V^p,n+1

)

,

k

, = 1 , ...,

N

, (b)

(2.17)

then

(

V^p+1

, μ

^p+1

) =

^limn→∞

(

V^p,n

, μ

^p,n

)

andX^p+1

=

^Xp

+ τ

V^p+1.

TheNAVcorrespondstoanadaptationofthemethoddevelopedbyMauryin[16].

3. Linearanalysis

3.1. Preliminaries

Undertheassumptionsconsideredintheprevioussection,theassociatedODEsystemsarepiecewisesmooth.Inpartic- ular,they aresmoothinaneighbourhood ofX,

¯

whichallowsustocarry outthelinearstabilityanalysisinordertostudy thelocalconvergenceofthesolutiontowardsasteadystate.

Weconsiderhereaphysicalsystemwhere N rigidspheresin

R

^{b attracteach otherthroughaglobalpotentialwhichis} givenbyaquadraticfunctionoftherelativedistance,

W

(

X

) = ¹ 2N

i,j∈{¹,...,N},ⁱ<j

|

^Xi

−

^Xj

|

²

. (3.1)

Definition3.1.Asteadystatex^∗ oftheODEsystemx

˙ =

^f

(

x

),

t

≥

^0,iscalled

•

^{stable(inthesenseofLyapunov)ifforall}

>

0,thereexistsa

δ >

0 suchthat

¯

^x

(

0

) −

^x∗

< δ

implies

¯

^x

(

t

) −

^x∗

<

, forallt

>

0 andforallsolution

¯

x;

•

asymptoticallystableifitisstableandlimt→∞

¯

^x

(

t

) −

^x∗

=

^0;

•

^{unstableifitisnotstable.}

Notethat thisdefinition assumesthat theinitial configuration ischosen closeenough to thesteadystate. Alternative notions ofstability could havebeen used [25,26].The one we consider hereallows usto get insight intothe behaviour ofthe algorithmasit isdescribed below.The next theoremallows ustoobtain conclusions abouttheoriginal nonlinear systemfromthecorrespondinglinearizedsystem.

Theorem3.2.ConsidertheODEsystemx

˙ =

^f

(

x

)

andasteadystatex^∗,where f issmoothatx^∗.Ifx^∗ isanasymptoticallystable (unstable)solutionofthelinearizedsystemaboutx^∗,i.e.,^x

˙˜ =

^f

(

x^∗

)(˜

^x

−

^x∗

)

,thenitisanasymptoticallystable(unstable)solutionof theoriginalsystem.

Proof. See[27],Thm. 2.42,p. 158.

2

InordertoensureconvergenceoftheODEsystemtowardsasteadystate,weonlyneedtoensurethattheeigenvaluesof f

(

x^∗

)

allhavenegativerealpart.Ifatleastoneeigenvaluehaspositiverealpart,thenx^∗isunstable,andifalleigenvalues arepure imaginary,then x^∗ isacenterequilibrium,i.e.ifa solutionstartsnearitthenitwillbe periodicaroundit.Inthe lattercase,we cannotconcludeanythingaboutthenonlinearsystem. Theanalysispresentednextismadeforthecaseof twospheresin

R

^.

3.2. TheArrow–Hurwiczalgorithm(AHA) 3.2.1. AHA-NS

Let

φ (

X

) =

^d

− |

^X

|

^{andconsider the potential (3.1). The ODE systemassociated to the DAHA-NS inthe case of two} spheresin

R

^{whereonesphereisfixedattheorigincanbewrittenas}

⎧ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎩

X

˙ = − α

1 − λ

|

^X

|

X

(a)

λ ˙ =

0 , if λ = ^{0 and}

^d

< |

^X

| β(

d

− |

^X

|), ^otherwise . ^(b)

(3.2)

Lemma3.3.Thesteadystatesofthesystem(3.2)(a)–(3.2)(b),

(

X^∗

, λ

^∗

) = (

d

,

d

)

and

(

X^∗

, λ

^∗

) = ( −

^d

,

d

)

,arebothasymptotically stable,forany

α

and

β

positive.

Proof. Sincethedynamicsaroundeach steadystateisidentical,weonlyneedtocarryouttheanalysisofthefirststeady state. Suppose X

>

0 andconsider thechangeofvariables Y

=

^X

−

^dand

μ ₌ λ −

^{d.The systemonthenewvariablesis} giveninmatrixformby

_Y

˙

˙ μ

=

^A

Y

μ

,

A

=

− α α

− β 0

.

We want the eigenvalues of matrix A to be real and negative in order to have a fast convergence to the steady state.

Therootsofthecharacteristicpolynomial

P (λ) = λ

²

+ α λ + β α

,havebothnegativerealpart,thereforethesteadystate is asymptoticallystable.

2

AnysolutiontotheODEsystem(3.2)(a)–(3.2)(b)convergestoasteadystateforall

α , β >

0 andthefastestconvergence is achieved when

α ₌

4

β

. Contrarily to the one dimensional case, in higher spatial dimensions the constraints are no longerpiecewiselinear.Consequently,wecannotdirectlyextrapolatetheconclusionsdrawninthissection.Inparticular,in dimensionb

=

^{2,thenumerical}simulationsshowoscillations aroundthesteadystate forN

>

3 withoutneverconverging toit.Thenon-convergenceinthiscaseisduetothenon-existenceofasaddle-pointoftheLagrangian[18].

3.2.2. AHA-S

Let

φ (

X

) =

^d2

−|

^X

|

^{2andconsiderthepotential(3.1).TheODEsystemassociatedtotheAHA-Sinthecaseoftwospheres} in

R

^{whereonesphereisfixedattheorigincanbewrittenas}

Fig. 2.Phaseportraitofthesystem(3.3)(a)–(3.3)(b)with(α,β,d)=(0.01,0.01,2)andinitialconditionX⁰=⁰.2.Thedynamicsdonotconvergetothe equilibrium(2,¹₂).

⎧ ⎪

⎨

⎪ ⎩

X

˙ = − α ( 1 − 2 λ)

X

(a) λ ˙ =

0 , if λ = ^{0 and}

^d

< |

^X

|

β(

d²

−

X²

), otherwise . ^(b)

(3.3)

Lemma3.4.Thesteadystatesofthesystemcorrespondingtothelinearizationof(3.3)(a)–(3.3)(b),

(

X^∗

, λ

^∗

) = (

d

,

1

/

2

)

and

(

X^∗

, λ

^∗

) = ( −

^d

,

1

/

2

)

,arebothcenterequilibria,forany

α

and

β

positive.

Proof. Asbefore,wewillonlycarryouttheanalysisofthefirststeadystate.

Suppose X

>

0 and considerthe change ofvariables Y

=

^X

−

^{d and}

μ = λ −

¹

/

2. The linearizedsystem onthe new variablesisgiveninmatrixformby

Y

˙

˙ μ

=

A

Y

μ

,

A

=

0 2d α

− 2d β 0

.

The rootsofthecharacteristicpolynomial

P(λ) = λ

²

+

^4d2

α β

are bothpurelyimaginary,thereforethesteadystateofthe linearizedsystemisacenterequilibrium.

2

The linearanalyses doesnot allow us to concludeanything aboutthe asymptotic behaviour of the nonlinear system (see Theorem 3.2).Nevertheless,thephaseportraitplottedinFig. 2revealsthatasolutiontothenonlinearsystemshould convergetowardsaperiodicorbitaroundthesteadystate.Aswewillseeinthenextsection,thedampingtermappliedto theArrow–Hurwiczsystem(2.7)(a)–(2.7)(b)ensuresasymptoticstabilityofthesteadystate,undercertainconditionsonthe parameters.

3.3. ThedampedArrow–Hurwiczalgorithm(DAHA) 3.3.1. DAHA-NS

Let

φ (

X

) =

^d

− |

^X

|

^{and consider the potential (3.1). The ODE system associated to the DAHA-NS in the case oftwo} spheresin

R

whereonesphereisfixedattheorigincanbewrittenas

⎧ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎩

X

¨ = − α

²

1 − λ

|

^X

|

X

+ α βλ(

d

− |

^X

|)

^X

|

^X

| −

^cX

˙ (a) λ ˙ =

0 , if λ = 0 and

d

< |

X

|,

β(

d

− |

X

|), otherwise . ^(b)

(3.4)

Lemma3.5.Let

α , β,

c

>

0.If

( α + β

d

)

c

− β α >

0,thenthesteadystatesofthesystem(3.4)(a)–(3.4)(b),

(

X^∗

,

X

˙

^∗

, λ

^∗

) = (

d

,

0

,

d

)

and

(

X^∗

,

X

˙

^∗

, λ

^∗

) = ( −

^d

,

0

,

d

)

,arebothasymptoticallystable.

Proof. Suppose X

>

0 andconsiderthechangeofvariablesY

=

^X

−

^{d, Z}

= ˙

^{Y and}

μ = λ −

^{d.Thelinearizedsystemonthe} newvariablesisgiveninmatrixformby

⎡

⎣

Y

˙

Z

˙

˙ μ

⎤

⎦ =

^A

⎡

⎣

^YZ

μ

⎤

⎦ ,

A

=

⎡

⎣ − α

²

− ⁰ α β

d

− ¹

^c

α ⁰

²

− β 0 0

⎤

⎦ .

Theeigenvaluesofmatrix A aretherootsofthecharacteristicpolynomialin

λ

,whichisgivenby

P (λ) = λ

³

+

^c

λ

²

+ ( α

²

₊

α β

d

)λ + β α

².Consider in generala cubic polynomial ofthe form

P(λ) = λ

³

+

^c2

λ

²

+

^c1

λ +

^c0,with c0

,

c1

,

c2

∈ R

^+.Let z₁

,

z₂ andz₃ bethe(complex)rootstothispolynomial.Wewanttoensurethatallrootshavenegativerealpart.Sinceall coefficientsarepositive,iftherootsarerealthentheymustbenegative.Supposenowthattworootsarecomplexconjugate,

forexample,z1

=

^a

+

^ib

,

z2

=

^a

−

^ib,a

,

b

∈ R

^andz3

∈ R

^{−.Inordertofindacondition onthecoefficientswhichensures} thataisnon-positive,westartbyidentifyingthecoefficientsoftheequationwithitsroots:

z1

+

z2

+

z3

= −

c2

,

z1z2

+

z1z3

+

z2z3

=

c1

,

z1z2z3

= −

c0 Rewritingintermsofa

,

bandz3 weget

2a +

^z3

= −

^c2

,

a²

+

^b2

+ ^2az

3

=

^c1

, (

a²

+

^b2

)

z₃

= −

^c0

(3.5)

From(3.5)wededucethatasatisfiesthecubicpolynomial

8a

³

+ ^8c

1a²

+ ² (

c₁

+

^c2₂

)

a

+

^c1c₂

−

^c0

= ⁰ .

Consequently,ifc₁c₂

−

^c0

>

0,thenaisnecessarilynegative.

Backto our case, we have c2

=

^{c, c}1

= α

²

₊ α β

d andc0

= β α

² anda sufficient condition forthe steady state to be asymptoticallystableis

( α

²

₊ α β

d

)

c

− β α

²

>

0,i.e.,

( α ₊ β

d

)

c

− β α >

0.Notethatsincethesteadystateisasymptotically stableasasolutiontothelinearizedsystem,thenitisalsoasymptoticallystable(seeTheorem 3.2).

2

3.3.2. DAHA-S

Let

φ (

X

) =

^d2

− |

^X

|

^{2 andconsiderthepotential(3.1).Forthecaseoftwospheresin}

R

^{whereonesphereisfixedatthe} origin,theODEsystemassociatedtotheDAHA-Scanbewrittenas

⎧ ⎪

⎨

⎪ ⎩

X

¨ = − α

²

( 1 − 2 λ)

X

+ 2 α βλ(

d²

− |

X

|

²

)

X

−

c_X

˙ _(a) λ ˙ =

0 , if λ = 0 and

d

< |

X

|

β(

d²

− |

X

|

²

), otherwise . ^(b)

(3.6)

Lemma3.6.Let

α , β,

c

>

0.Ifc

−

²

α >

0,thenthesteadystatesofthesystem(3.6)(a)–(3.6)(b),

(

X^∗

,

X

˙

^∗

, λ

^∗

) = (

d

,

0

,

1

/

2

)

and

(

X^∗

,

X

˙

^∗

, λ

^∗

) = ( −

^d

,

0

,

1

/

2

)

,arebothasymptoticallystable.

Proof. Asbefore,suppose X

>

0 andconsiderthechangeofvariables Y

=

^X

−

^{d, Z}

= ˙

^{Y and}

μ = λ −

¹

/

2.The linearized systemonthenewvariablesisgiveninmatrixformby

⎡

⎣

Y

˙

Z

˙

˙ μ

⎤

⎦ =

^A

⎡

⎣

^YZ

μ

⎤

⎦ ,

A

=

⎡

⎣ − 2 α ⁰ β

d²

− ¹

c

2d ⁰ α

²

− ^2d β 0 0

⎤

⎦ .

TheeigenvaluesofmatrixA aretherootsofthecharacteristicpolynomialin

λ

:

P (λ) = λ

³

+

^c

λ

²

+ ² α β

d²

λ + ^4d

²

β α

²

Usingthesamereasoningasbeforewehavec2

=

^c,c1

=

²

α β

d² andc0

=

^4d2

β α

².Asufficientconditionforthesteady statetobeasymptoticallystableis2c

α β

d²

−

^4d2

β α

²

>

0,i.e.,c

−

²

α >

0.

2

Remark3.1.We seethat aslong as the damping coefficient, c, islarge enough, the sufficient conditionsfor stability of boththeDAHA-NSandDAHA-Sarefulfilled.Furthermore,the parameterspacecorresponding tothestability ofDAHA-NS islargerthantheoneoftheDAHA-S.

ThecorrespondinganalysesfortheNAPandtheNAValgorithmsarepresentedintheAppendix A.

4. Numericalresults

InthissectionweinvestigateandcomparethenumericalresultsobtainedfromthedampedArrow–Hurwiczalgorithms (DAHA-NS,DAHA-S)andthenestedalgorithms(NAP-NS,NAP-SandNAV-NS)forthepotentialdefinedin(3.1).Duetothe difficultyinfindingtheoptimalparameters

( α , β)

foreachmethodandforeachN,wehaverestrictedthisstudytothe cases N

=

^{7 and N}

=

^{100 in two spatial dimensions (i.e.b}

=

^{2). We address the} convergence time andthe robustness ofthe convergencetimewithrespecttotheinitial configurations.Additionally,we comparethe accuracyofthemethods forthe caseN

=

^{7 only.Indeed,inthecaseN}

=

^{7,thestablesteadystateofthedynamicalsystemsassociatedtothealgorithmsis} unique(apartfromtranslations,rotationsandreflections) andisrepresentedinFig. 1b.Thisguaranteesthatallalgorithms converge to the same minimum for any initial configuration. In particular, thisallows us to assess the accuracy of the algorithms by comparing thecomputedminimum withthe exactone. We finally show some examples ofconfigurations obtainedwiththeDAHA-SforthecaseN

=

^{2000 intwoandthree}dimensions.

Inorder toadjustthe spatial dimensions, thenumericalparameters mustsatisfy

α , β,

c

∼ O (

1

)

forthemethods with thenon-smoothformoftheconstraintfunctionsand

α ,

c

∼ O(

¹

)

and

β ∼ O(

¹

/

d²

)

forthemethodswiththesmoothform oftheconstraintfunctions.Inthefollowingwehaveconsideredd

=

^1.

InordertobeabletocomparethenestedalgorithmswiththeDAHAregardingconvergencetime,weonlyconsiderthe evolutionofXandVandwedonotconsidertheevolutionof

λ

.Wedenoteby

·

^{theEuclideannormin}

R

^bN.Foragiven smallandpositive

,thestoppingcriterionfortheminimizationalgorithmsassociatedtotheNAP,isgivenbythefollowing conditionontherelativeerror

^Xn+1

−

^Xn

^Xn

<

inner

. (4.1)

Forthecaseoftheminimizationproblemformulatedintermsofthevelocities,thestoppingcriterionissimilarbutinstead ofXwewriteVandinsteadofnormalizingbyVⁿ,wenormalizebyXⁿ,yielding

^Vn+1

−

^Vn

Xⁿ

<

^inner

τ ^{. (4.2)}

By usingthe Eulerstep Xⁿ⁺¹

=

^Xp

+ τ

Vⁿ⁺¹ we show that thetwo conditions(4.1) and(4.2) areequivalent. As we will see,in ordertogeta fastconvergence withthe nestedalgorithms, onedoesnot needtowait forthe convergenceofthe inner-loop. Weintroduce a newparameter, Iinner,which standsfor themaximumnumber ofiterations ofthe inner-loop allowed perouter-loopiteration.Finally,thestoppingcriterionforboththeouter-loopoftheNAPandtheNAV,aswellas, fortheDAHAreads

^Xp+1

−

^Xp

^Xp

< . (4.3)

Theassessmentandcomparisonofthemethodswillbemadethroughthecomparisonofstatisticalindicatorsobtained fromaveragingcertainquantitiesoverasetofdifferentinitialconfigurations.Theseindicatorsareintroducedbellow.

Definition4.1.Consider a set ofm initial configurations forwhich an algorithm converges,i.e., the stopping criterion is satisfied ina finitenumberof iterations.Let T be the numberofiterations neededforthe algorithmto convergewhen starting withthe

th initialconfiguration.Let Ai j be theoverlapping areaofspheres i and j atconvergenceand Atotal

=

N

π (

d

/

2

)

².

Wedefine thefollowingstatisticalindicatorsmeanconvergencetime,varianceoftheconvergencetimeandthemean proportionofoverlappingareapersphereas

T

= ¹

m

m

=1

T

, σ

²

₌ ¹

m

− ¹

m

=1

(

T

−

^T

)

²

and

A

= ¹

mN A_total

i,j∈{¹,...,N},i<j

A_{i j}

,

respectively.

The indicator T measuresthe efficiencyofanalgorithm withrespect totheconvergencetime, A andW measure the accuracy of the final configurationand

σ

² measures the robustness of the convergencetime withrespect to the initial configurations. Forsimplicity we assume that the time interval between iterations is constant andinvariant among the differentalgorithms.Asaconsequenceofthissimplification,wewillusethenumberofiterationsasthetimeunitofT. 4.1. CaseN

=

⁷

WepresentadetailednumericalstudyforthecaseofN

=

^{7 spheresindimensionb}

=

^2.The20 differentinitialconfigu- rationsconsideredinthissectionweregeneratedfromastandardGaussiandistribution.Wechoosethetolerances

=

¹⁰⁻⁶ and

inner

=

^{10−9 andthemaximumnumberofiterationsofthe inner-loopI}inner

=

^{10.Inorderto studytherelation be-} tweenthedampingparameterc andtheconvergencetimeoftheDAHAwithsmoothandnon-smoothconstraints,weplot in Fig. 3 the maximumnumberof iterations over 20 differentrandomly generated initialconfigurations asa function of c

∈ (

0

,

10

]

^{.We observethatthelower} convergencetimeisattainedwhenc

≈

^{2,forboth theDAHAwiththe smoothand} withthenon-smoothconstraints.InFig. 4weplottherelativeerrorasafunctionofiterationnumber,n,fordifferentvalues of c.Ifc

=

^{0 weobserve thattherelative erroroscillatesandneverdropsbellow 10−1.As weincrease c the} oscillations tend to diminish. In the following we have used c

=

^{2. Note that thischoice for c eliminates the dependenceon Xn−1} in(2.10)(a)–(2.10)(b),inthiscase,theDAHAcanbeseenasadiscretizationofthefollowingfirst-orderODEsystem:

⎧ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎨

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎩

X

˙

_i

= − ¹

2 α

²

_[∇

X_iW

(

X

) +

k,∈{¹,...,N},k<

λ

_k

∇

X_i

φ

_k

(

X

) ]

− ¹

2 α β

k,∈{¹,...,N},k<

φ

_k

(

X

)λ

_k

∇

X_i

φ

_k

(

X

),

i

= 1 , ...,

N

λ ˙

_k

=

0 , if λ

k

= ^{0 and} φ

k

(

X

) < 0

βφ

k

(

X

), otherwise ,

k

, = ¹ , ...,

N

,

k

< .