A method to improve Standard PSO

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A method to improve Standard PSO

Maurice Clerc

To cite this version:

Maurice Clerc. A method to improve Standard PSO. 2009. �hal-00394945�

Tehnialreport. DRAFTMC2009-03-13

MaurieCler

Abstrat

Inthisreport,IpresentmypersonalmethodtodesignamoreaurateversionofPSO,assumingweknowwhatkindof

problemswewillhavetosolve. Toillustratethemethod,Ionsideralassofproblemsthataremoderatelymultimodal

andnotofveryhighdimensionality(typiallysmallerthan30). ThestartingversionisStandardPSO2007(SPSO07).

Iuseamodiedveloityupdateequationisused,speitotheonsideredlassofproblems.Then,inordertoimprove

therobustnessofthealgorithm,moregeneralmodiationsare added(inpartiular,betterinitialisations,andthe use

of two kindof partiles). Onthe whole, the resulting algorithm is indeed animprovement of SPSO07, though the

improvementisnotalwaysveryimpressive. Inpassing,Igiveapreisedenitionofbetterthan,andexplainwhythe

lassialmeanbestperformaneriterionmayeasily beompletelymeaningless.

1 Why this paper?

AquestionIoftenfaeisthis: YouarealwaysreferringtoStandardPSO2007(SPSO07,[1℄). Isn'titpossible

to designa better version?. The answeris ofourse yes. Itis even written in thesoure ode available on

line:

ThisPSOversiondoesnotintendtobethebest oneonthemarket

Ontheonehand,itisindeednotverydiulttowriteabetterPSO.Ontheotherhand,asareviewerIreada

lotofpapersinwhihtheauthorslaimtheyhavesueededindoingthat,thoughitisnotreallytrue.,mainly

fortworeasons:

•

^{theystartfromabadversionofPSO(usuallyaglobalbest one). Althoughtheymayreallyimproveit,}

theresultingalgorithmisstillnotasgoodasSPSO07.

•

^{orthey start from areasonablygood version(or even from SPSO 07 itself), but they update it having}

apparentlynolearideaofwhatthemodiationisfor. Atypialexampleistheusuallaimwewantto

improvetheexploitation/explorationbalane withoutgivinganyrigorousdenitionofwhatthisbalane

is. Asaresult,themodiedalgorithmmayormaynotbebetter,moreorlessin arandomway,andwe

donoeventknowwhy.

Also,ofourse,thereisthewellknownproblemofomparisonoftwoalgorithms. Often,wendsomethinglike

ournewalgorithmBisthereforebetterthanalgorithmA ... withoutanyleardenitionofbetterthan. Let

usrstsayafewwordsaboutthispoint.

2 My algorithm is better than yours

Whatdoessuhalaimmean? First,suhalaimisvalidonlyifweagreeonaommonbenhmark. Letussay,

forexample,weonsiderasubsetoftheCEC2005benhmark[2℄. Then,wehavetoagreeonagivenriterion.

For examplefor eah funtion, the riterionmay be

Cr

^{= suess rateover}

N

^{runs,for agiven auray}

ε

^,

after at most

F

^tnessevaluationsforeahrun, or,in aseofatile, meanofthe best results. Notethat the riterionmaybeaprobabilistione(forexample

Cr ^′

^=theprobabilitythat

A

^isbetterthan

B

^{onthisproblem}

aordingto theriterion

Cr

^{isgreater than0.5). What isimportant is tolearly dene themeaning of}

A

is betterthan

B

^{on one problem. A lot of researhersindeed use suh an approah, and then perform nie}

statistialanalyses(nullhypothesis,p-test,andsoon),inordertodeideinprobability whetheranalgorithm

AisbetterthananalgorithmB,onthewholebenhmark.

However, in the proess, they miss a simple and frustrating fat: there is no omplete order in

R ^D

^{, for}

all

D > 1

^{. Why is that important? It maybe usefulto express it moreformally. Letus say thebenhmark}

ontains

P

^{problems. We build twoomparison vetors. First}

C A,B = (c A,B,1 , . . . , c A,B,P )

^with

c A,B,i = 1

^if

A

^{is better than}

B

^{on problem}

i

^{(aording to the unique riterion dened),}

c A,B,i = 0

^{otherwise. Seond}

C B,A = (c B,A,1 , . . . , c B,A,P )

^with

c B,A,i = 1

^if

B

^{is betterthan}

A

^ontheproblem

i

^,and

c B,A,i = 0

^otherwise.

Wehave toompare thetwonumerialvetors

C A,B

^and

C B,A

^{. Now, preisely beause there isno omplete}

orderin

R ^D ,

^weansaythat

A

^isbetterthan

B

^{ifandonlyifforany}

i

^wehave

c A,B,i ≥ c B,A,i

^forall

i

^,andif

thereexists

j

^sothat

c A,B,j > c B,A,j

^.

This is similar to the denition of the lassialPareto dominane. As we have

P

^{valuesof oneriterion,}

theproessofomparing

A

^and

B

^anbeseenasamultiriterion(or multiobjetive)problem. Itimpliesthat most of the time no omparison is possible, exept by using an aggregation method. For example, here, we

ouldountthenumberof

1

^{s in eah vetor,andsay that theone withthe largersumwins. But thepoint}

isthat anyaggregationmethod isarbitrary,i.e. foreah methodthere isanotheronethatleadstoadierent

onlusion 1

.

Letusonsider anexample:

•

^{thebenhmarkontains5unimodalfuntions}

f 1

^to

f 5

^{,and5multimodalones}

f 6

^to

f 10

•

^thealgorithm

A

^{is extremelygoodonunimodalfuntions(veryeasy,sayforagradientmethod)}

•

^thealgorithm

B

^{isquitegood formultimodalfuntions,butnotforunimodalones.}

You nd

c A,B,i = 1

^for

i = 1, 2, 3, 4, 5

^{, and also for}

6

^{(just beause, for example,the attration basin of the}

globaloptimumisverylarge,omparedtotheonesoftheloal optima),and

c B,A,i = 1

^for

i = 7, 8, 9, 10

^{. You}

saythen"

A

^isbetterthan

B

^{". Anusertrustsyou,andhoosesAforhisproblems. Andasmostof}interesting realproblemsaremultimodal,hewillbeverydisappointed.

So,wehavetobebothmoremodestandmorerigorous. Thatiswhytherststepinourmethodofdesigning

an improvedPSO is thehoie ofasmall benhmark. Butwewill saythat

A

^{is betterthan}

B

^{only ifit is}

trueforalltheproblemsofthisbenhmark.

3 Step 1: a small representative benhmark

Thisisthemostspeipartofthemethod,foritisdependsonthekindorproblemswedowanttosolvelater

withourimprovedPSO.Letusonsiderthefollowinglassofproblems:

•

^moderatelymultimodal,orevenunimodal(butofoursewearenotsupposed toknowitinadvane)

•

^notoftoohighdimensionality(saynomorethan30)

Forthis lass, to whih alot of realproblems belong, I havefoundthat agood small benhmarkmaybethe

followingone(seethetable 1):

•

^{CEC2005Sphere(unimodal)}

•

^{CEC2005Rosenbrok(oneglobaloptimum,atleastoneloaloptimumassoonasthedimensionisgreater}

than3)

•

^{Tripod(twoloaloptima,oneglobaloptimum,verydeeptive[3℄)}

Thesethreefuntionsaresupposedtoberepresentativeofourlassofproblem. Ifwehaveanalgorithmthat

is good onthem, then itisveryprobablyalso good foralot ofother problems ofthesamelass. Ouraim is

thentodesignaPSOvariantthatisbetterthanSPSO07forthesethreefuntions. OurhopeisthatthisPSO

variantwillindeed bealsobetterthanSPSO 07onmoreproblemsof thesamekind. Andifitistrueevenfor

somehighlymultimodalproblems,and/orforhigherdimensionality,well,weanonsiderthatasaniebonus!

1

Forexample,itispossibletoassigna"weight"toeahproblem(whihrepresentshow"important"isthiskindofproblemfor

theuser) andtolinearly ombinethe

c A,B,i

^and

c B,A,i

^{. Butif,forasetof(nonidential)weights,}

A

^isbetterthan

B

^,thenit

alwaysexistsanotheroneforwhih

B

^isbetterthan

A

^.

Tab.1: Thebenhmark. Moredetailsaregivenin 9.1

Searh Required Maximumnumber

spae auray oftnessevaluations

CEC2005Sphere

[ − 100, 100] ³⁰

^{0.000001 15000}

CEC2005Rosenbrok

[ − 100, 100] ¹⁰

^{0.01 50000}

Tripod

[ − 100, 100] ²

^{0.0001 10000}

4 Step 2: a highly exible PSO

MymaintoolisaPSOversion(Code),whihisbasedonSPSO07. HoweverIhaveaddedalotofoptions,in

ordertohaveaveryexibleresearhalgorithm. Atually,Ioftenmodifyit, butyoualwaysanndthelatest

version(namedBalanedPSO)onmytehnial site[4℄. WhenIused itforthis paper,themain optionswere:

•

^{two kind of randomness (KISS [5℄, and the standard randomnessprovided in LINUX C ompiler). In}

what follows,IalwaysuseKISS,sothattheresultsanbemorereproduible

•

^seveninitialisationmethodsforthepositions(in partiularavariantoftheHammersley'sone[6℄)

•

^sixinitialisationmethodsfortheveloities(zero,ompletelyrandom,randomaround aposition,et.)

•

^{twolampingoptionsfortheposition(atually, justlampinglikeinSPSO 07,ornolampingandno}

evaluation)

•

possibilitytodeneasearhspaegreaterthanthefeasiblespae. Ofourse,ifapartileiesoutsidethe feasiblespae,itstnessisnotevaluated

•

^{sixloalsearhoptions(noloalsearhasinSPSO07,uniformin thebestloalarea,et.). Notethatit}

impliesarigorousdenitionofwhat aloal areais. See[7℄

•

^{twooptionsfortheloopoverpartiles}(sequentialorat random)

•

^{sixstrategies}

Thestrategiesarerelatedtothelassialveloityupdateformula

v (t + 1) = wv (t) + R (c 1 ) (p (t) − x (t)) + R (c 2 ) (g (t) − x (t))

⁽¹⁾

One anuse dierentoeients

w

^{, and dierent random} distributions

R

^{. Themost} interesting point is thatdierentpartilesmayhavedierentstrategies.

In theCsoureode, eah optionhasanidentier toeasily desribethe optionsused. Forexample, PSO

P1V2means: SPSO07, inwhihtheinitialisationofthepositionsisdonebymethod1,andtheinitialisation

oftheveloitiesbythemethod2. Pleaserefertotheonlineodeformoredetails. Inourase,wewillnowsee

nowhowaninterestingPSOvariantan bedesignedbyusing justthreeoptions.

5 Step 3: seleting the right options

Firstofall,wesimulateSPSO07,bysettingtheparametersandoptionstotheorrespondingones. Theresults

over500runs aregivenin thetable 2. Inpassing,itisworthnotingthat theusualpratieoflaunhingonly

25oreven100runs isnotenough,forreally badrunsmayourquiterarely. ThisisobviousforRosenbrok,

asweansee from the table 3. Any onlusionthat is drawn after just 100runs is risky, partiularly if you

onsiderthemeanbest value. Thesuessrateismorestable. Moredetails aboutthispartiularfuntion are

givenin9.5.

Tab.2: StandardPSO2007. Resultsover500runs

Suessrate Meanbest

CEC2005Sphere 84.8%

10 ^{− 6}

CEC2005Rosenbrok 15% 12.36

Tripod 46% 0.65

Tab.3:ForRosenbrok,themeanbestvalueishighlydependingonthenumberofruns(50000tnessevaluations

foreahrun). Thesuessrateismorestable

Runs Suess rate Meanbestvalue

100 16% 10.12

500 15% 12.36

1000 14,7% 15579.3

2000 14% 50885.18

5.1 Applying a spei improvement method

Whenweonsider thesurfaes oftheattrationbasins,theresultforTripodisnotsatisfying(thesuessrate

should begreater then50%). Whatoptions/parametersould we modify in order to improvethe algorithm?

Letusallthethree attrationbasinsas

B 1

^,

B 2

^{, and}

B 3

^{. Theproblemisdeeptivebeausetwoofthem,say}

B 2

^and

B 3

^{, leadto only loal optima. If, for aposition}

x

ⁱⁿ

B 1

^{(i.e. in the basinof theglobal optimum) the}

neighbourhood best

g

^{is eitherin}

B 2

^{or in}

B 3

^{, then, aordingto theequation 1,evenifthe distanebetween}

x

^and

g

^{is high, the position}

x

^{may be easily modiedsuh that it isnot in}

B 1

^{anymore. This isbeausein}

SPSO07theterm

R (c 2 ) (g (t) − x (t))

^issimply

U (0, c 2 ) (g (t) − x (t))

^,where

U

^istheuniformdistribution.

However,weareinterestedonfuntionswithasmallnumberofloaloptima,andthereforewemaysuppose

that thedistane betweentwooptimais usuallynotverysmall. So,in ordertoavoidtheabovebehaviour,we

usetheideaisthatthefurtheraninformeris,thesmallerisitsinuene(thisanbeseenasakindofnihing).

Wemaythentrya

R (c 2 )

^{that is infat a}

R (c 2, | g − x | )

^{,and dereasingwith}

| g − x |

^{. TheoptionalformulaI}

usetodothatin myexible PSOis

R (c 2, | g − x | ) = U (0, c 2 )

1 − | g − x | x max − x min

λ

(2)

Experimentssuggestthat

λ

^{should notbetoohigh, beauseinthat ase,although thealgorithmbeomes}

almost perfet for Tripod, the result for Sphere beomes quite bad. In pratie,

λ = 2

^{seemsto bea good}

ompromise. With thisvaluetheresultfor Sphereisalsoimprovedasweanseefrom thetable4. Aording

to ournomenlature, this PSO is alled PSO R2. Theresultfor Rosenbrokmaybenowslightlyworse, but

we haveseenthat wedo notneed toworry toomuh aboutthe meanbest,ifthe suessrate seemsorret.

Anyway,wemaynowalsoapplysomegeneral improvementoptions.

Tab.4: ResultswithPSO R2(distanedereasing distribution,aordingto theequation2

Suess rate Meanbest

CEC2005Sphere 98.6%

0.14 × 10 ^{− 6}

CEC2005Rosenbrok 13.4% 10.48

Tripod 47.6% 0.225

5.2 Applying some general improvement options (initialisations)

Theaboveoptionwasspeiallyhoseninordertoimprovewhatseemedtobetheworstresult,i.e. theonefor

theTripod funtion. Now,wean triggersomeotheroptionsthat areoftenbeneial,at leastformoderately

multimodalproblems:

•

^{modiedHammersleymethodforthe}initialisationofthepositions

x

•

^{One-rand method for the} initialisation of the veloity of the partile whose initial position is

x

^{, i.e.}

v = U (x min , x max ) − x

^{. Notethatin SPSO07,themethodistheHalf-di one,i.e.}

v = 0.5 (U (x min , x max ) − U (x min , x max ))

ThismodiedalgorithmisPSOR2P2V1. Theresultsaregiveninthetable5,andarelearlybetterthanthe

onesofSPSO07. Theyarestillnotompletelysatisfying(f. Rosenbrok),though. So,weantryyetanother

option,whihanbealled bi-strategy.

Tab.5: Resultswhenapplyingalsodierentinitialisations,forpositionsandveloities(PSOR2P2V1)

Suess rate Meanbest

CEC2005Sphere 98.2%

0.15 × 10 ^{− 6}

CEC2005Rosenbrok 18.6% 31132.29

Tripod 63.8% 0.259

5.3 Bi-strategy

The basiidea is verysimple: we usetwo kindsof partiles. In pratie, during the initialisation phase, we

assignoneoftwopossiblebehaviours,with aprobabilityequalto0.5. Thesetwobehavioursaresimply:

•

^{theoneofSPSO07. Inpartiular,}

R (c 2 ) = U (0, c 2 )

•

^{ortheoneofPSOR2(i.e. byusingequation2)}

TheresultingalgorithmisPSO R3P2V1. As weanseefrom thetable6,forallthethree funtionsnowwe

obtainresultsthatarealsolearlybetterthantheonesofSPSO07. Suess ratesareslightlyworseforSphere

andRosenbrok,slightlybetterforTripod,sonolearomparisonispossible. Howevermoretests(notdetailed

here) showthat this variant is morerobust, aswe anguess by looking at themean best values, sowe keep

it. Twoquestions,though. Isitstillvalidfordierentmaximumnumberoftnessevaluations(searheort).

Andisittrueformoreproblems,eveniftheyarenotreallyin thesamelass,in partiulariftheyarehighly

multimodal? Bothanswersarearmative,astakledinnextsetions.

Tab.6: Resultsbyaddingthebi-strategyoption(PSOR3P2V1)

Suessrate Meanbest

CEC2005Sphere 96.6%

< 10 ^{− 10}

CEC2005Rosenbrok 18.2% 6.08

Tripod 65.4% 0.286

6 Now, let's try

6.1 Suess rate vs Searh eort

Here, on the same three problems, we simply onsider dierent maximum numbers of tness evaluations

(

F E max

^{),andweevaluatethesuessrateover500runs. Asweanseefrom thegure1,forany}

F E max

^the

suessrate ofour variantis greaterthanthe oneofSPSO 07. SO,weansafelysaythat it is reallybetter,

at least on this small benhmark. Of ourse, it is not alwaysso obvious. Giving along list of resultsis out

of thesopeof this paper, whih isjust aboutadesign method, but weanneverthelesshavean ideaof the

performaneonafewmoreproblems.

6.2 Moderately multimodal problems

Table7andgure2areaboutmoderatelymultimodalproblems. Thisisasmallseletion,toillustratedierent

ases:

•

^learimprovement,i.e. nomatterwhatthenumberoftnessevaluationsis,buttheimprovementissmall (Shwefel,PressureVessel). AtuallySPSO07isalreadyprettygoodontheseproblems(forexample,for

PressureVessel,SOMAneedsmorethan50000tnessevaluationstosolveit[8℄),sooursmallmodiations

annotimproveitalot.

•

questionableimprovement,i.e. dependingonthenumberoftnessevaluations(Compressionspring)

•

^{lear big}improvement(Geartrain). Forthisproblem, andafter 20000tness evaluations, not onlythe suessrateofPSOR3P2V1is92.6%,butitndstheverygoodsolution

(19, 16, 43, 49)

^{(oranequivalent}

permutation),85timesover500runs. Thetnessofthissolutionis

2.7 × 10 ^{− 12}

^{(SOMAneedsabout200,000}

evaluationstondit).

Even when the improvement is not very important, the robustness is inreased. For example, for Pressure

Vessel, with 11000tness evaluations, themean best is 28.23 (standard dev. 133.35)with SPSO 07, asit is

18.78(standarddev. 56.97)withPSOR3P2V1.

Tab. 7: Moremoderatelymultimodalproblems. See9.2fordetails

Searh Required

spae auray

CEC2005Shwefel

[ − 100, 100] ¹⁰

^0.00001

Pressurevessel 4variables 0.00001

(disreteform) objetive7197.72893

Compressionspring 3variables 0.000001

objetive2.625421

(granularity0.001for

x 3

⁾

Geartrain 4variables

10 ^{− 9}

6.3 Highly multimodal problems

Table8andgure3areforhighlymultimodalproblems. ThegoodnewsisthatourmodiedPSOisalsobetter

evenforsomehighly multimodalproblems. It isnottrueallthetime(seeGriewankorCellularphone),butit

wasnotitsaim,anyway.

7 Claims and suspiion

We have seen that it is possible to improve Standard PSO 2007 by modifying the veloity update equation

and the initialisationshemes. However,this improvementis not valid arossall kindsof problems, and not

valid aross all riterions (in partiular, it may be depending on the number of tness evaluations). Also,

the improvement is not always very impressive. Thus, this study inites us to be suspiious when reading

an assertion like My PSO variant is far better than Standard PSO. Suh a laim has to be veryarefully

supported,byarigorousdenitionofwhatbettermeans,andbysigniantresultsonagoodrepresentative

benhmark,onalargerangeofmaximumnumberoftnessevaluations. Also,wehavetobeveryarefulwhen

using the meanbest riterionfor omparison, for it maybe meaningless. And, of ourse, theproposed PSO

variantshouldbeompared totheurrent StandardPSO,andnotto anoldbad version.

(a)Sphere

(b)Rosenbrok

()Tripod

Fig.1: SuessprobabilityvsSearheort. Forany

F E max

^{thevariantisbetter}

(a)Shwefel (b)Pressurevessel

()Compressionspring (d)Geartrain

Fig.2: OntheShwefeland Pressurevessel problemsPSOR3P2V1isslightlybetterthanSPSO 07for any

numberoftnessevaluations. OntheCompressionspringproblem, itistrueonlywhenthenumberoftness

evaluationsisgreaterthanagivenvalue(about19000). So,onthisproblem, eitherlaimSPSO 07isbetter

orPSOR3P2V1isbetter iswrong

(a)Rastrigin

(b)Griewank

()Akley (d)Cellularphone

Fig.3: Suessprobabilityforsomehighlymultimodalproblems. Althoughdesignedformoderatelymultimodal

problems,PSOR3P2V1isevensometimesgoodfortheseproblems. Butnotalways

Tab.8: Highlymultimodalproblems. See9.3fordetails

Searh Required

spae auray

CEC2005Rastrigin

[ − 5, 5] ¹⁰

^0.01

CEC2005Griewank

[ − 600, 600] ¹⁰

^0.01

(notrotated)

CEC2005Akley

[ − 32, 32] ¹⁰

^0.0001

(notrotated)

Cellularphone

[0, 100] ²⁰ 10 ^{− 8}

8 Home work

Thespei improvementmodiationof SPSO07usedherewasformoderatelymultimodalproblems,in low

dimension. LetusallthemM-problems. Now,whatouldbeaneetivespeimodiationforanotherlass

of problems? Take, for examplethe lassof highly multimodal problems, but still in low dimension(smaller

than30). Letusall themH-problems.

First,wehavetodeneasmallrepresentativebenhmark. Hint: inludeGriewank10D,fromtheCEC2005

benhmark(no needto usethe rotatedfuntion). Seond, wehaveto understand in whihwaythe diulty

ofanH-problemisdierentfromthatofanM-problem. Hint: onanH-problem,SPSO07isusuallylesseasily

trapped into a loal minimum, just beause theattration basins are small. Onthe ontrary, if apartile is

insidethegood attrationbasin(theoneoftheglobaloptimum),itmayevenleaveitprematurely. Andthird,

we haveto nd what optionsare neededto opewith the foundspei diulty(ies). Hint: just makesure

the urrent attration basin is well exploited, aquik loal searh may be useful. A simple way is to dene

aloal areaaround thebest known position, andto sample itsmiddle (PSOL4) 2

. With just this option,an

improvementseemspossible,asweanseefrom gure4fortheGriewankfuntion. However,itdoesnotwork

verywellforRastrigin.

Allthiswillprobablybethetopiofafuturepaper,butforthemoment,youanthinkatityourself. Good

luk!

2

Let

g = (g _1, g _2, . . . , g D )

^{be the best known position. On eah dimension}

i

^{, let}

p i

^and

p ^′ _i

^{are the nearest oordinates of}

knownpoints,"ontheleft",and"onthe right"of

g i

^{. Theloal area}

H

^isthe

D

^-retangle(hyperparallelepid)artesianprodut

⊗ _i

g i − α (g i − p i ) , g i + α p ^′ _i − g i

with,inpratie,

α = 1/3

^{.Thenitsenterissampled. Usually,itisnot}

g

^.

Fig.4: Griewank,omparisonbetweenSPSO07andPSOL4. Forahighlymultimodalproblem,averysimple

loalsearhmayimprovetheperformane.

9 Appendix

9.1 Formulae for the benhmark

Tab.9: Benhmarkdetails

Formula

Sphere

− 450 +

30

X

d=1

(x d − o d ) ²

^{Therandomosetvetor}

O = (o 1 , · · · , o 30 )

isdenedbyitsCode.

Thisisthesolutionpoint.

Rosenbrok

390 +

10

X

d=2

100 z _d− ² 1 − z d 2

+ (z d − 1 − 1) ²

Therandomosetvetor

O = (o 1 , · · · , o 10 )

with

z d = x d − o d + 1

^{isdenedbyitsCode.}

Thisisthesolutionpoint

Thereis alsoaloalminimumon

(o 1 − 2, · · · , o 30 )

^{. Thetnessvalueisthen}

394.

Tripod

1 − sign(x 2 )

2 ( | x 1 | + | x 2 + 50 | )

+ ^1+sign(x ₂ ^{2 ) 1 −sign(x} ₂ ^{1 )} (1 + | x 1 + 50 | + | x 2 − 50 | ) + ^1+sign(x ₂ ^{1 )} (2 + | x 1 − 50 | + | x 2 − 50 | )

sign (x) = − 1 x ≤ 0

^{Thesolutionpointis}

(0, − 50)

Oset forSphere/Parabola(Csoureode)

statidoubleoset_0[30℄={-3.9311900e+001,5.8899900e+001,-4.6322400e+001,-7.4651500e+001,-1.6799700e+001,

-8.0544100e+001, -1.0593500e+001, 2.4969400e+001, 8.9838400e+001, 9.1119000e+000, -1.0744300e+001, -

2.7855800e+001,-1.2580600e+001,7.5930000e+000,7.4812700e+001,6.8495900e+001,-5.3429300e+001,7.8854400e+001,

-6.8595700e+001, 6.3743200e+001, 3.1347000e+001, -3.7501600e+001, 3.3892900e+001, -8.8804500e+001, -

7.8771900e+001,-6.6494400e+001,4.4197200e+001,1.8383600e+001,2.6521200e+001,8.4472300e+001};

Oset forRosenbrok (Csoureode)

statidoubleoset_2[10℄={8.1023200e+001,-4.8395000e+001,1.9231600e+001,-2.5231000e+000,7.0433800e+001,

4.7177400e+001,-7.8358000e+000,-8.6669300e+001,5.7853200e+001};

9.2 Formulae for the other moderately multimodal problems

9.2.1 Shwefel

Thefuntionto minimiseis

f (x) = − 450 +

10

X

d=1 d

X

k=1

x k − o k

! ²

Thesearhspaeis

[ − 100, 100] ¹⁰

^{. Thesolutionpointistheoset}

O = (o 1 , . . . , o 10 )

^,where

f = − 450

^.

Oset (Csoureode)

statidoubleoset_4[30℄=

{3.5626700e+001,-8.2912300e+001,-1.0642300e+001,-8.3581500e+001,8.3155200e+001,4.7048000e+001,

-8.9435900e+001,-2.7421900e+001,7.6144800e+001,-3.9059500e+001};

9.2.2 Pressure vessel

Justinshort. Formoredetails,see[9,10,11℄. Therearefourvariables

x 1 ∈ [1.125, 12.5]

granularity

0.0625 x 2 ∈ [0.625, 12.5]

granularity

0.0625 x 3 ∈ ]0, 240]

x 4 ∈ ]0, 240]

andthree onstraints

g 1 := 0.0193x 3 − x 1 ≤ 0 g 2 := 0; 00954x 3 − x 2 ≤ 0

g 3 := 750 × 1728 − πx ₃ ² x 4 + ⁴ ₃ x 3

≤ 0

Thefuntion tominimiseis

f = 0.06224x 1 x 3 x 4 + 1.7781x 2 x ² ₃ + x ² ₁ (3.1611x + 19.84x 3 )

Theanalytialsolutionis

(1.125, 0.625, 58.2901554, 43.6926562)

^{whihgivesthetnessvalue}7,197.72893. To taketheonstraintsinto aount,apenaltymethodisused.

9.2.3 Compression spring

Formoredetails,see[9,10,11℄. Therearethreevariables

x 1 ∈ { 1, . . . , 70 }

granularity

1 x 2 ∈ [0.6, 3]

x 3 ∈ [0.207, 0.5]

granularity

0.001

andveonstraints

g 1 := ^{8C f F} _πx ^max 3 ^{x 2}

3 − S ≤ 0

g 2 := l f − l max ≤ 0 g 3 := σ p − σ pm ≤ 0 g 4 := σ p − ^F K ^p ≤ 0 g 5 := σ w − ^{F max} _K ^{−F p} ≤ 0

with

C f = 1 + 0.75 _x ^{x 3}

2 − x 3 + 0.615 _x ^{x 3}

2

F max = 1000 S = 189000

l f = ^{F max} _K + 1.05 (x 1 + 2) x 3

l max = 14 σ p = ^F _K ^p σ pm = 6

F p = 300

K = 11.5 × 10 ⁶ _8x ^{x 4 3}

1 x ³ ₂

σ w = 1.25

andthefuntion tominimiseis

f = π ² x 2 x ² ₃ (x 1 + 1) 4

The best known solution is

(7, 1.386599591, 0.292)

^{whih gives the tness value} 2.6254214578. To take the onstraintsinto aount,apenaltymethodisused.

9.2.4 Geartrain

Formoredetails,see[9,11℄. Thefuntion tominimiseis

f (x) = 1

6.931 − x 1 x 2

x 3 x 4

²

Thesearhspaeis

{ 12, 13, . . . , 60 } ⁴

^{. Thereareseveralsolutions,dependingontherequiredpreision. For}

example

f (19, 16, 43, 49) = 2.7 × 10 ^{− 12}

9.3 Formulae for the highly multimodal problems

9.3.1 Rastrigin

Thefuntionto minimiseis

f = − 230 +

10

X

d=1

(x d − o d ) ² − 10 cos (2π (x d − o d ))

Thesearhspaeis

[ − 5, 5] ¹⁰

^{. Thesolutionpointistheoset}

O = (o 1 , . . . , o 10 )

^,where

f = − 330

^.

Oset (Csoureode)

statidoubleoset_3[30℄=

{1.9005000e+000,-1.5644000e+000,-9.7880000e-001,-2.2536000e+000,2.4990000e+000,-3.2853000e+000,

9.7590000e-001,-3.6661000e+000,9.8500000e-002,-3.2465000e+000};

9.3.2 Griewank

Thefuntionto minimiseis

f = − 179 + P 10

d=1 (x d − o d ) ²

4000 −

10

Y

d=1

cos

x d − o d

√ d

Thesearhspaeis

[ − 600, 600] ¹⁰ .

^{Thesolutionpointis theoset}

O = (o 1 , . . . , o 10 )

^,where

f = − 180

^.

Oset (Csoureode)

statidoubleoset_5[30℄=

{-2.7626840e+002,-1.1911000e+001,-5.7878840e+002,-2.8764860e+002,-8.4385800e+001,-2.2867530e+002,

-4.5815160e+002,-2.0221450e+002,-1.0586420e+002,-9.6489800e+001};

9.3.3 Akley

Thefuntionto minimiseis

f = − 120 + e + 20e ^{− 0.2} q

1 D

P ¹⁰

d=1 (x d − o d ) ²

− e ^{D 1} P 10

d=1 cos(2π(x d −o d ))

Thesearhspaeis

[ − 32, 32] ¹⁰ .

^{Thesolutionpointis theoset}

O = (o 1 , . . . , o 10 )

^,where

f = − 140

^.

Oset (Csoureode)

statidoubleoset_6[30℄=

{-1.6823000e+001,1.4976900e+001,6.1690000e+000,9.5566000e+000,1.9541700e+001,-1.7190000e+001,

-1.8824800e+001,8.5110000e-001,-1.5116200e+001,1.0793400e+001};

9.3.4 Cellular phone

Thisproblemarisesinarealappliation,onwhihIhaveworkedintheteleommuniationsdomain. However,

here, all onstraints has been removed, exept of ourse the ones given by the searh spae itself. We have

a square at domain

[0, 100] ²

^{, in whih we want to put}

M

^{stations. Eah station}

m k

^{has two oordinates}

(m k,1 , m k,2 )

^{. These arethe}

2M

^{variablesoftheproblem. Weonsider eah integer pointofthedomain, i.e.}

(i, j) , i ∈ { 0, 1, . . . , 100 } , j ∈ { 0, 1, . . . , 100 }

^{. Oneahintegerpoint,theeldinduedbythestation}

m k

^isgiven

by

f i,j,m k )=

1

(i − m k,1 ) ² + (j − m k,2 ) ² + 1

andwewanttohaveatleastoneeld thatisnottooweak. Finally,thefuntion tominimiseis

f = 1

P 100 i=1

P 100

j=1 max k (f i,j,m k )

In this paper, we set

M = 10

^{. Therefore the dimension of the problem is 20. The objetive value is}

0.005530517. This is not the true minimum, but enough from an engineering point of view. Of ourse, in

reality we do not know the objetivevalue. We just run the algorithm several times for a given number of

tnessevaluations,andkeepthebestsolution. Fromthegure5weanseeasolutionfoundbySPSO07after

20000 tness evaluations. Atually, for this simplied problem, more eient methods do exist (Delaunay's

tessellation, for example), but those an not be used as soon as we introdue a third dimension and more

onstraints,sothattheeldisnotspherialanymore.

Fig. 5: Cellularphone problem. A possible (approximate) solution for 10 stations, found by SPSO 07 after

20000tnessevaluations

9.4 A possible simpliation

Wemaywonderwhetherthetwoinitialisationmethodsused in5.2are reallyusefulornot. Letus tryjust the

bi-strategyoption,by keepingthe initialisations ofSPSO 07. Resultsare in thetable 10. Whenweompare

theresultswiththosegiveninthetable6,weanseethat forthethree funtions,theresultsarenotasgood.

However,theyarenotbadatall. So,forsimpliity,itmaybeperfetly aeptabletousejustPSO R3.

Tab.10: Resultswithjustthebi-strategyoption(PSOR3)

Suessrate Meanbest

CEC2005Sphere %

CEC2005Rosenbrok %

Tripod 60.6% 0.3556

9.5 When the mean best may be meaningless

OntheRosenbrokfuntion,wehavequiklyseenthatthemeanbest dependsheavilyonthenumberof runs

(see table 3), and therefore is not an aeptable performane riterion. Here is a moredetailed explanation

of this phenomenon. First we show experimentally that the distribution of the errors for this funtion is

not Gaussian, and, more preisely, that the probability of a verybad run (i.e. a veryhigh tness value) is

not negligible. Then, and more generally, assuming that for agivenproblem this property is true, asimple

probabilistianalysisexplainswhythesuessrateisamorereliableriterion.

9.5.1 Distributionof the errorsforRosenbrok

We runthe algorithm 5000 times, with 5000 tness evaluations for eah run,i.e. just enough to havea non

zerosuessrate. Eahtime,wesavethebest valuefound. Weanthenestimatetheshapeofthedistribution

of these 5000 values, seen as ourrenes of a random variable. Contrary to what is sometimes said, this

distributionisfar fromnormal(Gaussian)one. Indeed,themain peak isveryaute, andthere aresomevery

highvalues. Evenifthesearerare,itimpliesthatthemeanvalueisnotreallyrepresentativeoftheperformane

of the algorithm. It would be better to onsider the value on whih the highest peak (the mode) lies. For

SPSO 07,itis about7(the rightvalueis0), and themean is25101.4(there areafewverybad runs). As we

ansee from gure6,wehaveaquite niemodel byusing theunion ofapowerlaw(onthe left ofthemain

peak),andaCauhylaw(ontheright).

f requency = α _class ^{m k} k+1

^if

class ≤ m

= ¹ _{π (class} ^γ

− m) ² +γ ²

^else

with

γ = 1.294

^,

m = 7

^,and

k = 6.5

^{. Note thataseondpowerlawfortherightpartofthe urve(instead}

of theCauhyone) would notbe suitable: although itould bebetterfor lassvaluessmaller thansay15, it

wouldforgettheimportantfatthattheprobabilityofhighvaluesisfarfromzero. Atually,eventheCauhy

modelisoverlyoptimisti,aswean seefromthemagniedversion(lasses40-70)ofthegure6,but atleast

theprobabilityisnotvirtuallyequaltozero,aswiththepowerlawmodel.

ForPSO R3P2V1,the mode isabout6,i. e. just slightlybetter. However, themean isonly3962.1. It

showsthat thisversionis abit morerobust(verybad runsdonotexist). Forbothalgorithms,thesmallpeak

(around10,astherightvalueis4)orrespondsto aloal optimum. Thesmallvalley (around3)is alsodue

totheloal optimum: sometimes(but veryrarely)theswarmisquiklytrappedintoit. It showsthatassoon

as there are loal optima the distribution has neessarily somepeaks, at least for asmall number of tness

evaluations.

9.5.2 Meanbestvs suessrateas riterion

Arun issaidtobesuessful ifthenalvalueissmallerthanasmall

ε

^{,andbad ifthenalvalueisgreater}

thanabig

M

^{. Foronerun,let}

p M

^betheprobabilityof thatrun beingbad. Then, theprobability, over

N

runs,thatatleastoneof theruns isbadis

p M,N = 1 − (1 − p M ) ^N

Thisprobabilityinreasesquiklywiththenumberofruns. Now,let

f i

^{bethenalvalueoftherun}

i

^{. The}

estimateofthemeanbest valueisusuallygivenby

µ N = P N

i=1 f i

N

Let us saythe suessrate is

ς

^{. It meanswehave}

ςN

^{suessful runs. Letus onsider another sequene}

of

N

^{runs,exatly thesame, exept that}

k

^{runs are replaed by bad ones. Let}

m

^{be themaximumof the}

orresponding

f i

^{in therstsequeneof}

N

^{runs. The}probabilityofthiseventis

p M,N,1 = p ^k _M (1 − p M ) ^{N − k}

Forthenewsuessrate

ς ^′

^,wehave

ς ≥ ς ^′ ≥ ς − k N

Forthenewestimate

µ ^′ _N

^{ofthemeanbest, wehave}

µ ^′ _N > µ N + k M − m N

Weimmediatelyseethatthereisaproblemwhenabigvalue

M

^{ispossiblewithanonnegligible}probability:

when thenumberofruns

N

^{inreasesthesuess ratemay slightlyderease,but then themean}dramatially inreases. Letussupposethat, foragivenproblemandagivenalgorithm,thedistributionoftheerrorsfollows

aCauhylaw. Thenwehave

(a)Globalshape

(b)Zoom"onlasses40to70

Fig.6: Rosenbrok. Distributionofthebest valueover5000runs. Onthezoom,weanseethat theCauhy

model, although optimisti, givesa better idea of thedistribution than the powerlawmodel forlass values

greaterthan40

p M = 0.5 − 1 π arctan

M γ

With theparametersof themodel of thegure6, wehavefor example

p 5000 = 8.3 × 10 ^{− 5}

^{. Over}

N = 30

runs,theprobabilitytohaveatleastonebadrun(tnessvaluegreaterthan

M = 5000

^)islow,just

2.5 × 10 ^{− 3}

^.

Letussaywendanestimateofthemeantobe

m.

^Over

N = 1000

^runs,theprobabilityis

0.08

^,whihisquite

high. It may easily happen. Insuh aase,eveniffor allthe otherruns the best valueisabout

m

^{, thenew}

estimateisabout

(4999m + 5000) /1000

^{,whihmaybeverydierentfrom}

m

^{. Inpassing,andifwelookatthe}

table3,thissimpliedexplanationshowsthat forRosenbrokaCauhylawbasedmodel isindeed optimisti.

In otherwords,ifthe numberof runs istoosmall, you may neverhaveabad one,andtherefore, wrongly

estimate the mean best, even when it exists. Note that in ertain asesthe mean maynot even exist at all

(for example, in ase of a Cauhy law), and therefore any estimate of a mean best is wrong. That is why

it is important to estimate the mean for dierent

N

^{values(but of ourse with the same number of tness}

evaluations). Ifitseemsnotstable, forgetthis riterion,and justonsider thesuessrate, or,asseenabove,

themode. Astherearealotof papersinwhihtheprobableexisteneof themeanisnotheked,itisworth

insistingonit: if thereisnomean,givinganestimate ofit isnottehnially orret. Worse,omparing two

algorithmsbasedonsuhanestimate issimplywrong.

Referenes

[1℄ PSC,PartileSwarmCentral,http://www.partileswarm.info.

[2℄ CEC, Congress on Evolutionary Computation Benhmarks,http://www3.ntu.edu.sg/home/epnsugan/,

2005.

[3℄ L.Gagne,Steadystateevolutionaryalgorithmwithanoperatorfamily, in EISCI,(Kosie,Slovaquie),

pp.373379,2002.

[4℄ M.Cler,MathStuaboutPSO,http://ler.maurie.free.fr/pso/.

[5℄ G.MarsagliaandA. Zaman,Thekissgenerator, teh.rep.,Dept.ofStatistis,U. ofFlorida,1993.

[6℄ T.-T. Wong, W.-S. Luk, and P.-A. Heng, Sampling with Hammersley and Halton points, Journal of

Graphis Tools,vol.2(2), pp.924,1997.

[7℄ M.Cler,Themythialbalane,orwhenPSOdoenotexploit,Teh.Rep.MC2008-10-31,2008.

[8℄ I.Zelinka,SOMA-Self-OrganizingMigratingAlgorithm,inNewOptimizationTehniquesinEngineering,

pp.168217,Heidelberg,Germany: Springer, 2004.

[9℄ E.Sandgren,Nonlinearintegeranddisreteprogramminginmehanialdesignoptimization, 1990.ISSN

0305-2154.

[10℄ M.Cler,PartileSwarmOptimization. ISTE(InternationalSientiandTehnialEnylopedia),2006.

[11℄ G.C.OnwuboluandB.V.Babu,NewOptimizationTehniquesinEngineering.Berlin,Germany: Springer,

2004.