CourseSwarmIntelligence

Chapter3:Assignments

WS16/17

SanazMostaghim

IntelligentSystemsGroupIKS,FIN

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Assignment1:SearchSpace/SoluFonSpace

ForatravelingsalesmanproblemofciFes,a)  Whatisthesizeofthe“SoluFonSpace”?b)  WewanttosolvethisproblemusingaPSOmethod.Note

thatPSOworksonconFnuousproblems.WhatshouldwechangeintheproblemformulaFon,sothatwecansolveTSPwithPSO?

c)  Whatisthesizeofthe“SearchSpace”fortheaboverepresentaFon?

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Assignment1:soluFona)  Whatisthesizeofthe“SoluFonSpace”?WehaveapermutaFonproblem,sothesizeofthesoluFonspaceis:

•  (𝑛−1)! forcyclicandnon-symmetricroutes

•  (𝑛−1)!/2  forcyclicandsymmetricroutes

Examplefor𝑛= 5:•  4!=24 forcyclicandnon-symmetricroutes

•  4!/2 =12 forcyclicandsymmetricroutes

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1-2-3-4-51-2-3-5-41-2-4-3-51-2-4-5-31-2-5-3-41-2-5-4-3

1-3-2-4-51-3-2-5-41-3-4-2-51-3-4-5-21-3-5-2-41-3-5-4-2

1-4-2-3-51-4-2-5-31-4-3-2-51-4-3-5-21-4-5-2-31-4-5-3-2

1-5-2-3-41-5-2-4-31-5-3-2-41-5-3-4-21-5-4-2-31-5-4-3-2

Symmetric

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Assignment1:soluFonb)  WewanttosolvethisproblemusingaPSOmethod.NotethatPSO

worksonconFnuousproblems.WhatshouldwechangeintheproblemformulaFon,sothatwecansolveTSPwithPSO?

c)  Whatisthesizeofthe“SearchSpace”fortheaboverepresentaFon?

SoluFon:ThestandardsoluFoncandidateforTSPisapermutaFon.HerewechangethesearchspacesothatwecanusePSOtosolvesuchaproblem.wedesignn-dimensionalreal-valuedsearchspaceForexample:n=5à(0.3,0.2,0.7,0.4,0.9)Toeachcityweassignareal-valuedasapriority,thentheciFesareorderedwithrespecttotheirprioriFes:(0.3,0.2,0.7,0.4,0.9)à5-3-4-1-212345

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Assignment2:FitnesslandscapesandPSO

DescribethemoFonofPSOinthefollowingenvironments.WhichofthedesignissuessuchasturbulencefactororstartpopulaFoncanhelptofindamaximuminsuchenvironments?

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Assignment2:soluFon(a)  RandomLandscape:PSOhasdifferentglobalbestparFclesateachiteraFon.TheswarmismovingalltheFme.-  StartpopulaFoncanbeiniFalizedusingrandom

samplingorGapsearch.InthiscasePSOcannotgetanybeberthanRandomSearchmethod!

-  Turbulencefactordoesn’thelp(b)  NeedleinHay-StackorFlatLandscapes:PSOdoesn‘tmove,ifwedonotapplyturbulencefactorandstartwithzerovelociFes.-  Weneedtoapplyturbulencefactortolettheswarm

moveanddoarandomsearchtobeabletofindthemaximumsoluFon

-  AwelldesignedstartpopulaFondoesn’thelpmuch

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Assignment2:soluFon(c)  PSOgoesveryquicklytowardsthewrongdirecFon.BothturbulencefactorandgoodstatpopulaFonneedtobeconsidered(d)  NiceLandscapes:PSOcanfindthemaximalpoint,butitcanhappenthatitistrappedintothelocalopFmum.-  Soweneedtoconsideraturbulencefactorto

avoidlocalopFmum-  AgoodstartpopulaFonisrequired

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Assignment3:NeighborhoodtopologyConsiderthefollowingpopulaFonofparFclesandaminimizaFonproblem.WhichparFclecanbeselectedasglobalbestfortheparFclewiththeindex=4:a)  usingthefullyconnectedtopology(knownasstandardordefault)?SoluFon:ParFclewithindex2->x=1.0a)  usingtheringtopology(indicesshowthetwoneighbors)?SoluFon:ParFclewithindex5->x=2.0

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Assignment4:PSO

PSOisdesignedtouseabracFonfuncFonstowardstheglobalbestandthepersonalbest.YouhavelearnedinChapter2thatwecandesignrepulsionfuncDons.HowwouldyouintegraterepulsionintoPSOsothatwedonotrequireturbulencefactor?WritethemoFonequaFonsforPSOwithrepulsion.

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Assignment4:soluFon

TurbulencefactorisappliedeitherrandomlyorwhenthevelociFesarezero(thereisnomoFon).Velocitygetszero,becausetheparFclesaresoclosetothe->WeaddrepulsionheretoavoidparFclestogotothesameposiFonasAbracFon-RepulsionfuncFon:Repelwhenclose,otherwiseabracFontowardsit

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Pg

vi(t+ 1) = wvi(t) + �1c1(Pi � xi(t))�

�(xi(t)� Pg)(a� b exp(

� || xi(t)� Pg ||2

c

))

Pg

xi(t+ 1) = xi(t) + vi(t+ 1)

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Assignment5:DynamicPSO

WritethemoFonequaFonsforthetwodifferentkindsofparFclesinthecharged-PSO(slideSI-3-53).SoluFon:1.  chargedparFcles:repeleachother->leadingtoacloudof

chargedparFclesaroundtheneutralparFcles2.  neutralparFcles:convergetothecurrentopFmum.

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ChargedparFcles

NeutralparFcles

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Assignment5:soluFon

1.  chargedparFclesinthepopulaFonPc:repulsiontoeachother!

isthenumberofparFclesinthechargedpopulaFonPc2.  neutralparFcles:convergetothecurrentopFmum(PSO).

MoFonequaFonsforallparFclesiintheneuralpopulaFonPn

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xi(t+ 1) = xi(t) + vi(t+ 1)

vi(t+ 1) = wvi(t) + �1c1(Pi � xi(t)) + �2c2(Pg � xi(t))

vi(t+ 1) =McX

j,j 6=i

Krep(xi(t)� xj(t))

Mc

xi(t+ 1) = xi(t) + vi(t+ 1)

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Assignment6:InteracFveOpFmizaFon

SupposethatauserobservestheopFmizaFonprocessonamonitor.a)HowcanheguideaPSOtowardshispreferredsoluDonasshownbelow?

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OpFmalsoluFon PreferredsoluFon

ParFcles

Feasiblearea x1

x2

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Assignment6:soluFonWeselectafullyconnectedtopologyanddefinethepreferredsoluFonastheglobalbestparFcle:Wecanselectφ1tobezero.SowecanforcetheopFmizaFontofocusmorearoundthepreferredsoluFon.iindicatesthei-thparFcle.

OpFmalsoluFon PreferredsoluFon

ParFcles

Feasiblearea x1

x2

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~

Pg = (x1g, x2g)

x1g

x2g

~vi(t+ 1) = w~vi(t) + �1c1(~Pi � ~xi(t)) + �2c2(~Pg � ~xi(t))

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Assignment6:InteracFveOpFmizaFon

b)Whatifhehasapreferredarea?

OpFmalsoluFon

Preferredarea

Feasiblearea x1

x2 x11x12

x21

x22

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Assignment6:soluFonLikein(a)weselectafullyconnectedtopology.WeselecttheparFcleclosesttothecenterofthepreferredarea:astheglobalbest:(Likein(a)wecanselectφ1tobezeroforabeberfocusingeffect)

Preferredarea

Feasiblearea

x11x12

x21

x22

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~xc = (x11 +x12 � x11

2, x21 +

x22 � x21

2)

~vi(t+ 1) = w~vi(t) + �1c1(~Pi � ~xi(t)) + �2c2(~Pg � ~xi(t))

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Assignment6:soluFon

c)Whathappensifthepreferredareaislocatedintheinfeasibleregion?

Preferredarea

Feasiblearea x1 x2

SoluFon:WefindtheclosestfeasibleparFcletothepreferredareaanddothesameasin(b).

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Assignment7:MOPs

Inthefigurebelowa)  findthesetofnon-dominatedparFcles.b)  rankalltheparFclesintermsofthenumberofparFclesthey

dominatec)  rankalltheparFclesintermsofthenumberofparFclesby

whichtheyaredominatedd)  whatcanwechangeintherankingfromslideSI-3-72,sothat

weavoidlargegapsonthefront.

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Assignment7:MOPs(a)and(b)

N=0

N=5

N=5

N=3

N=3

N=2

N=1 N=0N=0

N=0

N=0

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Assignment7:MOPs(c)

N=0

N=0

N=0

N=0

N=2

N=2

N=3 N=5N=6

N=1

N=1

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Assignment7:MOPs(d)

N=0

N=0N=0

N=0

N=2

N=2

N=3 N=5N=6

N=1

N=1

S=0

S=5

S=5

S=3

S=3

S=2

S=1 S=0S=0

S=0

S=0

Wecalltherankingfrom(b)asstrength(𝑆(𝑖)).

(b) (c)

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Assignment7:MOPs(d)

R=0

R=5R=5

R=3

R=13

R=12

R=16 R=16R=16

R=3

R=3

S=0

S=5

S=5

S=3

S=3

S=2

S=1 S=0S=0

S=0

S=0

(b) (d)

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Weusetherankingsin(b)tofindtherankofthesoluFons.ParFclei’srankisthesumoverthestrengthoftheparFcleswhichdominateitplusitsownstrength.Inthiscasethesmallertherank,thebeberisthefitness:

R(i) =X

j�i

s(j) + s(i)

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Assignment8:ConeDominaFon

LetuschangethedefiniFonofdominaFontothefollowing.ParFcle𝑖dominatesalltheotherparFclesasshowninthefollowingfigure.a)  MarkthedominatedsoluFonsinthefigurebelow.b)  Whatwillchangein(a)ifhavethefollowingdefiniFonsfor

dominaFon?

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Assignment8:ConeDominaFon

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Assignment8:ConeDominaFon(a)

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Assignment8:ConeDominaFon(b)

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Assignment8:ConeDominaFon(b)

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Assignment8:ConeDominaFon(b)

Conclusion:

1.  andfindcomplementarynon-dominatedsoluFons:2.  findsthekneesofafront

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Assignment9:MO-PSO

SupposethatalltheparFclesinthecurrentpopulaFonofaMO-PSOarenon-dominatedasshowninFigurebelow.NotethatourgoalistofindasetofdiversesoluFons.a)  HowdoesaMO-PSOwithRankingLeaderSelecFonMethod

wouldworkhere?b)  Proposearankingmethodtodealwiththeproblem.

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Assignment9:soluFon

a)  HowdoesaMO-PSOwithRankingLeaderSelecFonMethodwouldworkhere?TheparFclecannotmove!Wehaveaflatlandscape

b)  Proposearankingmethodtodealwiththeproblem:InordertoobtainagooddiversesetofsoluFons,weneedtofavorthesoluFonsclosetothegaps(markedby):

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Assignment9:soluFonT-5-31

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Assignment 9: MO-PSO

• We sort the particles in terms of one of the objectives.

• We compute the Euclidian distances between the particle 𝑖 and its two neighbors 𝑖 − 1 and 𝑖 + 1: 𝐷(𝑖, 𝑖 − 1) and 𝐷(𝑖, 𝑖 + 1)

𝑅𝑎𝑛𝑘(𝑖)

= 𝐷(𝑖, 𝑖 − 1) + 𝐷(𝑖, 𝑖 + 1)

• Here the larger the rank, the better is the solution

• The extreme solutions will get the highest ranks (𝑅 1 = 𝑅 𝑀 = ∞ )

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Assignment10:FocusingMO-PSO

SupposethattheuserhasgivenhispreferencesintheobjecFvespaceasinpicturesbelow.HowcanwechangetheMO-PSOleaderselecFonmechanismthattheswarmfocusesthesearchinthegivenpartintheobjecFvespace?

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Assignment10:FocusingMO-PSO

Inthiscase,alltheparFcleswillselecttheirglobalbestfromthenon-dominatedparFcleswhichareinthearea.IfthereisnosoluFoninthearea,allofthemwillselectoneleader,whichistheclosesttothearea.

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Assignment10:FocusingMO-PSO

WeselectthesigmamethodforleaderselecFon.TheparFclesselecttheparFclewiththesigmavalueclosesttozeroastheleader.

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Assignment10:FocusingMO-PSO

Weusesigmamethod.TheparFcleswiththesigmavaluesbetweenthetwosigmavaluesofthebordersofthepreferredarea(asshownabove)areselectedastheleaders.

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Assignment10:FocusingMO-PSO

Herethepreferredareaisnotfeasible.Itistheso-calledIdealpoint.Onlythenon-dominatedparFclesclosesttotheidealpointareselectedastheglobalbest.ItcouldonlybeoneparFcleinthepopulaFon/archivewhocanfulfillthis.

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Assignment11:Archiving

ProposeanarchivingmethodforMO-PSOwhichmustkeepafixednumberofN=5soluFonsandaddiFonallyhelptofindasetofnon-dominatedparFclesascloseaspossibletotheidealpointasshowninthefigurebelow?

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Assignment11:soluFon

•  Inthiscasethediversityisnotimportant.SowealwayskeeptheNclosetsparFclestotheIdealpoint

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Assignment12:Hypervolume(HV)

a)  WhichoneofthefollowingsoluFonshasthelargestmarginalHV?

b)  WhichonehasthesmallestmarginalHV?

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Assignment12:soluFon

a)  A(NoteitisnotthepointC!,why?)b)  B

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A

B

C

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Assignment12:Hypervolume(HV)

c)  SupposewearecomparingtwosetsAandBwitheachotherandobtainHV(A)<HV(B).Whichoneofthefollowingscenarioscanbetrueandwhy?

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(1) (2)

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Assignment12:soluFon

c)  HV(A)<HV(B)->Scenario(1)“A”musthaveboththeworstdiversityandconvergence.

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(1) (2)

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Assignment13:𝜖-dominaFon

a)  WhichoneofthefollowingparFclescanbenon-𝜖-dominated,ifweselect𝜖=1?a)  Whatabout𝜖 =3?

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Assignment13:soluFon

a)  𝜖=1

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f/1+𝜖

f

f1

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Assignment13:soluFon

b) 𝜖=3

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f/1+𝜖

f

f1

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Assignment14:Archiving

SupposewehaveapopulaFonofMparFclesandwanttofindanarchiveofnon-dominatedparFclebyinserFngthenon-dominatedparFclesfromthepopulaFonintothearchive.WriteafastmechanismforidenFfyingthenon-dominatedparFcleswiththeleastnumberofoperaFonsaspossible.

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P(t)M

A(t+1)

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Assignment14:soluFon

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Assignment 14: solution Input: Given set 𝑃 of size 𝑀 Output: The non-dominated set 𝐴

Step 1: Set counter 𝑖 = 1 and create an empty non-dominated set A. Step 2: For a particle 𝑗 ∈ 𝑃 (but 𝑗 ≠ 𝑖), check if particle 𝑗 dominates particle

𝑖. If yes, go to Step 4. Step 3: If more particles are left in 𝑃, increment 𝑗 by one and go to Step 2;

otherwise, set A = 𝐴 ∪ {𝑖}. Step 4: Increment 𝑖 by one. If 𝑖 ≤ 𝑀, go to Step 2; otherwise stop and declare

A as the non-dominated set.

P M

A Perform maximal 𝑀2 comparisons and insert the found non-dominated particle in the archive

Naive and Slow

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Assignment14:soluFon

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Assignment 14: solution Input: Given set 𝑃 of size 𝑀 Output: The non-dominated set 𝐴

Step 1: Set counter 𝑖 = 1 and include first member in A. Step 2: For each solution 𝑗 ∈ 𝑃 (but 𝑗 ∉ 𝐴), Take one solution at a time Include 𝑗 in 𝐴 (𝐴 = 𝐴 ∪ {𝑗}) Include 𝑗 temporarily in 𝐴 For each 𝑖 ∈ 𝐴 𝑖 ≠ 𝑗 if 𝑗 dominates 𝑖 then 𝐴 = 𝐴\{𝑖} elseif 𝑖 dominates 𝑗 then 𝐴 = 𝐴\{𝑗}

Fast Non-dominated Sorting

P M

A Insert one particle into A

Compare the members of A to each other and make it dominate free P

MAInsertoneparFcle

intoA

ComparethemembersofAtoeachotherandkeepitdominated-free

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Assignment15:D-WARranking

RanktheparFclesA–FaccordingtotheirfuncFonsvaluesshownbelow,byusingtheDistance-basedWARranking.A=(1,1,1,1,1)B=(0,1,1,1,1)C=(1,0,1,1,1)D=(1,1,0,1,1)E=(1,1,1,0,1)F=(1,1,1,1,0)

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Assignment15:soluFon

RanktheparFclesA–FaccordingtotheirfuncFonsvaluesshownbelow,byusingtheDistance-basedWARranking.rankA=(1,1,1,1,1)2B=(0,1,1,1,1)1C=(1,0,1,1,1)1D=(1,1,0,1,1)1E=(1,1,1,0,1)1F=(1,1,1,1,0)1DWAR=5.00006.65696.65696.65696.65696.6569

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