Multi-ObjectiveEvolutionaryAlgorithms

KalyanmoyDeba

KanpurGeneticAlgorithmLaboratory(KanGAL)

IndianInstituteofTechnologyKanpur

Kanpur,Pin208016INDIA

deb@iitk.ac.in

http://www.iitk.ac.in/kangal/deb.html

aCurrentlyvisitingTIK,ETHZurich

1

OverviewoftheTutorial

•Multi-objectiveoptimization

•Classicalmethods

•Historyofmulti-objectiveevolutionaryalgorithms(MOEAs)

•Non-elitstMOEAs

•ElitistMOEAs

•ConstrainedMOEAs

•ApplicationsofMOEAs

•Salientresearchissues

2

Multi-ObjectiveOptimization

•Weoftenfacethem

B

C

Comfort

Cost10k100k

90%

1

2

A

40%

3

MoreExamples

Acheaperbutinconvenient

flight

Aconvenientbutexpensive

flight

4

WhichSolutionsareOptimal?

Domination:

x(1)

dominatesx(2)

if

1.x(1)

isnoworsethanx(2)

inall

objectives

2.x(1)

isstrictlybetterthanx(2)

inatleastoneobjective

f

1

14

3

(maximize) f1

610 218

6

3

5

4

1

2

5

2

(minimize)

5

Pareto-OptimalSolutions

Non-dominatedsolutions:Among

asetofsolutionsP,thenon-

dominatedsetofsolutionsP′

arethosethatarenotdominated

byanymemberofthesetP.

O(MN2)algorithmsexist.

Pareto-Optimalsolutions:When

P=S,theresultingP′isPareto-

optimalset

(maximize)

14

2

3

f1

610 218

1

f

5

(minimize) 2

1

4

5

3

Non−dominatedfront

Anumberofsolutionsareoptimal

6

Pareto-OptimalFronts

Min−−Max

2

f

2

1

1

Min−−Min

f

Max−−MinMax−−Max

f2

f

1

f2

f1

f

f

7

Preference-BasedApproach

optimization problem

Minimize f

Single−objectiveoptimization problem

a composite function

or

112MM 2

Multi−objective

Minimize f

......

Minimize fw

Higher−level 1

2

M(w.. 1w2)

information

M

F = w f + w f +...+ w f

One optimumsolution

optimizer

Single−objective

Estimate arelativeimportancevector

subject to constraints

•Classicalapproachesfollowit

8

ClassicalApproaches

•NoPreferencemethods(heuristic-based)

•Posteriorimethods(generatingsolutions)

•Apriorimethods(onepreferredsolution)

•Interactivemethods(involvingadecision-maker)

9

WeightedSumMethod

•Constructaweightedsumof

objectivesandoptimize

F(x)=M∑

m=1

wmfm(x).

•Usersuppliesweightvectorw

Feasible objective space

Pareto−optimal front

b

1

a

w

cd

A

w

1

2

f2

f

10

DifficultieswithWeightedSumMethod

•Needtoknoww

•Non-uniformityinPareto-

optimalsolutions

•Inabilitytofindsome

Pareto-optimalsolutions

Pareto−optimal front

D

Feasible objective space

B

C

ab

f

2f

1

A

11

ε-ConstraintMethod

•Optimizeoneobjective,

constrainallother

Minimizefµ(x),

subjecttofm(x)≤εm,m6=µ;

•Usersuppliesaεvector1

B

ε1

D

11 εεεabcd

C

1

f2

f

•Needtoknowrelevantεvectors

•Non-uniformityinPareto-optimalsolutions

12

DifficultieswithMostClassicalMethods

•Needtorunasingle-

objectiveoptimizermany

times

•Expectalotofproblem

knowledge

•Eventhen,gooddistribu-

tionisnotguaranteed

•Multi-objectiveoptimiza-

tionasanapplicationof

single-objectiveoptimiza-

tion

1

f

f

2

frontPareto−optimal

D

A

B

C

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

−0.100.10.20.30.40.50.60.70.80.91

13

IdealMulti-ObjectiveOptimization

Minimize f

Step 1

Multiple trade−offsolutions found

......Minimize fMinimize f

Step 2

subject to constraints

Multi−objectiveoptimization problem

M

1

2

Choose onesolution

Higher−levelinformation

Multi−objectiveoptimizer

IDEAL

Step1FindasetofPareto-optimalsolutions

Step2Chooseonefromtheset

14

AdvantagesofIdealMulti-ObjectiveOptimization

•Decision-makingbecomeseasierandlesssubjective

•Single-objectiveoptimizationisadegen-

eratecaseofmulti-objectiveoptimiza-

tion

–Step1findsasinglesolution

–NoneedforStep2

•Multi-modaloptimizationisaspecial

caseofmulti-objectiveoptimization

15

TwoGoalsinIdealMulti-ObjectiveOptimization

1.ConvergeonthePareto-

optimalfront

2.Maintainasdiverseadistri-

butionaspossible

f1

2f

16

WhyEvolutionary?

•Populationapproachsuitswelltofindmultiplesolutions

•Niche-preservationmethodscanbeexploitedtofinddiverse

solutions

2

1

1

0.8

0.6

0.4

0.2

1 0.8 0.6 0.4 0.2 00

f

f1

2f

1

0.8

0.6

0.4

0.2

0

1 0.8 0.6 0.4 0.2 0f

17

HistoryofMulti-ObjectiveEvolutionary

Algorithms(MOEAs)

•Earlypenalty-basedap-

proaches

•VEGA(1984)

•Goldberg’ssuggestion

(1989)

•MOGA,NSGA,NPGA

(1993-95)

•ElitistMOEAs(SPEA,

NSGA-II,PAES,MOMGA

etc.)(1998–Present)

Number of Studies

Year

0

20

40

60

80

100

120

140

19921993199419951996199719981999 1991 1990and before

18

19

WhattoChangeinaSimpleGA?

•Modifythefitnesscomputation

Initialize Population

Cond?

Begin

Reproduction

Mutation

Crossover

t = t + 1

t = 0

Stop

No

YesEvaluation

Assign Fitness

20

IdentifyingtheNon-dominatedSet

Step1Seti=1andcreateanemptysetP′.

Step2Forasolutionj∈P(butj6=i),checkifsolutionj

dominatessolutioni.Ifyes,gotoStep4.

Step3IfmoresolutionsareleftinP,incrementjbyoneandgo

toStep2;otherwise,setP′=P′∪i.

Step4Incrementibyone.Ifi≤N,gotoStep2;otherwisestop

anddeclareP′asthenon-dominatedset.

O(MN2)computationalcomplexity

21

AnEfficientApproach

Kungetal.’salgorithm(1975)

Step1Sortthepopulationindescend-

ingorderofimportanceoff1

Step2,Front(P)If|P|=1,

returnPastheoutput

ofFront(P).Otherwise,

T=Front(P(1)

−−P(|P|/2)

)and

B=Front(P(|P|/2+1)

−−P(|P|)).

Ifthei-thsolutionofBisnotdom-

inatedbyanysolutionofT,create

amergedsetM=T∪i.Return

MastheoutputofFront(P).

T1

B1

T2

B2

Merging

O(

N(logN)M−2

)

forM≥4andO(NlogN)forM=2and3

22

ASimpleNon-dominatedSortingAlgorithm

•Identifythebestnon-dominatedset

•Discardthemfrompopulation

•Identifythenext-bestnon-dominatedset

•Continuetillallsolutionsareclassified

•WediscussaO(MN2)algorithmlater

23

Non-ElitistMOEAs

•VectorevaluatedGA(VEGA)(Schaffer,1984)

•VectoroptimizedEA(VOES)(Kursawe,1990)

•WeightbasedGA(WBGA)(HajelaandLin,1993)

•MultipleobjectiveGA(MOGA)(FonsecaandFleming,1993)

•Non-dominatedsortingGA(NSGA)(SrinivasandDeb,1994)

•NichedParetoGA(NPGA)(Hornetal.,1994)

•Predator-preyES(Laumannsetal.,1998)

•Othermethods:DistributedsharingGA,neighborhood

constrainedGA,NashGAetc.

24

Non-DominatedSortingGA(NSGA)

•Anon-dominatedsortingofthepopulation

•Firstfront:FitnessF=Ntoall

•Nichingamongallsolutionsinfirstfront

•Noteworstfitness(sayF1w)

•Secondfront:FitnessF1w−ε1toall

•Nichingamongallsolutionsinsecondfront

•Continuetillallfrontsareassignedafitness

25

Non-DominatedSortingGA(NSGA)

f1f2Fitness

xFrontbeforeafter

−1.502.2512.2523.003.00

0.700.491.6916.006.00

4.2017.644.8423.003.00

2.004.000.0016.003.43

1.753.060.0616.003.43

−3.009.0025.0032.002.004

1

52

3

1

6

215

20

25

30

05

10

2530

f

5

15 100

20f

•Nichinginparameterspace

•Non-dominatedsolutionsareemphasized

•Diversityamongthemismaintained

26

Vector-EvaluatedGA(VEGA)

•DividepopulationintoMequalblocks

•Eachblockisreproducedwithoneobjectivefunction

•Completepopulationparticipatesincrossoverandmutation

•Biastowardstoindividualbestobjectivesolutions

•Anon-dominatedselection:Non-dominatedsolutionsare

assignedmorecopies

•Mateselection:Twodistant(inparameterspace)solutionsare

mated

•Bothnecessaryaspectsmissinginonealgorithm

27

Multi-ObjectiveGA(MOGA)

•Countthenumberofdomi-

natedsolutions(sayn)

•Fitness:F=n+1

•Afitnessrankingadjust-

ment

•Nichinginfitnessspace

•Restallaresimilarto

NSGA

FAsgn.Fit.

1232.5

2165.0

3222.5

4155.0

5145.0

6311.0

28

NichedParetoGA(NPGA)

•Solutionsinatournamentarecheckedfordominationwith

respecttoasmallsubpopulation(tdom)

•Ifonedominatedandothernon-dominated,selectsecond

•Ifbothnon-dominatedorbothdominated,choosetheonewith

smallernichecountinthesubpopulation

•Algorithmdependsontdom

•Nevertheless,ithasbothnecessarycomponents

29

NPGA(cont.)

YX

XY

Parameter Space

Check fordomination

t_dom

Population

30

ShortcomingofNon-ElitistMOEAs

•Elite-preservationismissing

•Elite-preservationisimportantforproperconvergencein

SOEAs

•SameistrueinMOEAs

•Threetasks

–Elitepreservation

–ProgresstowardsthePareto-optimalfront

–Maintaindiversityamongsolutions

31

ElitistMOEAs

Elite-preservation:

•Maintainanarchiveofnon-dominatedsolu-

tions

ProgresstowardsPareto-optimalfront:

•Preferringnon-dominatedsolutions

Maintainingspreadofsolutions:

•Clustering,niching,orgrid-basedcompeti-

tionforaplaceinthearchive

Elite EA

(maximize)

14

2

3

f1

610 218

1

f

5

(minimize) 2

1

4

5

3

Non−dominatedfront

f2

f1

clusters

32

ElitistMOEAs(cont.)

•Distance-basedParetoGA(DPGA)(OsyczkaandKundu,

1995)

•ThermodynamicalGA(TDGA)(Kitaetal.,1996)

•StrengthParetoEA(SPEA)(ZitzlerandThiele,1998)

•Non-dominatedsortingGA-II(NSGA-II)(Debetal.,1999)

•Pareto-archivedES(PAES)(KnowlesandCorne,1999)

•Multi-objectiveMessyGA(MOMGA)(Veldhuizenand

Lamont,1999)

•Othermethods:Pareto-convergingGA,multi-objective

micro-GA,elitistMOGAwithcoevolutionarysharing

33

ElitistNon-dominatedSortingGeneticAlgorithm

(NSGA-II)

Non-dominatedsorting:O(MN2)

•Calculate(ni,Si)foreach

solutioni

•ni:Numberofsolutions

dominatingi

•Si:Setofsolutionsdomi-

natedbyi

f

f

1

2

3

10

11

9

4

87

6

5

(5, Nil)

(0, 9,11)

2

1

34

NSGA-II(cont.)

Elitesarepreserved

sortingCrowding

3

t

distancesorting

Non−dominated

t+1

F

F1

2

F

Q

R

P

Rejectedt

tP

35

NSGA-II(cont.)

Diversityismaintained:O(MNlogN)

Cuboid

f

f

1

2

ii-1

i+1

0

l

OverallComplexity:O(MN2)

36

NSGA-IISimulationResults

1

2

NSGA−II (binary−coded)

0.8

0.7

0.4

0.9

0.5

f

1.1

1

0.6

0.3

0.2

0.1

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

f

0

NSGA−II

−14 −15 −16 −17 −18 −19 −20−12

2

0

−2

−4

−6

−8

−10

f2

f1

37

StrengthParetoEA(SPEA)

•Storesnon-dominatedsolutionsexternally

•Pareto-dominancetoassignfitness

–Externalmembers:Assignnumberofdominatedsolutions

inpopulation(smaller,better)

–Populationmembers:Assignsumoffitnessofexternal

dominatingmembers(smaller,better)

•Tournamentselectionandrecombinationappliedtocombined

currentandelitepopulations

•Aclusteringtechniquetomaintaindiversityinupdated

externalpopulation,whensizeincreasesalimit

38

SPEA(cont.)

•Fitnessassignmentandclusteringmethods

Function Space

xxxxx

PopulationExt_popFitness Assignment

x

x

xx

Function Space

Clustering (d and p_max)

39

ParetoArchivedES(PAES)

•An(1+1)-ES

•ParentptandchildctarecomparedwithanexternalarchiveAt

•IfctisdominatedbyAt,pt+1=pt

•IfctdominatesamemberofAt,deleteitfromAtandinclude

ctinAtandpt+1=ct

•If|At|<N,includectandpt+1=winner(pt,ct)

•If|At|=NandctdoesnotlieinhighestcounthypercubeH,

replacectwitharandomsolutionfromHand

pt+1=winner(pt,ct).

Thewinnerisbasedonleastnumberofsolutionsinthehypercube

40

NichinginPAES-(1+1)

front

2

Pareto−optimal

f

f1

3

1

2

Offspring

Parent

frontPareto−optimal1

2

f

f

ParentOffspring

41

ConstrainedHandling

•Penaltyfunctionapproach

Fm=fm+RmΩ(~g).

•Explicitprocedurestohandleinfeasiblesolutions

–Jimenez’sapproach

–Ray-Tang-Seow’sapproach

•Modifieddefinitionofdomination

–FonsecaandFleming’sapproach

–Debetal.’sapproach

42

Constrain-DominationPrinciple

Asolutioniconstrained-

dominatesasolutionj,ifanyis

true:

1.Solutioniisfeasibleandso-

lutionjisnot.

2.Solutionsiandjarebothin-

feasible,butsolutionihasa

smalleroverallconstraintvi-

olation.

3.Solutionsiandjarefeasible

andsolutionidominatesso-

lutionj.

2

f

f

2

1

1

3

56

4

3

4

5

0.80.91 0.7 0.6 0.5 0.4 0.3 0.2 0.1

10

8

6

4

2

0

Front 1

Front 2

43

ConstrainedNSGA-IISimulationResults

2

1f

10

8

6

4

2

10

8

6

4

2

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.10

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.10

f

1

2

1.4

1.2

1

0.8

0.6

0.4

0.2

1.4 1.2 1 0.8 0.6 0.4 0.2

f

0

f

0

44

ApplicationsofMOEAs

•Space-crafttrajectoryoptimization

•Engineeringcomponentdesign

•Microwaveabsorberdesign

•Ground-watermonitoring

•Extruderscrewdesign

•Airlinescheduling

•VLSIcircuitdesign

•Otherapplications(referDeb,2001andEMO-01proceedings)

45

SpacecraftTrajectoryOptimization

•Coverstone-Carrolletal.(2000)withJPLPasadena

•Threeobjectivesforinter-planetarytrajectorydesign

–Minimizetimeofflight

–Maximizepayloaddeliveredatdestination

–MaximizeheliocentricrevolutionsaroundtheSun

•NSGAinvokedwithSEPTOPsoftwareforevaluation

46

Earth–MarsRendezvous

11.522.533.540

100

200

300

400

500

600

700

800

900

1000 22

36

132

72

Transfer Time (yrs.)

73

44

Mass Delivered to Target (kg.)

Individual 44

Earth09.01.05

Mars10.16.06

Individual 73

Mars09.22.07

Earth09.01.05

Individual 72

Mars08.25.08

Earth09.01.05

Individual 36

Mars02.04.09

Earth09.01.05

47

SalientResearchTasks

•ScalabilityofMOEAstohandlemorethantwoobjectives

•Mathematicallyconvergentalgorithmswithguaranteedspread

ofsolutions

•Testproblemdesign

•Performancemetricsandcomparativestudies

•Controlledelitism

•DevelopingpracticalMOEAs–Hybridization,parallelization

•Applicationcasestudies

49

HybridMOEAs

•CombineEAswithalocalsearchmethod

–Betterconvergence

–Fasterapproach

•Twohybridapproaches

–LocalsearchtoupdateeachsolutioninanEApopulation

(IshubuchiandMurata,1998;Jaskiewicz,1998)

–FirstEAandthenapplyalocalsearch

97

PosterioriApproachinanMOEA

MOEAProblem

local searchesMultiple

Non−dominationcheck Clustering

•Whichobjectivetouseinlocalsearch?

98

ProposedLocalSearchMethod

•Weightedsumstrategy(oraTchebycheffmetric)

F=∑

i

wi∗fi

•fiisscaled

•Weightwichosenbasedonlocationofiintheobtainedfront

wj=(f

maxj−fj(x))/(f

maxj−f

minj)

∑

Mk=1(fmax

k−fk(x))/(fmaxk−fmin

k)

•Weightsarenormalized

∑

i

wi=1

99

FixedWeightStrategy

•Extremesolutionsareas-

signedextremeweights

•Linearrelationbetween

weightandfitness

•Manysolutioncanconverge

tosamesolutionafterlocal

searchlocal search

max

min

set after

MOEA solution set

1f

f

f

max min

2

2

f1f1

ba

A

2f

100

DesignofaCantileverPlate

100 mm

60 mm

P

Baseplate

2

4

6

8

10

12

14

16

18

202530354045505560Scaled deflection

Weight

Non−dominated solutions

2

4

6

8

10

12

14

16

18

202530354045505560

Clustered solutions

2

4

6

8

10

12

14

16

18

202530354045505560

Scaled deflection

Weight

NSGA−II

2

4

6

8

10

12

14

16

18

202530354045505560

Scaled deflection

Weight

Local search

Weight

Scaled deflection

Ninetrade-offsolutionsarechosen

102

Trade-offSolutions

(1.00,0.00)(0.60,0.40)(0.50,0.50)

(0.43,0.57)(0.38,0.62)(0.35,0.65)

(0.23,0.77)(0.14,0.86)(0.00,1.00)

103

Conclusions

•Idealmulti-objectiveoptimizationisgenericandpragmatic

•Evolutionaryalgorithmsareidealcandidates

•Manyefficientalgorithmsexist,moreefficientonesareneeded

•Withsomesalientresearchstudies,MOEAswillrevolutionize

theactofoptimization

•EAshaveadefiniteedgeinmulti-objectiveoptimizationand

shouldbecomemoreusefulinpracticeincomingyears

106