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Multi-ObjectiveEvolutionaryAlgorithms
KalyanmoyDeba
KanpurGeneticAlgorithmLaboratory(KanGAL)
IndianInstituteofTechnologyKanpur
Kanpur,Pin208016INDIA
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