Evolutionary Multi-Objective Optimization and Interactive

TDDA Lecture at HUT, Espoo (6 September 2007)

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Evolutionary MultiEvolutionary Multi--Objective Objective Optimization and Interactive Optimization and Interactive

DecisionDecision--MakingMaking

KalyanmoyKalyanmoy DebDebProfessor of Mechanical EngineeringProfessor of Mechanical Engineering

Kanpur Genetic Algorithms Laboratory (Kanpur Genetic Algorithms Laboratory (KanGALKanGAL))Indian Institute of Technology KanpurIndian Institute of Technology Kanpur

Email: Email: [email protected]@iitk.ac.inhttp://http://www.iitk.ac.in/kangal/deb.htmwww.iitk.ac.in/kangal/deb.htm

Currently a Finnish Distinguished Professor at Currently a Finnish Distinguished Professor at Helsinki School of EconomicsHelsinki School of Economics


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OverviewOverview

Evolutionary optimization principlesEvolutionary optimization principlesEEvolutionary volutionary MMultiulti--objective objective OOptimization (ptimization (EMOEMO))Some EMO methodologiesSome EMO methodologiesEMO and interactive decisionEMO and interactive decision--making making (I(I--MODE)MODE)Other uses of EMOOther uses of EMOConclusions Conclusions


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Nature as an OptimizerNature as an OptimizerNature as structural engineerNature as structural engineer

Stem, Bamboo, insect Stem, Bamboo, insect trachea, beetrachea, bee--hivehive

Nature as a CFD solverNature as a CFD solverBirds, fishesBirds, fishes

Nature as a drag reducerNature as a drag reducerPenguin bodyPenguin body


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Evolutionary Algorithms (Evolutionary Algorithms (EAsEAs) Follow ) Follow Natural Evolutionary PrinciplesNatural Evolutionary Principles

begin begin Solution Representation Solution Representation t := 0; // t := 0; // generation countergeneration counterInitializationInitialization P(tP(t); ); Evaluation Evaluation P(tP(t); ); while notwhile not Termination Termination dodo

P'(tP'(t) := ) := Selection Selection ((P(tP(t));));P''(tP''(t) := ) := Variation Variation ((P'(tP'(t));));Evaluation Evaluation P''(tP''(t););P(t+1):= P(t+1):= Survivor Survivor ((P(t),P''(tP(t),P''(t));));t := t+1;t := t+1;

ododendend

Mean approaches Mean approaches optimumoptimum

Variance reducesVariance reduces


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Main EA PrinciplesMain EA Principles

Population of solutions in an iterationPopulation of solutions in an iterationSelectionSelection operator exploits better solutionsoperator exploits better solutionsVariationVariation (recombination, mutation etc.) (recombination, mutation etc.) operators use them to explore new solutionsoperators use them to explore new solutionsSurvivorSurvivor operator maintains better solutionsoperator maintains better solutionsAsymptotic convergence proof (Rudolph, Asymptotic convergence proof (Rudolph, 1996)1996)Flexible, hence needs a good understanding Flexible, hence needs a good understanding of above interactionsof above interactions


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Simulation of an EASimulation of an EA


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Advantages with a Population Advantages with a Population Compare solutions for an efficient searchCompare solutions for an efficient search

Handling constraintsHandling constraintsChanging objective/constraint (relative Changing objective/constraint (relative evaluation)evaluation)

Find multiple optimaFind multiple optimaMultiMulti--objective optimizationobjective optimizationMultimodal optimizationMultimodal optimization

Enable use of parallel processors for Enable use of parallel processors for faster computationfaster computationEnsemble learning Ensemble learning


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Advantages with EA OperatorsAdvantages with EA OperatorsFlexible and can be tailorFlexible and can be tailor--made for a made for a problemproblem`Respectful`Respectful’’ operators through building operators through building block processingblock processingHowever, flexibility comes with onus on However, flexibility comes with onus on fixing appropriate operatorsfixing appropriate operatorsSelfSelf--adaptiveadaptive operators can adjust operators can adjust extent of their use depending on extent of their use depending on populationpopulation


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Advantages with RepresentationAdvantages with RepresentationFlexible and can be tailorFlexible and can be tailor--made for a made for a problemproblemMixed representation allowedMixed representation allowed

Discrete, continuous, permutation, codeDiscrete, continuous, permutation, codeScheduling through fixed codingScheduling through fixed coding

CodingCoding--Variation operator interaction is Variation operator interaction is importantimportant


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A PenaltyA Penalty--ParameterParameter--less less Constraint Handling ApproachConstraint Handling Approach

Tournament selection:Tournament selection:Pick two from population Pick two from population and keep betterand keep better

Modified tournament Modified tournament selsel..Feasible is better than Feasible is better than infeasibleinfeasibleFor two feasible solutions, For two feasible solutions, choose the one with better choose the one with better ffFor two infeasible solutions, For two infeasible solutions, choose the one with smaller choose the one with smaller constraint violationconstraint violation

( )( ) ( )

( )max 1

, if 0,

, otherwise

j

Jjj

f x g x j JF x

f g x=

≥ ∀ ∈⎧⎪= ⎨+⎪⎩ ∑


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Customized Customized EAsEAs for Complex for Complex ProblemsProblems

Initialization and EA operators must use Initialization and EA operators must use problem problem knowledgeknowledgeFaster convergence is expectedFaster convergence is expectedNecessary for realNecessary for real--world problem solvingworld problem solving

A case study involving A case study involving millions of variablesmillions of variables (Deb (Deb and Reddy, 2001)and Reddy, 2001)Casting schedulingCasting schedulingInteger linear program Integer linear program (ILP)(ILP)


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ScaleScale--Up ResultsUp Results

KnowledgeKnowledge--augmented GA has subaugmented GA has sub--quadratic quadratic complexity and up to complexity and up to one millionone million variables (Deb variables (Deb and Reddy, 2001)and Reddy, 2001)


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Evolutionary Multi-Objective Optimization Books (Since 2001)


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Evolutionary MultiEvolutionary Multi--Objective Objective Optimization (EMO)Optimization (EMO)

Principle:Principle:Find multiple representative ParetoFind multiple representative Pareto--optimal optimal solutions simultaneouslysolutions simultaneously

Three main applications:Three main applications:Better decisionBetter decision--makingmakingUnveil salient optimality properties of solutionsUnveil salient optimality properties of solutionsHelpful in solving other optimization problemsHelpful in solving other optimization problems


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Two Goals in an Ideal Two Goals in an Ideal MultiMulti--Objective OptimizationObjective Optimization

Converge to the Converge to the ParetoPareto--optimal optimal frontfront

Maintain as Maintain as diverse a diverse a distribution as distribution as possiblepossible


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Evolutionary MultiEvolutionary Multi--Objective Objective Optimization (EMO)Optimization (EMO)

Step 1 :Step 1 :

Find a set of Find a set of ParetoPareto--optimal optimal solutionssolutions

Step 2Step 2 ::

Choose one from Choose one from the setthe set

•• Ideal for an EAIdeal for an EA

(Deb, 2001)(Deb, 2001)


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Comparison with Comparison with Classical Generating MethodsClassical Generating Methods

OneOne--atat--aa--time and time and repeatrepeatPopulation approachesPopulation approaches

TimmelTimmel’’ss methodmethodSchafflerSchaffler’’ss methodmethod

Absence of parallel Absence of parallel search is a drawbacksearch is a drawbackEMO finds multiple EMO finds multiple solutions with an solutions with an implicit parallel searchimplicit parallel search


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History of Evolutionary MultiHistory of Evolutionary Multi--Objective OptimizationObjective Optimization

Early penaltyEarly penalty--based based approachesapproaches

VEGA (1984)VEGA (1984)Goldberg's (1989) Goldberg's (1989) suggestionsuggestion

MOGA, NSGA, NPGA MOGA, NSGA, NPGA (1993(1993--95) used Goldberg's95) used Goldberg'ssuggestionsuggestion

Elitist EMO (SPEA, NSGAElitist EMO (SPEA, NSGA--II,II,PAES, MOMGA etc.) (1998 PAES, MOMGA etc.) (1998

---- Present)Present)

EMOO Web site (as of 4 JulyEMOO Web site (as of 4 July’’06)06)603 journal, 1371 conference603 journal, 1371 conference136 PhD theses (136 PhD theses (CoelloCoello’’ss site)site)


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Elitist NonElitist Non--dominated Sorting dominated Sorting Genetic Algorithm (NSGAGenetic Algorithm (NSGA--II)II)

NSGANSGA--II can II can extract Paretoextract Pareto--optimal frontieroptimal frontierAlso find a wellAlso find a well--distributed set of distributed set of solutionssolutionsiSIGHTiSIGHT and and modeFrontiermodeFrontieradopted NSGAadopted NSGA--IIII

FastFast--Breaking Paper in Engineering by ISI Web of Science Breaking Paper in Engineering by ISI Web of Science (Feb(Feb’’04), Thomson Citation Laureate Award 200604), Thomson Citation Laureate Award 2006


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NSGANSGA--II Procedure II Procedure (Deb et al., 2001)(Deb et al., 2001)

Elites are preservedElites are preservedNonNon--dominated solutions are emphasizeddominated solutions are emphasized


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NSGANSGA--II (cont.)II (cont.)

Overall Complexity Overall Complexity O(O(N logN logMM--11NN))

Diversity is maintainedDiversity is maintained

Higher ComplexityHigher Complexity


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Simulation on ZDT1Simulation on ZDT1


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Simulation on ZDT3Simulation on ZDT3


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Theoretically Accurate Solutions Theoretically Accurate Solutions Using NSGAUsing NSGA--IIIIEA EA solution(ssolution(s) ) improved with local improved with local search (classical or search (classical or hillhill--climbing)climbing)If derivative exists, If derivative exists, verify the solution to verify the solution to be a KKT pointbe a KKT pointFor every point, For every point, calculate a norm calculate a norm stating extent of KKT stating extent of KKT condition satisfactioncondition satisfaction

∑ ∑= =

∇+∇=∇J

j

K

kkkjjii hgfX

1 1μμλ

Norm can be used as termination criteria Norm can be used as termination criteria


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EMO Applications:EMO Applications:1. Better Decision1. Better Decision--MakingMaking

Identify different tradeIdentify different trade--off solutions for choosing off solutions for choosing one (Better and more confident decisionone (Better and more confident decision--making)making)InterInter--planetary trajectoryplanetary trajectorydesigndesign


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2.2. InnovizationInnovization::Discovery of Innovative design principles through Discovery of Innovative design principles through optimizationoptimization

Example: Electric motor Example: Electric motor design with varying design with varying ratings, say 1 to 10 kWratings, say 1 to 10 kW

Each will vary in size Each will vary in size and power and power Armature size, Armature size, number of turns etc.number of turns etc.

How do solutions vary?How do solutions vary?Any common principles!Any common principles!

Understand important design principles in Understand important design principles in a routine design scenarioa routine design scenario


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GearGear--Box DesignBox DesignA multiA multi--spindle gearspindle gear--box box design design (Deb and Jain, 2003)(Deb and Jain, 2003)28 variables (integer, 28 variables (integer, discrete, realdiscrete, real--valued)valued)101 non101 non--linear constraintslinear constraints

Important insights Important insights obtained obtained (larger module for more (larger module for more power)power)


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InnovizedInnovized PrinciplesPrinciplesModule varies proportional to squareModule varies proportional to square--root of powerroot of powerKeep other 27 variables more or less the sameKeep other 27 variables more or less the same


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EMO for DecisionEMO for Decision--MakingMaking

Use where multiple, repetitive Use where multiple, repetitive applications are soughtapplications are soughtUse where, instead of a point, a tradeUse where, instead of a point, a trade--off region is soughtoff region is soughtUse for finding points with specific Use for finding points with specific properties (nadir point, knee point, etc.)properties (nadir point, knee point, etc.)Use for robust, reliable or other frontsUse for robust, reliable or other frontsUse EMO for an idea of the front, then Use EMO for an idea of the front, then decisiondecision--making (Imaking (I--MODE)MODE)


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Reference Point Based EMOReference Point Based EMO

WierzbickiWierzbicki, 1980, 1980A PA P--O solution closer O solution closer to a reference pointto a reference point

Multiple runsMultiple runsToo structuredToo structured

Extend for EMOExtend for EMOMultiple reference Multiple reference points in one runpoints in one runA distribution of A distribution of solutions around each solutions around each reference pointreference point


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Making Decisions:Making Decisions:Reference Point Based EMOReference Point Based EMO

Ranking based on Ranking based on closeness to each closeness to each reference pointreference pointClearing within each Clearing within each niche with niche with εε

Population Population AdvantageAdvantage

(Deb and Sundar 2006)(Deb and Sundar 2006)


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EMO for Making Decisions:EMO for Making Decisions:Reference Direction based EMOReference Direction based EMO((KorhonenKorhonen and students, 1996and students, 1996--))

Choose a direction dChoose a direction dSolve for different t: Solve for different t: min min max(z_imax(z_i--q_i)/w_iq_i)/w_is.ts.t. z=. z=q+tq+t*d*dChoose most preferred Choose most preferred solutionsolutionIf different from If different from previous, continueprevious, continue

d

q

Instead of solving several opt. problems, use Instead of solving several opt. problems, use EMO onceEMO once

z i


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Reference Direction based EMO Reference Direction based EMO (Deb and Kumar, 2007)(Deb and Kumar, 2007)

Multiple directions togetherMultiple directions together


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Reference Direction Based EMO Reference Direction Based EMO (cont.)(cont.)

1010--objective problem with two directionsobjective problem with two directions(0.8,0.8,0.8,0.8,0.8,0.2,0.2,0.2,0.2,0.2)(0.8,0.8,0.8,0.8,0.8,0.2,0.2,0.2,0.2,0.2)(0.2,0.2,0.2,0.2,0.2,0.8,0.8,0.8,0.8,0.8)(0.2,0.2,0.2,0.2,0.2,0.8,0.8,0.8,0.8,0.8)


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EMO for Finding EMO for Finding Nadir PointNadir Point

DTLZ problems DTLZ problems extended up to 20 extended up to 20 objectivesobjectives

Modifying crowdingdistance computation

(Deb, Chaudhuri & Miettinen, 2006)(Deb, Chaudhuri & Miettinen, 2006)


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EMO for Finding Knee Solutions EMO for Finding Knee Solutions ((BrankeBranke et al., 2004)et al., 2004)

Find only the knee or nearFind only the knee or near--knee solutions knee solutions


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MultiMulti--Objective Robust FrontierObjective Robust Frontier

Not all ParetoNot all Pareto--optimal optimal points may be robustpoints may be robustA is robust, but B is notA is robust, but B is not

f_2

x_1

x_2

x_3

f_1

B

A

B

A

Objectivespace

Decision space

(δ=0.007)

η=0.4

η=0.5

η=0.6

η=0.7Type I robustfront

Original front

0

0.5

1

1.5

2

2.5

0 0.2 0.4 0.6 0.8 1

f_2

f_1

Type-II Robustness:


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MultiMulti--Objective ReliabilityObjective Reliability--Based Based OptimizationOptimization (Deb et al., 2006)(Deb et al., 2006)

Reliable fronts appear due to uncertainty in Reliable fronts appear due to uncertainty in variablesvariables


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II--MODE: EMO with Interactive MODE: EMO with Interactive DecisionDecision--Making Making (Deb and (Deb and ChaudhuriChaudhuri, 2006), 2006)

GUIGUI--based based interactive EMO on interactive EMO on linuxlinux platformplatformStart with a NSGAStart with a NSGA--II II front front Analyze solutions Analyze solutions using DM aidesusing DM aidesChoose Choose region(sregion(s) for ) for further analysisfurther analysisAn iterative An iterative procedureprocedure


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II--MODE Developed at MODE Developed at KanGALKanGAL

Links with evaluation Links with evaluation softwaressoftwares (FEM, (FEM, MatlabMatlabetc.)etc.)


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II--MODE to Welded Beam Design MODE to Welded Beam Design


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ConclusionsConclusions

Evolutionary algorithms are flexible and Evolutionary algorithms are flexible and good candidates for multigood candidates for multi--objective objective optimizationoptimization

Population + stochastic aspectsPopulation + stochastic aspects

EMO can be useful in decisionEMO can be useful in decision--makingmakingSome ideas are worked out, many must be Some ideas are worked out, many must be donedone

Interactive EMO+MCDMInteractive EMO+MCDMII--MODE is a start, other ideas must be worked MODE is a start, other ideas must be worked outout

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Evolutionary Multi-Objective Optimization and Interactive