06061.debkalyanmoy.slides1

Dagstuhl Seminar, 5-10 February, 2006

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Functional Decomposition of NSGA-II and Functional Decomposition of NSGA-II and Various Problem-Solving StrategiesVarious Problem-Solving Strategies

Kalyanmoy DebKalyanmoy DebProfessor of Mechanical EngineeringProfessor of Mechanical EngineeringIndian Institute of Technology KanpurIndian Institute of Technology KanpurDirector, Kanpur Genetic Algorithms Director, Kanpur Genetic Algorithms

Laboratory (KanGAL)Laboratory (KanGAL)Email: [email protected]: [email protected]

http://www.iitk.ac.in/kangal/deb.htmhttp://www.iitk.ac.in/kangal/deb.htm


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Overview Essentials of multi-objective

optimization NSGA-II platform Different multi-objective problem-

solving tasks Omni-optimizer

Degeneracy to various single and multi-objective tasks

Conclusions


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Multi-Objective Optimization:Multi-Objective Optimization:Handling multiple conflicting objectivesHandling multiple conflicting objectives

We often face themWe often face them


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Which Solutions are Optimal?Which Solutions are Optimal? Relates to the concept Relates to the concept

of dominationof domination xx(1)(1) dominates x dominates x(2)(2), if , if

xx(1)(1) is no worse than x is no worse than x(2)(2) in in all objectivesall objectives

xx(1)(1) is strictly better than is strictly better than xx(2)(2) in at least one in at least one objectiveobjective

Examples: Examples: 3 dominates 23 dominates 2 3 does not dominate 53 does not dominate 5


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Pareto-Optimal SolutionsPareto-Optimal Solutions P’=Non-dominated(P) P’=Non-dominated(P)

Solutions which are Solutions which are not dominated by any not dominated by any

member of the set Pmember of the set P O(N log N) O(N log N) algorithms existalgorithms exist Pareto-Optimal set Pareto-Optimal set = Non-dominated(S)= Non-dominated(S) A number of solutions A number of solutions

are optimalare optimal


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Pareto-Optimal Fronts Depends on

the type of objectives

Definition of domination takes care of possibilities

Always on the boundary of feasible region


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Local Versus Global Pareto-Optimal Fronts Local Pareto-optimal Front: Domination check is

restricted within a neighborhood (in decision space) of P


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Some Terminologies Ideal point (z*)

nonexistent, lower bound on Pareto-optimal set

Utopian point (z**) nonexistent

Nadir point (znad) Upper bound on

Pareto-optimal set Normalization:


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Differences with Single-Objective Optimization One optimum versus multiple optima Requires search and decision-making Two spaces of interest, instead of one


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Ideal Multi-Objective Ideal Multi-Objective OptimizationOptimizationStep 1 :Step 1 : Find a set of Pareto-Find a set of Pareto-optimal solutionsoptimal solutions

Step 2Step 2 :: Choose one from Choose one from the setthe set


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Two Goals in Ideal Multi-Objective Two Goals in Ideal Multi-Objective OptimizationOptimization

Converge to the Converge to the Pareto-optimal frontPareto-optimal front

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


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

NSGA-II can NSGA-II can extract Pareto-extract Pareto-optimal frontieroptimal frontierAlso find a well-Also find a well-distributed set of distributed set of solutionssolutionsiSIGHT and iSIGHT and modeFrontier modeFrontier adopted NSGA-IIadopted NSGA-II

Fast-Breaking Paper in Engineering by ISI Web of Science (Feb’04)Fast-Breaking Paper in Engineering by ISI Web of Science (Feb’04)


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Functional Decomposition Convergence:

Emphasize non-dominated solutions

Diversity: Prefer less-crowded solutions

Elite-preservation For ensuring convergence

properties


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An Iteration of NSGA-IIAn Iteration of NSGA-IIEl

ite-p

rese

rvat

ion

Elite

-pre

serv

atio

n

ConvergenceConvergence Diversity-maintenanceDiversity-maintenance


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NSGA-II: Crowding DistanceNSGA-II: Crowding Distance

Overall Complexity Overall Complexity O(O(N logN logM-1M-1NN))

Diversity is maintainedDiversity is maintained

Improve diversity byImprove diversity by• k-mean clusteringk-mean clustering• Euclidean distance Euclidean distance measuremeasure• Other techniquesOther techniques


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Simulation on ZDT1Simulation on ZDT1


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Simulation on ZDT3Simulation on ZDT3


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Changing Dominance Relation Alter the meaning of Pareto-optimal

points Constrained optimization (Fonseca and

Fleming, 1996, Deb et al., 2000) Cone dominance (guided dominance,

Branke et al., 2000) Distributed EMO (Deb et al., 2003) Epsilon-MOEA (Laumanns et al., 2003;

Deb et al., 2005) Robust and reliability-based EMO (Deb

and Gupta, 2005)


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Constraint-Domination PrincipleConstraint-Domination Principle

1.1. ii is feasible and is feasible and jj is not is not2.2. ii and and jj are both are both

infeasible, but infeasible, but ii has a has a smaller overall smaller overall constraint violationconstraint violation

3.3. ii and and jj are feasible and are feasible and ii dominates dominates jj

A solution A solution ii constraint-constraint-dominatesdominates a solution a solution jj, if any , if any is true:is true:


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Constrained NSGA-II Simulation Constrained NSGA-II Simulation ResultsResults

22 )( xxf MinimizeMinimize 11 )( xxf 1

22

1)(xxxf

1969

12

12

xxxx WhereWhere

5.05.05.0

0tan16cos1011

22

21

2

1122

21

xx

xxxx

MinimizeMinimize11 )( xxf

WhereWhere


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Simulation on TNKSimulation on TNK


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Simulation on CTP5Simulation on CTP5


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Cone DominanceCone DominanceUsing a DM’s Using a DM’s preference (not a preference (not a solution but a region)solution but a region)

Guided domination Guided domination principle: Biased principle: Biased niching approachniching approach

Weighted domination Weighted domination approachapproach


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Distributed Computing of Distributed Computing of Pareto-Optimal SetPareto-Optimal Set

Guided domination concept to search different parts Guided domination concept to search different parts of Pareto-optimal region of Pareto-optimal region Distributed computing of different partsDistributed computing of different parts


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Distributed computing: Distributed computing: A Three-Objective ProblemA Three-Objective Problem

Spatial computing, not temporalSpatial computing, not temporal

TheoryTheory NSGA-II SimulationsNSGA-II Simulations


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εε-MOEA: Using -MOEA: Using εε-Dominance-Dominance

EA and archive EA and archive populations evolve populations evolve One EA and one One EA and one archive member are archive member are mated mated Archive update using Archive update using εε-dominance -dominance EA update using EA update using usual dominanceusual dominance


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Comparative Study on Comparative Study on Three-Objective DTLZ ProblemsThree-Objective DTLZ Problems


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Test Problem DTLZ2Test Problem DTLZ2


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Multi-Objective Robust Multi-Objective Robust SolutionsSolutions

Not all Pareto-Not all Pareto-optimal points may optimal points may be robustbe robustA is robust, but B is A is robust, but B is notnotDecision-makers will Decision-makers will be interested in be interested in knowing robust part knowing robust part of the frontof the front


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Domination Based on Aggregate Domination Based on Aggregate FunctionsFunctions

Functions averaged over a delta-Functions averaged over a delta-neighborhodneighborhod

Alternate Strategy: (Type II Robustness)Alternate Strategy: (Type II Robustness)


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Effect of Effect of δδ-neighborhood Size-neighborhood Size Theory and NSGA-II simulationTheory and NSGA-II simulation Larger Larger δδ, more shift from original front, more shift from original front Some part is more sensitive than othersSome part is more sensitive than others


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Effect of Effect of δδ-neighborhood Size-neighborhood Size Theory and NSGA-II simulationTheory and NSGA-II simulation Larger Larger δδ, more shift from original front, more shift from original front Some part is no more robustSome part is no more robust


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Robust Front as Robust Front as Partial Global and Partial LocalPartial Global and Partial Local

Theory:Theory:

For global frontFor global front


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Simulation Using NSGA-IISimulation Using NSGA-IISimulation:Simulation:


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Reliability-Based Optimization Deterministic

optimum often not reliable

Due to uncertainities in decision variables/problem parameters

Find the reliable solution for a specified Reliability


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Constrained Domination for Reliability Consideration Chance

constraints: P(g(x)≤0) ≥ β

β depends on chosen reliability

Prefer reliable solutions

Indicates how P-O front moves away with β


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Goal Programming Using EMO Target function

values are specified

Convert them to objectives and perform domination


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Goal Programming Using EMO Target function values

are specified Convert them to

objectives and perform domination check with them


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Preferred Diversity Find a subset of Pareto-optimal points

dictated by preference information Biased EMO (Branke and Deb, 2005) Reference-point based EMO (Deb and

Sundar, 2006) Knee-based EMO (Branke et al., 2004) Nadir point and EMO (Deb and

Chaudhuri, 2006) Multi-modal EMO (Deb and Reddy, 2003) Variable versus objective space niching


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Preference-Based EMOPreference-Based EMOEMO (NSGA-II) not efficient for many objectivesEMO (NSGA-II) not efficient for many objectives

Large number of points neededLarge number of points neededDomination-based methods are slowDomination-based methods are slow


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EMO for a Biased DistributionChoose a hyper-planeProject points on itCompute two distances: d and d’Compute D=d(d’/d)^aPoint b has small DPoint a has large D


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Biased Distribution in NSGA-IIZDT1ZDT1 ZDT2: a=100ZDT2: a=100


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Biased NSGA-II (cont.)Three-objective Problems: a=0 and a=500Three-objective Problems: a=0 and a=500


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Reference Point Based EMO Wierzbicki, 1980 A P-O solution closer to

a reference point Multiple runs Too structured

Extend for EMO Multiple reference

points in one run A distribution of

solutions around each reference point


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Reference Point Based EMO (cont.) Ranking based on

closeness to each reference point

Clearing within each niche with ε


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More Results Five-objective with

two reference points (z1-5=0.5 & z1-4=0.2, z5=0.8)

A engineering design problem with three reference points


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Knee Based EMOKnee Based EMOFind only the knee or Find only the knee or near-knee solutions near-knee solutions Knees are important Knees are important solutionssolutions

Not much motivation to Not much motivation to move out from kneesmove out from kneesA large gain for a small A large gain for a small lossloss in any pair of in any pair of objectivesobjectives

Non-convex frontNon-convex front No knee pointNo knee point Extreme solutions are Extreme solutions are

attractorsattractors


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Finding Knee SolutionsFinding Knee SolutionsBranke et al. (2004) for more detailsBranke et al. (2004) for more details


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Nadir Point and EMONadir Point and EMOImportant for Important for knowing range and knowing range and normalization of normalization of objectivesobjectivesDifficult to find Difficult to find using classical using classical methodmethod

Pay-off table Pay-off table method does not method does not workwork


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EMO for Finding Nadir PointEMO for Finding Nadir PointEmphasize only extreme Emphasize only extreme pointspointsM≤3 find complete M≤3 find complete front, else use front, else use extremized crowded extremized crowded NSGA-IINSGA-II


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EMO for Finding Nadir Point (cont.)EMO for Finding Nadir Point (cont.)DTLZ problems extended up to 20 objectivesDTLZ problems extended up to 20 objectives


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Multi-Modal EMOsMulti-Modal EMOsDifferent solutions having identical objective values Different solutions having identical objective values Multi-modal Pareto-optimal solutions: Design, Multi-modal Pareto-optimal solutions: Design, Bioinformatics Bioinformatics


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Multiple Gene Subsets for Multiple Gene Subsets for Leukemia SamplesLeukemia Samples

Deb and Reddy Deb and Reddy (BioSystems, 2003)(BioSystems, 2003)

Multiple (26) four-Multiple (26) four-gene combinations gene combinations for 100% for 100% classificationclassification

Discovery of some Discovery of some common genes common genes


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Parameter Versus Objective-space NichingDistribution depends on the space niching is Distribution depends on the space niching is performedperformed


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Redefining Elites To aid in better diversity

Controlled Elitist EMO (Deb and Goel, 2001)


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Controlled Elitism Keep solutions

from dominated fronts in GP


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Controlled Elitism (cont.)ZDT4 has many

local P-O frontsg()=1 is globalControlled elitism

can come closer to global P-O front


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Omni-Optimizer:Omni-Optimizer:Motivation from ComputationMotivation from Computation

Multiple is a generic case, single is specific Multiple is a generic case, single is specific Single objective as a degenerate case multi-Single objective as a degenerate case multi-objective case objective case One algorithm for single and multi-objective One algorithm for single and multi-objective problem solving problem solving ((Deb and Tiwari, 2005Deb and Tiwari, 2005))Accommodating NFL theorem, not violating itAccommodating NFL theorem, not violating it

Single-objective, uni-optimum problemsSingle-objective, uni-optimum problemsSingle-objective, multi-optima problemsSingle-objective, multi-optima problemsMulti-objective, uni-optimal front problemsMulti-objective, uni-optimal front problemsMulti-objective, multi-optimal front problemsMulti-objective, multi-optimal front problems


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Structure of Omni-optimizer Very much like NSGA-II

Epsilon-dominance Variable-space and

objective space niching Use maximum of both crowding distances


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Single-Objective, Uni-Optimum Dominance reduced to simple ‘<‘ Epsilon-dominance to fa< fb-ε

Allows multiple solutions within ε to exist Elite-preservation is similar to CHC and

(μ+λ)-ES


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Shinn et al.’s 12 Problems12 problems


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Single-Objective, Multi-Optima Variable-space niching help find multiple

solutions Weierstrass function

16 minima with f=0


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101044SinSin22(x): 20 Minima(x): 20 Minima


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Himmelblau’s Function: 4 MinimaHimmelblau’s Function: 4 Minima


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Multi-Objective, Uni-Pareto frontMulti-Objective, Uni-Pareto front Constrained and

unconstrained test problems


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More Results Comparable

performance to existing EMO methods


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Multi-Objective, Multi-Optima Multi-Objective, Multi-Optima

Nine regions leading to the same Pareto-optimal front

Multiple solutions cause a single Pareto-optimal point


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Nine Optimal RegionsOm

ni-o

ptim

izer

Omni

-opt

imize

rNSGA-IINSGA-II


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Nine Optimal Fronts Nine Optimal Fronts


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ConclusionsConclusionsFunctional decomposition of NSGA-IIFunctional decomposition of NSGA-II

Non-domination for convergenceNon-domination for convergenceNiching for diverse set of solutionsNiching for diverse set of solutionsElite-preservation for reliable convergenceElite-preservation for reliable convergence

For a new problem-solving, find the For a new problem-solving, find the suitable place to changesuitable place to changeMany different problem-solving tasks Many different problem-solving tasks achieved with NSGA-IIachieved with NSGA-IIOmni-optimizer provides a holistic Omni-optimizer provides a holistic approach for optimizationapproach for optimization


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Thank You for Your AttentionThank You for Your Attention

For further information:For further information:http://www.iitk.ac.in/kangalEmail: [email protected]: [email protected]

Acknowledgement:Acknowledgement: KanGAL students, staff and collaboratorsKanGAL students, staff and collaborators

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

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

06061.debkalyanmoy.slides1

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