tutorial on evolutionary multiobjective optimization

7/30/2019 Tutorial on Evolutionary Multiobjective Optimization

1/34

A Tutorial on EvolutionaryA Tutorial on EvolutionaryMultiobjectiveMultiobjective OptimizationOptimization

Eckart Zitzler

Computer Engineering and Networks Lab

Swiss Federal Institute of Technology (ETH) Zurich

Computer EngineeringComputer Engineering

and Networks Laboratoryand Networks Laboratory


2/34

Eckart Zitzler

ETH Zrich

MOMH 200

Tutorial on EMOverviewOverview

n The Big Picture:Optimization and evolutionary computation

o The Construction Kit:

Design issues and algorithmic concepts

p The Pieces Put Together:

Example of an algorithm variantq The Big Question:

Performance of evolutionary algorithms

r The Challenge:

Standard interface for search algorithms

sThe End:Conclusions and outlook


3/34

Eckart Zitzler

ETH Zrich

MOMH 200

Tutorial on EMDecision and Objective SpaceDecision and Objective Space

y2

y1

x2

x1

(x1, x2, , xn) f (y1, y2, , yk)

decisionspace

objectivespace

search evaluation

Pareto setPareto set approximation Pareto frontPareto front approximation


4/34

Eckart Zitzler

ETH Zrich

MOMH 200

Tutorial on EMOptimization and Decision MakingOptimization and Decision Making

y1

y2 Pareto optimality:defines set of optimal trade-offs

(all objectives equally important)

Decision making:

choose best compromise

(based on preference information)

n Decision making before search (define single objective)

o Decision making after search (find/approximate Pareto set first)

p Decision making during search (guide search interactively)

q Combinations of the above


5/34

Eckart Zitzler

ETH Zrich

MOMH 200

Tutorial on EMStochastic Search AlgorithmsStochastic Search Algorithms

selection

environmental

selection

mating

selection

variationmemory

EA 1 both 1 N : M

1 randomized

TS 1 no mating selection 1 1 : M

1 deterministic

SA 1 no mating selection 1 1 : M

1 randomized

ACO 1 neither 1 1 : 11 randomized


6/34

Eckart Zitzler

ETH Zrich

MOMH 200

Tutorial on EMOutline of a Simple Evolutionary AlgorithmOutline of a Simple Evolutionary Algorithm

0100

0011 = 1 solution

fitness

evaluation

0111 fitness = 19

mating selection

recombination

0011

0000

mutation

0011

1011

environmentalselection


7/34

Eckart Zitzler

ETH Zrich

MOMH 200








r The Challenge:Standard interface for search algorithms

sThe End:Conclusions and outlook


8/34

Eckart Zitzler

ETH Zrich

MOMH 200

Tutorial on EMIssues in EMOIssues in EMO

y2

y1

Diversity

Convergence

How to maintain a diverse

Pareto set approximation?

odensity estimation

How to prevent nondominated

solutions from being lost?

p environmental selection

How to guide the population

towards the Pareto set?

nmating selection

i i i


9/34

Eckart Zitzler

ETH Zrich

MOMH 200

Tutorial on EMFitness Assignment StrategiesFitness Assignment Strategies

y1

y2

y1

y2y2

y1

aggregation-based criterion-based dominance-based

parameter-orientedscaling-dependent set-orientedscaling-independent

weighted sum VEGA SPEA2

iD i d kiB d R ki


10/34

Eckart Zitzler

ETH Zrich

MOMH 200

Tutorial on EMDominanceDominance--Based RankingBased Ranking

Types of information:dominance rank by how many individuals is an

individual dominated?

dominance count how many individuals does an

individual dominate?

dominance depth at which front is an individual

located?

Examples:MOGA, NPGA dominance rank

NSGA/NSGA-II dominance depth

SPEA/SPEA2 dominance count + rank

E k t Zit l MOMH 200

Di i P iDi it P ti


11/34

Eckart Zitzler

ETH Zrich

MOMH 200

Tutorial on EMDiversity PreservationDiversity Preservation

Density estimation techniques: [Silverman: 1986]

ff

f

Kernel

MOGA

density estimate=

sum of f values

where f is afunction of the

distance

Nearest neighbor

SPEA2

density estimate=

volume of the

sphere defined bythe nearest

neighbor

Histogram

PAES

density estimate=

number of

solutions in thesame box

Eckart Zitzler MOMH 200

E i t l S l tiE i t l S l ti


12/34

Eckart Zitzler

ETH Zrich

MOMH 200

Tutorial on EMEnvironmental SelectionEnvironmental Selection

old population offspring

newpopulation

offspring archive

newarchive

old population

newpopulation

Variant 1: without archive Variant 2: with archive

Selection criteria:

Dominance: only nondominated solutions are kept

Density: less crowded regions are preferred to crowded regions

Time: old archive members are preferred to new solutions

Chance: each solution has the same probability to enter the archive


O iO i


13/34

Eckart Zitzler

ETH Zrich

MOMH 200









s The End:

Conclusions and outlook


SPEA2 Al ithSPEA2 Algorithm


14/34

Eckart Zitzler

ETH Zrich

MOMH 200

Tutorial on EM

Step 1: Generate initial population P0 and empty archive

(external set) A0. Set t = 0.

Step 2: Calculate fitness values of individuals in Pt and At.

Step 3: At+1 = nondominated individuals in Pt and At.

If size of At+1 > N then reduce At+1, else ifsize of At+1 < N then fill At+1 with dominated

individuals in Pt and At.

Step 4: If t > T then output the nondominated set of At+1.Stop.

Step 5: Fill mating pool by binary tournament selection

with replacement on At+1.Step 6: Apply recombination and mutation operators to

the mating pool and set Pt+1 to the resulting

population. Set t = t + 1 and go to Step 2.

SPEA2 AlgorithmSPEA2 Algorithm

[Zitzler, Laumanns, Thiele: 2001]


Pareto Fitness AssignmentPareto Fitness Assignment


15/34

ETH Zrich Tutorial on EMPareto Fitness AssignmentPareto Fitness Assignment

y2

y1

2

14

5+45+7

5+2

SPEA

y2

y1

0

00

4+3+2

2+1+4+3+2

2

4

4+3

SPEA2

S (strength) =

#dominated solutions

R (raw fitness) = N +

strengths of dominators

S (strength) =

#dominated solutions

R (raw fitness) =

strengths of dominators


Diversity PreservationDiversity Preservation


16/34

ETH Zrich Tutorial on EMDiversity PreservationDiversity Preservation

Density Estimation

k-th nearest neighbor method:

Fitness = R + 1 / (2 + Dk)

Dk = distance to the k-th

nearest individual

k = popsize + archivesize

Truncation

Incremental approach:

Remove individual A for which

A


17/34

ETH Zrich Tutorial on EMOverviewOverview








s The End:



lPerformance Assessment: ApproachesPerformance Assessment: Approaches


18/34

ETH Zrich Tutorial on EMPerformance Assessment: ApproachesPerformance Assessment: Approaches

n Theoretically (by analysis): difficult

Limit behavior

Is the Pareto set found, if there are unlimited run-timeresources?

Run-time analysis

How long does it take to generate the Pareto set with high

probability?

o Empirically (by simulation): standard


T t i l EMLimit BehaviorLimit Behavior


19/34

ETH Zrich Tutorial on EMLimit BehaviorLimit Behavior

Basic assumptions:Every solution can be generated from every other solution by

mutation

The number of iterations tgoes to infinity (t)

Studies:

Convergence: [Hanne: 1999][Rudolph, Agapie: 2000]

Diversity: e.g., [Knowles, Corne: 2000][Deb et al.: 2001]

Convergence + diversity:

Unlimited memory resources [Rudolph and Agapie: 2000]

Limited memory resources [Laumanns et al.: 2002]

Eckart Zitzler

ETH Z i h

MOMH 200

Tutorial on EMEpsilon DominanceEpsilon Dominance


20/34

ETH Zrich Tutorial on EMEpsilon DominanceEpsilon Dominance

Definition 1: -DominanceA -dominates B iff f(A) f(B)

(known since 1987)

Definition 2: -Pareto set

subset of the Pareto set

which -dominates all Pareto-optimal solutions

-dominated

dominated

Pareto front

-Pareto front

Eckart Zitzler

ETH Zrich

MOMH 200

Tutorial on EMAchieving Convergence and DiversityAchieving Convergence and Diversity


21/34

ETH Zrich Tutorial on EMAchieving Convergence and DiversityAchieving Convergence and Diversity

Goal: Maintain -Pareto setIdea: -grid, i.e. maintain a

set of nondominated

boxes (one solutionper box)

Algorithm: (-update)

Accept a child if

n the corresponding box is not

dominated by any box that

contains an individual

AND

o any other individual in the samebox is dominated by the new

solution

y2

y1

2

2

3

3

1

1

Eckart Zitzler

ETH Zrich

MOMH 200

Tutorial on EMRunRun--Time Analysis of aTime Analysis of a MultiobjectiveMultiobjective EAEA


22/34

ETH Zrich Tutorial on EMRunRun Time Analysis of aTime Analysis of a MultiobjectiveMultiobjective EAEA

Basic question: [Laumanns et al.: 2002]

What is the worst case run-time of a multiobjective EA to find

the Pareto set with high probability?

Scenario:Problem: leading ones,

trailing zeros (LOTZ)

Variation: single

point mutation

1 1 0 1 0 0 0

1 1 1 0 0 0

1 1 1 0 0 01

0

y2

y1

0 0 0 0 0 0 0

trailing 0s

leading 1s

1 1 1 1 1 1 1

1 1 1 1 0 0 0

Eckart Zitzler

ETH Zrich

MOMH 200

Tutorial on EMTwo SimpleTwo Simple Multiobjective EAsMultiobjective EAs


23/34

ETH Zrich Two SimpleTwo Simple Multiobjective EAsMultiobjective EAs

select

individualfrom archive

insert

into archive

if not dominated

remove

dominatedfrom archive

flip

randomlychosen bit

Variant 1: SEMOEach individual in the

archive is selected with

the same probability

(uniform selection)

Variant 2: FEMOSelect individual with

minimum number of

children

(fair selection)

Eckart Zitzler

ETH Zrich

MOMH 200

Tutorial on EMResults of AnalysisResults of Analysis


24/34

ETH Zrich Results of AnalysisResults of Analysis

Simple multiobjective EA with uniform selection (SEMO): (n )To get to the Pareto front requires n steps

To cover the entire front needs n steps

Simple multiobjective EA with fair selection (FEMO): (n log n)

Fair selection helps to spread over the Pareto front

Multistart single-objective optimizer: (n )

In average, one out ofn mutations successful

To get to the Pareto front, n successful mutations needed

Overall n Pareto-optimal solutions have to be found

3

3

2

2

3

multiobjective EA faster than multistart strategy

Eckart Zitzler

ETH Zrich

MOMH 200

Tutorial on EMEmpirical Performance AssessmentEmpirical Performance Assessment


25/34

Empirical Performance AssessmentEmpirical Performance Assessment

Issues: quality measures, statistical testing, benchmark problems,visualization,

Popular approach: unary quality measures

Assign each outcome a real number

Outcomes are compared by comparing the corresponding values

Hypervolume

[Zitzler, Thiele: PPSN1998]

Generational Distance

[Van Veldhuizen: 1999]

Eckart Zitzler

ETH Zrich

MOMH 200

Tutorial on EMQuality Measures: Theoretical Results IQuality Measures: Theoretical Results I


26/34

Quality Measures: Theoretical Results IQ y

There is no combination of unary quality measures such thatS is better than T in all criteria is equivalent to S dominates T

ST

application of

quality measures

Basic question: Can we say on the basis of the quality measures

whetheror thatan algorithm outperforms another?

hypervolume 432.34 420.13

distance 0.3308 0.4532

diversity 0.3637 0.3463

spread 0.3622 0.3601

cardinality 6 5

S T

Unary quality measures usually do not tell thatS dominates T; at maximum that S does not dominate T

[Zitzler et al.: 2002]

Eckart Zitzler

ETH Zrich

MOMH 200

Tutorial on EMQuality Measures: Theoretical Results IIQuality Measures: Theoretical Results II


27/34

Q yQ y

Many popular quality measures are not compliantwith the dominance relation

[Hansen, Jaszkiewicz: 1998][Knowles, Corne: 2002][Zitzler et al.: 2002]

Example: diversity measures

Needed: appropriate binary quality measures that indicatewhetheran outcome dominates another, e.g., -measure

ST

S better diversityvalue than T

ST

T better diversityvalue than S

Eckart Zitzler

ETH Zrich

MOMH 200

Tutorial on EMEpsilon Quality MeasureEpsilon Quality Measure


28/34

p Q yp Q y

Definition 3: single solutionsI(A,B) = minimum such that

A -dominates B

Definition 4: sets of solutions

I(S,T) = minimum such thateach solution in T is

-dominated by at leastone solution in S

AB

S

T[Zitzler et al.: 2002]

Eckart Zitzler

ETH Zrich

MOMH 200



29/34

nThe Big Picture:Optimization and evolutionary computation




Example of an algorithm variant

q The Big Question:



s The End:


Eckart Zitzler

ETH Zrich

MOMH 200

Tutorial on EMImplementation: Current SituationImplementation: Current Situation


30/34

pp

Application engineerknowledge in the algorithm

domain necessary

state-of-the-art algorithms getmore and more complex

many algorithms

Algorithm designercomparison to competing

algorithms mandatory

tests on various benchmarkproblems necessary

algorithms and applications

become increasingly complex

high implementation effort / risk of implementation errors

Programming libraries:

valuable tools to tailor a particular technique to a specific application

exchange of optimization algorithm or application still difficult

Eckart Zitzler

ETH Zrich

MOMH 200

Tutorial on EMThe Concept of PISAThe Concept of PISA


31/34

SPEA2

NSGA-II

PAES

Algorithms Applications

knapsack

TSP

network

design

text-based

Platform and programming language independent Interface

for Search Algorithms [Bleuler et al.: 2002]

Eckart Zitzler

ETH Zrich

MOMH 200

Tutorial on EMPISA: ImplementationPISA: Implementation


32/34

selector

process

selector

process

text

files

sharedfile system

sharedfile system

variator

process

variator

process

application independent:

mating / environmentalselection

individuals are described

by IDs and objective

vectors

handshake protocol:

state / actionindividual IDs

objective vectors

parameters

application dependent:

variation operatorsstores and manages

individuals

Eckart Zitzler

ETH Zrich

MOMH 200

Tutorial on EMConclusionsConclusions


33/34

Why using an evolutionary algorithm?Flexibility: problem formulation can be easily modified /

extended (minimum requirements)

Multiple objectives: the solution space can be explored in asingle optimization run

Feasibility: EAs are applicable to complex and huge search

spacesWhy multiobjective optimization?

Robustness: aggregation of several objectives into a single one

requires setting of parametersConfidence: it is easier to select a solution if alternatives are

known

Main application of EMO: design space exploration

Eckart Zitzler

ETH Zrich

MOMH 200

Tutorial on EMMore onMore on EMOEMO


34/34

Links:

EMO mailing list:http://w3.ualg.pt/lists/emo-list/

EMO bibliography:

http://www.lania.mx/~ccoello/EMOO/PISA website:http://www.tik.ee.ethz.ch/pisa/

Events:

Conference on Evolutionary Multi-Criterion Optimization (EMO 2003),

April 8-11, 2003, Algarve, Portugal:http://conferences.ptrede.com/emo03/

Acknowledgments:

Stefan Bleuler, Marco Laumanns, Lothar Thiele

tutorial on evolutionary multiobjective optimization

Documents