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Computer EngineeringComputer Engineeringand and NetworksNetworks LaboratoryLaboratory
TwoTwo DecadesDecades of EMO:of EMO:A A GlanceGlance Back and A Look Back and A Look AheadAhead
Eckart ZitzlerEckart Zitzler
IEEE Symposium on CI in MCDMIEEE Symposium on CI in MCDM, 5 April 20075 April 2007
Two Decades of EMO 2© Eckart Zitzler ETH Zurich
Evolutionary Multi-Criterion Optimization (EMO)
Key issues:How to formalize what a goodPareto set approximation is?
How to search for a goodPareto set approximation?
How to use the informationprovided by an approximation?
f2
f1
EMO = evolutionary algorithms and other randomized search heuristics
... applied to problems involving multiple objectives (in general)
... used to approximate the Pareto-optimal set (mainly)
Two Decades of EMO 3© Eckart Zitzler ETH Zurich
A Brief History of EMO Research
1984
1990
1995
2000
2007
first EMO approaches
dominance-based EMO algorithms with diversity preservation techniques
elitist EMO algorithms
quantitative performance assessment
attainment functions
EMO algorithms based on set quality measures
preference articulation convergence proofs
running time analyses quality measure designuncertainty and robustness
statistical performance assessment
test problem design
high-dimensional objective spaces
multiobjectivization
dominance-based population ranking
Two Decades of EMO 4© Eckart Zitzler ETH Zurich
EMO: A Fast Growing Field
Statistics of the EMO repositorymaintained by C. A. Coello Coello
Overall: 2615 references by 11/2006
http://www.lania.mx/~ccoello/EMOO/EMOOstatistics.html
Two Decades of EMO 5© Eckart Zitzler ETH Zurich
The EMO Community
The EMO conference series:
EMO2001 EMO2003 EMO2005 EMO2007Zurich Faro Guanajuato Matsushima
Switzerland Portugal Mexico Japan
45 / 87 56 / 100 59 / 115 65 / 124
Many further activities:special sessions, special issues, workshops, tutorials, ...
Two Decades of EMO 6© Eckart Zitzler ETH Zurich
A Personal View: Four Main Lessons Learned
Lesson 1: EMO provides information about a problem(search space exploration)
Lesson 2: EMO can help in single-objective scenarios(multiobjectivization)
Lesson 3: EMO is part of the decision making process(preference articulation)
Lesson 4: EMO for large n is different from n = 2(high-dimensional objective spaces)
Two Decades of EMO 7© Eckart Zitzler ETH Zurich
A Personal View: Four Main Lessons Learned
Lesson 1: EMO provides information about a problem(search space exploration)
Lesson 2: EMO can help in single-objective scenarios(multiobjectivization)
Lesson 3: EMO is part of the decision making process(preference articulation)
Lesson 4: EMO for large n is different from n = 2(high-dimensional objective spaces)
Two Decades of EMO 8© Eckart Zitzler ETH Zurich
A Personal View: Four Main Lessons Learned
Lesson 1: EMO provides information about a problem(search space exploration)
Lesson 2: EMO can help in single-objective scenarios(multiobjectivization)
Lesson 3: EMO is part of the decision making process(preference articulation)
Lesson 4: EMO for large n is different from n = 2(high-dimensional objective spaces)
Two Decades of EMO 9© Eckart Zitzler ETH Zurich
Lesson 1: EMO provides information about a problem
The question:Why at all should one try to approximate the entire Pareto-optimal set?
An answer:Because it provides useful information about the problem...
and
...we know how to do that for a small number of objectives!
ProblemProblem
DecisionMaking
DecisionMaking
EMOEMO
ModelModel
SolutionSolution
Two Decades of EMO 10© Eckart Zitzler ETH Zurich
Note: good in terms of set quality = good in terms of search?
A General Scheme of a Dominance-Based MOEA
(archiv)population offspring
environmental selection (greedy heuristic)environmental selection (greedy heuristic)
mating selection (stochastic)mating selection (stochastic) fitness assignmentpartitioning into
dominance classes
rank refinement withindominance classes
fitness assignmentpartitioning into
dominance classes
rank refinement withindominance classes
+
Two Decades of EMO 11© Eckart Zitzler ETH Zurich
Ranking of the Population Using Dominance
... goes back to a proposal by David Goldberg in 1989.
... is based on pairwise comparisons of the individuals only.
dominance rank: by howmany individuals is anindividual dominated?MOGA, NPGAdominance count: how manyindividuals does an individualdominate?SPEA, SPEA2dominance depth: at whichfront is an individual located?NSGA, NSGA-II
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dominancecount
dominancerank
dominance depth
Two Decades of EMO 12© Eckart Zitzler ETH Zurich
Refinement of Dominance Rankings
Goal: rank incomparable solutions within a dominance class
Density information (good for search)
Quality indicator (good for set quality): later...
ff
f
Kernel method
density =function of the
distances
k-th nearest neighbor
density =function of distance
to k-th neighbor
Histogram method
density =number of elements
within box
Two Decades of EMO 13© Eckart Zitzler ETH Zurich
EMO and Population Synergies
One MOEA run can be more effective than aggregation + multiple runs
MOEAs
Theoretical evidence:[Laumanns et al. 2004]
Empirical evidence:
[Zitzler, Thiele 1999]
Two Decades of EMO 14© Eckart Zitzler ETH Zurich
Application: Design Space Exploration
Cost
Latency Power
SpecificationSpecification OptimizationOptimization ImplementationImplementation
EnvironmentalSelection
EnvironmentalSelectionMutation
Mutation
x2
x1
f
MatingSelection
MatingSelectionRecombination
Recombination
EvaluationEvaluation
Two Decades of EMO 15© Eckart Zitzler ETH Zurich
Application: Design Space Exploration
Cost
Latency Power
SpecificationSpecification OptimizationOptimization ImplementationImplementation
EnvironmentalSelection
EnvironmentalSelectionMutation
Mutation
x2
x1
f
MatingSelection
MatingSelectionRecombination
Recombination
EvaluationEvaluation
Water resourcemanagement[Siegfried et al. 2006]
Water resourcemanagement[Siegfried et al. 2006]
Two Decades of EMO 16© Eckart Zitzler ETH Zurich
Application: Trade-Off Analysis
Module identification from biological data [Calonder et al. 2006]
Find group of genes w.r.t.different data types:
similarity of geneexpression profiles
overlap of proteininteraction partners
metabolic pathwaymap distances
Two Decades of EMO 17© Eckart Zitzler ETH Zurich
Application: Approximation Set Analysis
Multiple disk clutch brake design [Deb, Srinivasan 2006]
Two Decades of EMO 18© Eckart Zitzler ETH Zurich
Lesson 2: EMO Helps in Single-Objective Scenarios
Have seen: EMO can help in single-objective aggregation scenariosEven better: EMO can help in general in single-objective scenarios
Multiobjectivization [Knowles et al. 2001]:1. Transform a single-objective problem into a multiobjective one2. Solve it using an MOEA
Types of multiobjectivization:Approach I:
Decompose objective function into several functionsApproach II:
Leave problem as it is, but add further objective functions
(in principle combination possible)
Two Decades of EMO 19© Eckart Zitzler ETH Zurich
Approach I: Decomposition Into Multiple Objectives
Empirical studies:
converting constraints into objectives[Coello 1999]transforming the H-IFF problem into a biobjective problem[Knowles et al. 2001]
Theoretical studies:
shortest path problem[Scharnow et al.2004]decomposing the spanning tree problem into two objectives[Neumann, Wegener 2005]
In all of the above studies, the MOEA outperformed its single-objective counterpart.
Two Decades of EMO 20© Eckart Zitzler ETH Zurich
Approach II: Together Is Easier
Running time analysis results:[Brockhoff et al. 2007]
(1+1)-EA on both functions:
Θ(n3)
Simple MOEA on biobjectiveproblem:
Θ(n2logn)
Note: search space and Pareto-optimal set unchanged
Problem:
Two Decades of EMO 21© Eckart Zitzler ETH Zurich
Approach II: Multiobjective Genetic Programming
Problem: trees grow rapidlypremature convergenceoverfitting of training data
Common methods:constraint(tree size limitation)penalty term(parsimony pressure) objective ranking(size post-optimization)
structure-based (ADF, etc.)
Multiobjective method:Optimize both error and size[Ekart, Nemeth 2001; deJong et al. 2001; Bleuler et al 2001]
Keep small trees(diversity)
error
tree size
Two Decades of EMO 22© Eckart Zitzler ETH Zurich
Multiobjective GP: Results
Algorithm:SPEA2 without density estimation
Benchmark:even-parity problem
Runs
Siz
e (#
of e
dges
)
SPEA2
Runs
Siz
e (#
of e
dges
)
Constant Parsimony
Two Decades of EMO 23© Eckart Zitzler ETH Zurich
Multiobjective GP: Why Does It Work?
Two Decades of EMO 24© Eckart Zitzler ETH Zurich
Lesson 3: EMO is based on preferences
What we thought: EMO is preference-less
What we learnt: EMO just uses weaker preference information
⇒ (almost) all MOEAs implicitly implement specific preferences
[Zitzler 1999]
A
B
preferable?environmentalselection
3 out of 6
Two Decades of EMO 25© Eckart Zitzler ETH Zurich
What is the Quality of a Pareto Set Approximation?
Problem: incomparability does not give a search directionNeeded: total ordering of the set of Pareto set approximations
One possibility: Quality indicators I: Ωm → ℜ
unary indicator: assign each approximation a real number I(A)binary indicator: assigns each approximation pair a real number I(A,B)
Example: unary indicators combined
AB
hypervolume 432.34 420.13distance 0.3308 0.4532diversity 0.3637 0.3463spread 0.3622 0.3601cardinality 6 5
A B
“A better”
Two Decades of EMO 26© Eckart Zitzler ETH Zurich
Set Quality Measures: Examples
Unary (absolute)Hypervolume indicator
Binary (relative)Coverage indicator
I(A)A
B
A
I(A) = 60%I(A,B) = 25%I(B,A) = 75%
Two Decades of EMO 27© Eckart Zitzler ETH Zurich
What Are Good Set Quality Measures?
There are three aspects [Zitzler et al. 2000]:
Wrong! [Zitzler et al. 2003]:
f2
f1
An infinite number of unary set measures is needed to detectin general whether A is better than B
Two Decades of EMO 28© Eckart Zitzler ETH Zurich
Order Compliance + Strict Monotonicity
Order preserving: the preference is refined and not violated
Strictly monotonic: sensitive to Pareto dominance
⇒ Uniqueness of optimum: Pareto front achieves maximum value
Bad news: the hypervolume is currently the only known unary setmeasure with these properties
Good news: preferences can be integrated into the hypervolumeindicator [Zitzler et al. 2007]
Two Decades of EMO 29© Eckart Zitzler ETH Zurich
Problems With Non-Compliant Indicators
Two Decades of EMO 30© Eckart Zitzler ETH Zurich
Incorporation of Preferences During Search
Refine/modify dominance relation, e.g.:
using goals, priorities, constraints[Fonseca, Fleming 1998]using different types of cones[Branke 2000]
Use quality indicators, e.g.:
based on reference points [Deb, Sundar 2006]based on the hypervolume indicator (later)based on binary quality indicators (now)
f2
f1
Two Decades of EMO 31© Eckart Zitzler ETH Zurich
Preference-Adaptive Search
Given:Preference information in terms of a binary quality indicator I,here binary epsilon indicator(≡ continuous extension of dominance relation)
Optimization goal:Find Pareto set approximation A such that
I(A, S)
is minimum (S = Pareto-optimal set)
Question:How to assign fitness values?
Two Decades of EMO 32© Eckart Zitzler ETH Zurich
Indicator-Based Fitness Assignment
Idea: measure for “loss in quality” if x1 is removed
Fitness A:
...corresponds to continuous extension of dominance rank(MOGA, Fonseca & Fleming 1993)
...blurrs influence of dominating and dominated individuals
Fitness B:
... parameter κ is problem- and indicator-dependent
... no additional diversity preservation mechanism
Two Decades of EMO 33© Eckart Zitzler ETH Zurich
Lesson 4: Two Objectives Are Not Many
Search:What are the effects of multiple objectives on the search spacestructure?How to guide the search towards the Pareto-optimal set (efficiently)?
Decision making:
Can objectives be omitted, is there redundancy in the set of objectives?Which are the most important objectives?
Most EMO publications focused on two or three objectives –what about many objectives?
Two Decades of EMO 34© Eckart Zitzler ETH Zurich
What Happens If Objectives Are Added?
Here: graph-based representation of dominance relation
Adding objectives can only remove edges, i.e.,(i) comparable → incomparable; or indifferent → (in)comparable
4 separate objectives 4 objectives combined
Two Decades of EMO 35© Eckart Zitzler ETH Zurich
The Effect of Adding Objectives
Observation:The number of incomparable solutions may increase with thenumber of objectives.
Winkler (1985): For random orders with n points and k dimensions, the exptected number of incomparable solutions is between
and
But: In general, adding objectives can have good and bad effects:
Neumann and Wegener (2006), Scharnow et al. (2002):Theoretical examples where more objectives helpBrockhoff et al. (2007):Adding an objective can have positive and/or negative effects
Two Decades of EMO 36© Eckart Zitzler ETH Zurich
Third Objective Makes Problem Easier / Harder
0
500
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2000
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3000
3500
4000
100 150 200 250 300 350 400 450 500
runt
ime
[gen
erat
ions
]
bitstring length
Making indifferent solutions comparable
LOTZ with additional function (i)2-dimensional LOTZ
LOTZ with additional function (ii)
Average runtimes for 10 IBEA runs with population size 200
Two Decades of EMO 37© Eckart Zitzler ETH Zurich
The Problem of Cycling Behavior
Observation: current density-based EMO algorithms fail for n > 3[Wagner et al. 2007]
Example:20-objective problemSPEA2 for 1000 generations
⇒ all solutions visited during the run are incomparable
Explanation:cyclic behavior, i.e.,preference informationill-defined
Two Decades of EMO 38© Eckart Zitzler ETH Zurich
Hypervolume ResearchEmpirical performance assessment:
(Zitzler, Thiele: 1998, 1999)many more...
Theoretical investigations of properties:(Knowles, Corne: 2002)(Fleischer: 2003)(Zitzler et al.: 2003, 2007)
Algorithm design:(Knowles et al.: 2003)(Zitzler, Künzli: 2004)(Emmerich et al.: 2005)(Igel et al.: 2007)
Computational issues:(While et al.: 2005, 2006)(Beume, Rudolph: 2006)(Fonseca et al.: 2006)
How to assign fitness values?
Fitness = loss in hypervolumeif individual is removed
How to assign fitness values?
Fitness = loss in hypervolumeif individual is removed
How to make the calculation fast?How to make the calculation fast?
Two Decades of EMO 39© Eckart Zitzler ETH Zurich
Hypervolume-Based Search: Proof of Principle
Two Decades of EMO 40© Eckart Zitzler ETH Zurich
Dimensionality Reduction
Key question: Are some objectives redundant or less important?
→ decision making easier→ search computationally less expensive
Objective reduction approaches:reduction based on correlation (PCA) [Deb, Saxena 2005]reduction based on ε-dominance [Brockhoff, Zitzler 2006]
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Two Decades of EMO 41© Eckart Zitzler ETH Zurich
Objective Reduction: Example
values values
omit
still the same relations
Key question: Can objectives be omitted without loosing too much?
all objectives pairwisely conflicting
Two Decades of EMO 42© Eckart Zitzler ETH Zurich
Approximation of efficient set computed by evolutionary algorithmused as
for and for
Objective reduction of 50% possible for various test problems
Dimensionality Reduction: What Is Possible?
Num
bero
f obj
ectiv
esin
min
imal
set
problemsDTLZ2 DTLZ5 DTLZ7 KP100 KP250 KP500
k=15k=25
Two Decades of EMO 43© Eckart Zitzler ETH Zurich
So Far So Good – And Now?
My main conclusion: EMO is part of the decision making process
1. Throw everything in2. Run your EMO tool3. Analyze results and learn about the problem4. Refine problem / preferences in a guided or automated fashion5. Go to 2
Many further research topics:Uncertainty and robustnessExpensive objective function evaluationsHybridization: EMO and OR methodsMulti-multiobjective problems...