multi object optimization
TRANSCRIPT
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Evolutionary Methods in Multi-ObjectiveEvolutionary Methods in Multi-Objective
OptimizationOptimization
- Why do they work ? -- Why do they work ? -
Lothar Thiele
Computer Engineering and Networks LaboratoryDept. of Information Technology and Electrical
EngineeringSwiss Federal Institute of Technology (ETH) Zurich
Computer EngineeringComputer Engineering
and Networks Laboratoryand Networks Laboratory
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OverviewOverview
introduction
limit behavior
run-time
performance measures
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?
Black-Box OptimizationBlack-Box Optimization
Optimization Algorithm:
only allowed to evaluate f
(direct search)
decision
vector xobjective
vector f(x(
objective function
(e.g. simulation model)
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Issues in EMOIssues in EMO
y2
y1
Diversity
Convergence
How to maintain a diverse
Pareto set approximation?
density estimation
How to preventnondominated
solutions from being lost?
environmental
selection
How to guide thepopulation
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Multi-objective OptimizationMulti-objective Optimization
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A Generic Multiobjective EAA Generic Multiobjective EA
archivepopulation
new population new archive
evaluatesamplevary
updatetruncate
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Comparison of Three ImplementationsComparison of Three Implementations
SPEA2
VEGA
extended VEGA
2-objective knapsack problem
Trade-off betweendistance and
diversity?
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Performance Assessment: ApproachesPerformance Assessment: Approaches
Theoretically (by analysis): difficult
Limit behavior (unlimited run-time resources)
Running time analysis
Empirically (by simulation): standard
Problems: randomness, multiple objectives
Issues: quality measures, statistical testing,benchmark problems, visualization, …
Which technique is suited for which problem class?
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The Need for Quality MeasuresThe Need for Quality Measures
A
B
AB
independent of user preferences
Yes (strictly) No
dependent on
user preferences
How much? In what aspects?
Is A better than B?
Ideal: quality measures allow to make both type of statements
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OverviewOverview
introduction
limit behavior
run-time
performance measures
A l i M i AA l i M i A t
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Analysis: Main AspectsAnalysis: Main Aspects
Evolutionary algorithms are random searchheuristics
Computation Time
(number of iterations)
ProbabilityOptimum found
∞
1
1/2
Qualitative:Limit behavior
for t → ∞
Quantitative:
Expected
Running
Time E(T)
A l g o r i t h m
A a p
p l i e d t
o P r o b
l e m B
A hi iA hi i
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ArchivingArchiving
optimizationoptimization archivingarchiving
generate update, truncatefinite
memory
finite
archive A
every solution
at least once
f 2
f 1
Requirements:
1. Convergence to Pareto front2. Bounded size of archive
3. Diversity among the stored
solutions
P bl D t i tiP bl D t i ti
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Problem: DeteriorationProblem: Deterioration
f 2
f 1
f 2
f 1
t t+1
new
solution
discarded
new
solution
New solution accepted in t+1 is dominated by asolution found previously (and “lost” during theselection process)
bounded
archive
(size 3)
P bl D t i tiProblem: Deterioration
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Goal: Maintain “good” front
(distance + diversity)
But: Most archiving
strategies may forget
Pareto-optimal
solutions…
Problem: DeteriorationProblem: Deterioration
NSGA-II
SPEA
Limit Beha ior Related WorkLimit Behavior: Related Work
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Limit Behavior: Related Work Limit Behavior: Related Work
Requirements for
archive:
1. Convergence
2. Diversity
3. Bounded Size
(impractical)
“store all”
[Rudolph
98,00]
[Veldhuizen 99]
“store one”
[Rudolph 98,00]
[Hanne 99]
(not nice)
in this work
Solution Concept: Epsilon DominanceSolution Concept: Epsilon Dominance
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Solution Concept: Epsilon DominanceSolution Concept: Epsilon Dominance
Definition 1: ε-Dominance
A ε-dominates B iff (1+ε)·f(A) ≥ f(B)
Definition 2: ε-Pareto set
A subset of the Pareto-optimalset which ε-dominates all Pareto-optimal solutions
ε-dominated
dominated
Pareto set ε-Pareto set
(known since 1987)
Keeping Convergence and DiversityKeeping Convergence and Diversity
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Keeping Convergence and DiversityKeeping Convergence and Diversity
Goal: Maintain ε-Pareto set
Idea: ε-grid, i.e.maintain a
set of non-dominated
boxes (one solutionper box)
Algorithm: (ε-update)
Accept a new solution f if
the corresponding box is not
dominated by any box
represented in the archive A
AND
any other archive member inthe same box is dominated by
the new solution
y2
y1
(1+ε)2
(1+ε)2
(1+ε)3
(1+ε)3
(1+ε)
(1+ε)
1
1
Correctness of Archiving Method
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Correctness of Archiving Method
Theorem:
Let F = (f 1, f 2, f 3, …) be an infinite sequence of objective
vectorsone by one passed to the ε-update algorithm, and Ft the
union of the first t objective vectors of F.
Then for any t > 0, the following holds: the archive A at time t contains an ε-Pareto front of Ft
the size of the archive A at time t is bounded by theterm
(K = “maximum objective value”, m = “number of objectives”)
1
)1log(
log−
+
m
K
ε
Correctness of Archiving Method
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Sketch of Proof:
3 possible failures for At not being an ε-Pareto set of Ft
(indirect proof)
at time k ≤ t a necessary solution was missed
at time k ≤ t a necessary solution was expelled
At contains an f ∉ Pareto set of Ft
Number of total boxes in objective space:Maximal one solution per box accepted
Partition into chains of boxes
Correctness of Archiving Method
m
K
+ )1log(
log
ε
1
)1log(
log−
+
m
K
ε
+ )1log(
log
ε
K
Simulation ExampleSimulation Example
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Simulation ExampleSimulation Example
Rudolph and Agapie, 2000
Epsilon- Archive
OverviewOverview
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OverviewOverview
introduction
limit behavior
run-time
performance measures
Running Time Analysis: Related WorkRunning Time Analysis: Related Work
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Running Time Analysis: Related Work Running Time Analysis: Related Work
Single-objective
EAs
Multiobjective
EAs
discrete
search
spaces
continuous
search
spaces
problem domain type of results
• expected RT
(bounds)
• RT with high
probability
(bounds)
[Mühlenbein 92]
[Rudolph 97]
[Droste, Jansen, Wegener 98,02]
[Garnier, Kallel, Schoenauer 99,00]
[He, Yao 01,02]
• asymptoticconvergence rates
• exact convergence
rates
[Beyer 95,96,…]
[Rudolph 97]
[Jagerskupper 03]
[none]
MethodologyMethodology
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MethodologyMethodology
Typical “ingredients”
of a Running TimeAnalysis:
Simple algorithms
Simple problems
Analytical methods & tools
Here:
⇒ SEMO, FEMO, GEMO (“simple”,
“fair”, “greedy”)
⇒ mLOTZ, mCOCZ (m-objectivePseudo-Boolean problems)
⇒ General upper bound technique
& Graph search process1. Rigorous results for specific algorithm(s) on specific
problem(s)
2. General tools & techniques
3. General insights (e.g., is a population beneficial at all?)
Three Simple Multiobjective EAsThree Simple Multiobjective EAs
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Three Simple Multiobjective EAsThree Simple Multiobjective EAs
select
individualfrom population
insertinto population
if not dominated
removedominatedfrom population
fliprandomly
chosen bit
Variant 1: SEMO
Each individual in thepopulation is selectedith the same probability
(uniform selection)
Variant 2: FEMO
Select individual withminimum number of mutation trials
(fair selection)
Variant 3: GEMO
Priority of convergenceif there is progress
(greedy selection)
SEMOSEMO
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SEMOSEMO
population P
x
x ’
uniform selection
single point
mutation
include,
if not dominated
remove dominated and
duplicated
Example Algorithm: SEMOExample Algorithm: SEMO
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Example Algorithm: SEMOExample Algorithm: SEMO
1. Start with a random solution
2. Choose parent randomly (uniform)
3. Produce child by variating parent
4. If child is not dominated then
• add to population
• discard all dominated
1. Goto 2
Simple Evolutionary Multiobjective Optimizer
,T ili Z )
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, Trailing Zeroes)
Given:
Bitstring
Objective:
Maximize # of “leading
ones”Maximize # of “trailingzeroes”
−=Ν
∑∏
∑∏
= =
= =n
i
n
i j
j
n
i
i
j
j
n
x
x
x
1
1 12
)1(
)LOTZ( , 1,0:LOTZ
1 1 1 0 1 1 0 01 1 1 0 0
Run-Time Analysis: ScenarioRun-Time Analysis: Scenario
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u e a ys s Sce a oy
Problem:leading ones, trailing zeros (LOTZ)
Variation:single point mutation
one bit per individual
1 1 0 1 0 0 0
1 1 1 0 0 0
1 1 1 0 0 01
0
The Good NewsThe Good News
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SEMO behaves like a single-objective EA until the Paretoset
has been reached...y2
y1
trailing 0s
leading 1s
Phase 1:
only one
solution stored
in the archive
Phase 2:
only Pareto-optimal
solutions stored
in the archive
SEMO: Sketch of the Analysis ISEMO: Sketch of the Analysis I
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yy
Phase 1: until first Pareto-optimal solution has beenfound
i → i-1: probability of a successful mutation ≥ 1/nexpected number of mutations = n
i=n → i=0: at maximum n-1 steps (i=1 not possible)expected overall number of mutations =
O(n2)
1 1 0 1 1 0 0
leading 1s trailing 0s
i = number of ‘incorrect’ bits
SEMO: Sketch of the Analysis IISEMO: Sketch of the Analysis II
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y
Phase 2: from the first to all Pareto-optimal solutions
j → j+1: probability of choosing an outer solution ≥ 1/j, ≤2/j
probability of a successful mutation ≥ 1/n , ≤
2/nexpected number T j of trials (mutations) ≥ nj/4,
≤ nj
Pareto set
j = number of optimal solutions in thepopulation
SEMO on LOTZSEMO on LOTZ
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Can we do even better ?Can we do even better ?
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Our problem is the exploration of the Pareto-front.
Uniform sampling is unfair as it samples early found
Pareto-points more frequently.
FEMO on LOTZFEMO on LOTZ
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Sketch of Proof
000000
100000
110000
111000
111100
111110
111111
Probability for each individual, that parent did not generate it with c/p log n
n individuals must be produced. The probability that one needs more
than c/p log n trials is bounded by n1-c.
FEMO on LOTZFEMO on LOTZ
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Single objective (1+1) EA with multistart strategy(epsilon-constraint method) has running time Θ (n3).
EMO algorithm with fair sampling has running timeΘ (n2 log(n)).
Generalization: Randomized Graph SearchGeneralization: Randomized Graph Search
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⇒ Pareto front can be modeled as a graph
⇒ Edges correspond to mutations
⇒ Edge weights are mutation probabilities
How long does it take to explore the whole graph?
How should we select the “parents”?
Randomized Graph SearchRandomized Graph Search
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Comparison to Multistart of Single-objective Method
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feasible region
constraint
feasible region
constraint
feasible region
constraint
Underlying concept:
Convert all objectives except of one into constraints
Adaptively vary constraints
f 2
f 1
maximize f 1
s.t. f i > ci, i ∈ 2,…,m
)()( 2nnT E ⋅Θ=
Find one
point
number of
points
Optimizer (GEMO)Optimizer (GEMO)
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1. Start with a random solution
2. Choose parent with smallest counter;
increase this counter
3. Mutate parent
4. If child is not dominated then
• If child not equal to anybody
• add to population
• else if child equal to somebody
• if counter = ∞: set back to 0
• if child dominates anybody
• set all other counters to ∞
• discard all dominated
1. Goto 2
Optimizer (GEMO)Optimizer (GEMO)
Idea:
Mutation counter for
each individual
I. Focus
search effort
II. Broaden
search effort
… without knowing where we are!
Running Time Analysis: Comparison
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Algorithms
Problems
Population-based approach can be more efficient than
multistart of single membered strategy
OverviewOverview
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introduction
limit behavior
run-time
performance measures
Performance Assessment: ApproachesPerformance Assessment: Approaches
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Theoretically (by analysis): difficult
Limit behavior (unlimited run-time resources)
Running time analysis
Empirically (by simulation): standard
Problems: randomness, multiple objectives
Issues: quality measures, statistical testing,benchmark problems, visualization, …
Which technique is suited for which problem class?
The Need for Quality MeasuresThe Need for Quality Measures
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A
B
AB
independent of user preferences
Yes (strictly) No
dependent onuser preferences
How much? In what aspects?
Is A better than B?
Ideal: quality measures allow to make both type of statements
Independent of User PreferencesIndependent of User Preferences
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weakly dominates = not worse in all objectives
sets not equal
dominates= better in at least one
objective
strictly dominates= better in all objectives
is incomparable to= neither set weakly better
A
B
C
D
A B
C D
Pareto set approximation (algorithm outcome) =set of incomparable
solutionsΩ = set of all Pareto set approximations
A C
B C
Dependent on User PreferencesDependent on User Preferences
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Goal: Quality measures compare two Pareto setapproximations A and B.
A
B
hypervolume 432.34 420.13distance 0.3308 0.4532
diversity 0.3637 0.3463
spread 0.3622 0.3601
cardinality 6 5
A B
“A better”
application of quality measures
(here: unary)
comparison andinterpretation of
quality values
Quality Measures: ExamplesQuality Measures: ExamplesUnary Binary
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Unary
Hypervolume measure
Binary
Coverage measure
performance
cheapness
S(A)A
performance
cheapness
B
A
[Zitzler, Thiele: 1999]
S(A) = 60%C(A,B) = 25%
C(B,A) = 75%
MeasuresMeasures
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Status:
Visual comparisons common until recently
Numerous quality measures have been proposedsince the mid-1990s[Schott: 1995][Zitzler, Thiele: 1998][Hansen, Jaszkiewicz: 1998][Zitzler: 1999][Van Veldhuizen, Lamont: 2000][Knowles, Corne , Oates: 2000][Deb et al.: 2000] [Sayin: 2000][Tan, Lee, Khor: 2001][Wu,
Azarm: 2001]…
Most popular: unary quality measures (diversity +distance)
No common agreement which measures should be
usedOpen questions:
What kind of statements do quality measures allow?
Can quality measures detect whether or that a Pareto
NotationsNotations
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Def: quality indicator
I: Ωm → ℜ
Def: interpretation function
E: ℜk × ℜk →false, true
Def: comparison method based on I = (I1, I2, …, Ik) and EC: Ω× Ω →false, true
where
A, B ℜk × ℜk false, true
quality
indicators
interpretation
function
RelationsRelations
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Compatibility of a comparison method C:
C yields true ⇒ A is (weakly, strictly) better than B
(C detects that A is (weakly, strictly) better than B)
Completeness of a comparison method C:
A is (weakly, strictly) better than B ⇒C yields true
Ideal: compatibility and completeness, i.e.,
A is (weakly, strictly) better than B ⇔C yields true
(C detects whether A is (weakly, strictly) betterthan B)
Limitations of Unary Quality MeasuresLimitations of Unary Quality Measures
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Theorem:
There exists no unary quality measure that is ableto detectwhether A is better than B.
This statement even holds, if we consider a finite
combinationof unary quality measures.
There exists no combination of unary measuresapplicable to any problem.
Theorem 2: Sketch of Proof Theorem 2: Sketch of Proof
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Any subset A ⊆ S is aPareto set approximation
Lemma: any A’, A’’ ⊆ Smust be mapped to adifferent point in ℜk
Mapping from 2S to ℜk mustbean injection, but
Cardinality of 2S greaterthan ℜk
Power of Unary Quality IndicatorsPower of Unary Quality Indicators
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dominates doesn’t weakly dominate doesn’t dominate weakly domin
Quality Measures: ResultsQuality Measures: Results
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There is no combination of unary quality measuressuch that
S is better than T in all measures is equivalent to S dominates T
ST
application of quality measures
Basic question: Can we say on the basis of thequality measures
whether or that an algorithm outperformsanother? hypervolume 432.34 420.13distance 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]
A New Measure:A New Measure: ε-Quality Measure-Quality Measure
Two solutions: Two approximations:
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Two solutions:
E(a,b) =max1≤ i ≤ n min
ε ε ⋅ f i(a) ≥
f i(b)
1 2 4
A
B2
1
E(A,B) = 2E(B,A) = ½
Two approximations:
E(A,B) =maxb ∈ B mina ∈ A E(a,b)
a
b2
1
E(a,b) = 2E(b,a) = ½
1 2
Advantages: allows all kinds of statements (complete andcompatible)
Selected ContributionsSelected Contributions
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Algorithms:° Improved techniques [Zitzler, Thiele: IEEE TEC1999]
[Zitzler, Teich, Bhattacharyya: CEC2000][Zitzler, Laumanns, Thiele:
EUROGEN2001]
° Unified model [Laumanns, Zitzler, Thiele: CEC2000][Laumanns, Zitzler, Thiele: EMO2001]
° Test problems [Zitzler, Thiele: PPSN 1998, IEEE TEC1999]
[Deb, Thiele, Laumanns, Zitzler: GECCO2002]
Theory:° Convergence/diversity [Laumanns, Thiele, Deb, Zitzler:
GECCO2002][Laumanns, Thiele, Zitzler, Welzl, Deb: PPSN-
How to apply (evolutionary) optimization algorithms tolarge-scale multiobjective optimization problems?