Evolutionary Methods in Multi-Objective Evolutionary Methods in Multi-Objective Optimization Optimization - Why do they work ? - - Why do they work ? - Lothar Thiele Computer Engineering and Networks Laboratory Dept. of Information Technology and Electrical Engineering Swiss Federal Institute of Technology (ETH) Zurich Computer Engineering Computer Engineering and Networks Laboratory and Networks Laboratory
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Evolutionary Methods in Multi-Objective Optimization - Why do they work ? -
Computer Engineering and Networks Laboratory. Evolutionary Methods in Multi-Objective Optimization - Why do they work ? -. Lothar Thiele Computer Engineering and Networks Laboratory Dept. of Information Technology and Electrical Engineering Swiss Federal Institute of Technology (ETH) Zurich. - PowerPoint PPT Presentation
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Evolutionary Methods in Multi-Objective Evolutionary 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
Definition 1: -DominanceA -dominates B iff (1+)·f(A) f(B)
Definition 2: -Pareto setA subset of the Pareto-optimalset which -dominates all Pareto-optimal solutions
-dominated dominated
Pareto set -Pareto set
(known since 1987)
Keeping Convergence and DiversityKeeping Convergence and Diversity
Goal: Maintain -Pareto set
Idea: -grid, i.e. maintain a
set of non-dominated
boxes (one solution
per box)
Algorithm: (-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 in the same box is dominated by the new solution
y2
y1
(1+)2
(1+)2 (1+)3
(1+)3
(1+)
(1+)
11
Correctness of Archiving MethodCorrectness of Archiving Method
Theorem:Let F = (f1, f2, f3, …) be an infinite sequence of objective vectorsone by one passed to the -update algorithm, and Ft the union ofthe first t objective vectors of F.
Then for any t > 0, the following holds: the archive A at time t contains an -Pareto front of Ft
the size of the archive A at time t is bounded by the term (K = “maximum objective value”, m = “number of objectives”)
1
)1log(
log
mK
Sketch of Proof:
3 possible failures for At not being an -Pareto set of Ft (indirect proof)at time k t a necessary solution was missedat time k t a necessary solution was expelledAt contains an f Pareto set of Ft
Number of total boxes in objective space:Maximal one solution per box acceptedPartition into chains of boxes
Correctness of Archiving MethodCorrectness of Archiving Method
mK
)1log(
log
1
)1log(
log
mK
)1log(
log
K
Simulation ExampleSimulation Example
Rudolph and Agapie, 2000
Epsilon- Archive
OverviewOverview
introduction
limit behavior
run-time performance measures
Running Time Analysis: Related WorkRunning Time Analysis: Related Work
General upper bound technique & Graph search process
1. 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
selectindividual
from population
insertinto population
if not dominated
removedominatedfrom population
fliprandomly
chosen bit
Variant 1: SEMO
Each individual in thepopulation is selected
with 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)
Previous Work on Multiobjective Quality Previous Work on Multiobjective Quality MeasuresMeasures
Status:Visual comparisons common until recentlyNumerous 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 used
Open questions:What kind of statements do quality measures allow?Can quality measures detect whether or that a Pareto set approximation is better than another?
[Zitzler, Thiele, Laumanns, Fonseca, Grunert da Fonseca: 2003]
Comparison Methods and Dominance Comparison Methods and Dominance RelationsRelations
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) better than B)
Limitations of Unary Quality MeasuresLimitations of Unary Quality Measures
Theorem:There exists no unary quality measure that is able to 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 measures
applicable to any problem.
Power of Unary Quality IndicatorsPower of Unary Quality Indicators