A Simulated Annealing Algorithm for Noisy Multi-Objective ...non-dominated solution is high Current...
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A Simulated Annealing Algorithm forNoisy Multi-Objective Optimization
Ville Mattila, Kai Virtanen, and Raimo P. HämäläinenSystems Analysis LaboratoryAalto University School of Science
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Overview
Noisy multi-objective optimization problemValues of objective functions are uncertain
Techniques for finding non-dominated solutionsFew take into account noiseMostly evolutionary algorithms (EAs), e.g, [Goh&Tan, 2007]
New simulated annealing (SA) algorithmFeatures for handling noise and generating candidatesolutionsPerformance superior to existing EAs in numericaltests
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Noisy multi-objective optimizationproblem
minl≤x≤u
[f1(x), . . . , fH(x)]
Values of objective functions fi(x) include uncertainty→ Optimization based on fi(x) + ωi where ωi ∼ N (0, σ2
i )
Decision variable x ∈ <n
Constraints l and u
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General steps of SA inmulti-objective optimization
0. Define performance measure E for candidate solutionsSelect temperature TGenerate a current solution x
1. Generate a candidate solution y from the neighborhood ofx
2. Set x = y with probability min[1, exp
(E(x)−E(y)
T
)]3. Solutions with best values of E into the non-dominated set4. Return to step 1 or stop iteration
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Features of the new SA algorithm
Performance of a solution, EBased on probabilistic dominance
→ Takes into account multiple objectives and noiseUsed previously in evolutionary algorithms [Hughes, 2001],not in SA
Generation of candidate solutionsUsing information on current non-dominated solutions
→ Increase likelihood of obtaining new non-dominatedsolutionsPreviously in deterministic single-objective SA[Sun et al., 2008]
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Probabilistic dominance
Probability that x dominates y
Product of probabilities thatobjective function values aresmaller for x
M samplesf̄i(x) and s2
i (x) sampleaverage and variance
P(x � y) =H∏
i=1
Φ
(f̄i (x)− f̄i (y)√
s2i (x)/M + s2
i (y)/M
)0
0
f1
f 2
x
yf1(x) − f1(y)
f2(x) − f2(y)
Φ cumulative distribution function of standard normal distribution
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Performance of candidate solution E(y)
Sum of probabilities thatcurrent non-dominatedsolutions dominatecandidate solution y
E(y) =∑z∈S
P(z � y)
0
0
f1
f 2
yNon-dominated set S
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Generation of candidate solutions
Use empirical data to samplenew value of decision variable x
Recent values innon-dominated setx(1), . . . , x(m) ∈ [a, b]Feasible neighborhood ofcurrent value [a, b]
Concentrates to regions wherelikelihood of obtaining a newnon-dominated solution is high
Current valueNew value
a x(1) x(2)x(3) x(4)x(5) bx
Probability
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Numerical testing
Reference algorithmMOEA-RF, Multi-Objective EA with Robust FeaturesOutperformed several existing EAs in [Goh&Tan, 2007]
Test problemsZDT1, ZDT4, ZDT6 [Zitzler et al., 2007], FON[Fonseca&Fleming, 1998], KUR [Kursawe, 1991]Four noise levels
Performance measures usedGenerational distanceSpacingMaximum spreadHypervolume ratio
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Non-dominated solutions for ZDT1
Highest noise levelSA algorithm produces solutions closer to actualnon-dominated set
0 10
5
f1
f 2
0 10
5
f1
f 2
� SA algorithm � MOEA-RF, 30 runs
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Generational distance for ZDT1
Distance of solutions to actual non-dominated setValues lower (better) for the SA algorithm
Low noise High noise0
0.05
0.1
0.15
� SA algorithm � MOEA-RF, 30 runs with each noise level
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Spacing for ZDT1
Distance between obtained solutionsValues similar for the algorithms (lower is better)
Low noise High noise0
0.2
0.4
� SA algorithm � MOEA-RF, 30 runs with each noise level
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Maximum spread for ZDT1
Range of objective function values for obtained solutionsValues higher (better) for the SA algorithm
Low noise High noise
0.8
0.9
1
� SA algorithm � MOEA-RF, 30 runs with each noise level
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Hypervolume ratio for ZDT1
Volume dominated by obtained solutionsValues higher (better) for the SA algorithm
Low noise High noise0
0.5
1
� SA algorithm � MOEA-RF, 30 runs with each noise level
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Summary of the numerical tests
Our SA algorithm superior accross test problems andnoise levels
Noise level low highPercentile 0 25 50 75 100 0 25 50 75 100 0 25 50 75 100 0 25 50 75 100
ZDT1ZDT4
GD ZDT6FONKURZDT1ZDT4
S ZDT6FONKURZDT1ZDT4
MS ZDT6FONKURZDT1ZDT4
HV ZDT6FONKUR
� SA algorithm superior � MOEA-RF superior
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Conclusions
New SA algorithm for noisy multi-objective optimizationPerformance of solutions based on probabilistic dominanceGeneration of candidate solutions using non-dominated set
Outperformed reference EA in numerical testsComputational requirements comparable to the EASuccessful application: Maintenance scheduling of aircraft[Mattila et al., 2011]Further development
Dynamic sample size for evaluating candidate solutions
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References
C. M. Fonseca and P. J. Fleming. Multiobjective optimization and multiple constrainthandling with evolutionary algorithms - Part II: Application example.IEEE Trans. Syst., Man, and Cybern. A, Syst. Humans, 28(1):38–47, 2000.
C. K. Goh and K. C. Tan. An Investigation on Noisy Environments in EvolutionaryMultiobjective Optimization.IEEE Trans. Evol. Comp., 11(3):354–381, 2007.
E. Hughes. Evolutionary Multi-objective Ranking with Uncertainty and Noise.Proc. 1st Int. Conf. Evol. Multi-Criterion Optim., 329–343, 2001.
F. Kursawe. A variant of evolution strategies for vector optimization.Proc. 1st Workshop on Parallel Problem Solving from Nature, 193–197, 1998.
V. Mattila, K. Virtanen, and R. P. Hämäläinen. Multi-objective simulation-optimization withsimulated annealing and decision analysis – Maintenance scheduling of a fleet of fighteraircraft.Manuscript, 2011.
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References II
S. Sun, F. Zhuge, J. Rosenberg, R. Steiner, G. Rubin, and S. Napel. Learning-enhancedsimulated annealing: Method, evaluation, and application to lung nodule registration.Appl. Intell., 28(1):83-99, 2008.
E. Zitzler, K. Deb, and L. Thiele. Comparison of multiobjective evolutionary algorithms:Empirical results.Evol. Comput., 8(2):173–195, 2000.
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