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IntroductionProjection-based Real Coded Genetic Algorithm
Experimental ProcedureExperimental Results
ConclusionThank you
Benchmarking Projection-Based Real CodedGenetic Algorithm on BBOB-2013 Noiseless
Function Testbed
Babatunde Sawyerr1 Aderemi Adewumi2 Montaz Ali3
1University of Lagos, Lagos, Nigeria2University of KwaZulu-Natal, Durban, South Africa
3University of The Witwatersrand, Johannesburg, South Africa
Workshop on Black Box Optimization Benchmarking, 2013
Sawyerr, Adewumi & Ali Benchmarking PRCGA on BBOB-2013 Noiseless Testbed
IntroductionProjection-based Real Coded Genetic Algorithm
Experimental ProcedureExperimental Results
ConclusionThank you
Outline
1 IntroductionProblem StatementGenetic Algorithms
2 Projection-based Real Coded Genetic AlgorithmProjectionThe PRCGA Algorithm
3 Experimental ProcedureExperimental Settings
4 Experimental ResultsEmpirical ResultsDiscussion
5 Thank you
Sawyerr, Adewumi & Ali Benchmarking PRCGA on BBOB-2013 Noiseless Testbed
IntroductionProjection-based Real Coded Genetic Algorithm
Experimental ProcedureExperimental Results
ConclusionThank you
Problem StatementGenetic Algorithms
Outline
1 IntroductionProblem StatementGenetic Algorithms
2 Projection-based Real Coded Genetic AlgorithmProjectionThe PRCGA Algorithm
3 Experimental ProcedureExperimental Settings
4 Experimental ResultsEmpirical ResultsDiscussion
5 Thank you
Sawyerr, Adewumi & Ali Benchmarking PRCGA on BBOB-2013 Noiseless Testbed
IntroductionProjection-based Real Coded Genetic Algorithm
Experimental ProcedureExperimental Results
ConclusionThank you
Problem StatementGenetic Algorithms
The Global Optimization ProblemReal Parameter Optimization
The task is to minimize an objective function f . Givenf : S → ℜ where S ⊂ ℜn, find x∗ ∈ S for which,
f (x∗) ≤ f (x), ∀x ∈ S. (1)
Black Box approach:gradients are not known or not useful.problem domain are rugged and ill-conditioned.
Goal:To find the global optimum, x∗ quickly.With the least search cost (function evaluations).
Sawyerr, Adewumi & Ali Benchmarking PRCGA on BBOB-2013 Noiseless Testbed
IntroductionProjection-based Real Coded Genetic Algorithm
Experimental ProcedureExperimental Results
ConclusionThank you
Problem StatementGenetic Algorithms
Outline
1 IntroductionProblem StatementGenetic Algorithms
2 Projection-based Real Coded Genetic AlgorithmProjectionThe PRCGA Algorithm
3 Experimental ProcedureExperimental Settings
4 Experimental ResultsEmpirical ResultsDiscussion
5 Thank you
Sawyerr, Adewumi & Ali Benchmarking PRCGA on BBOB-2013 Noiseless Testbed
IntroductionProjection-based Real Coded Genetic Algorithm
Experimental ProcedureExperimental Results
ConclusionThank you
Problem StatementGenetic Algorithms
Genetic Algorithms
Developed by John Holland in 1975.
Goal: Develop robust and adaptive systems.
Solutions are represented internally as genetic encoding ofpoints.Reproduction of offspring via:
mutation,recombination.
Selection methods: initially Fitness-proportional method.
Model: Generational or Steady state.
Sawyerr, Adewumi & Ali Benchmarking PRCGA on BBOB-2013 Noiseless Testbed
IntroductionProjection-based Real Coded Genetic Algorithm
Experimental ProcedureExperimental Results
ConclusionThank you
Problem StatementGenetic Algorithms
Real Coded Genetic Algorithms
Real valued representation are used as genetic encodingsof points.
They are better adapted to numerical optimization ofcontinuous problems.
They can also be easily hybridized with other searchmethods.
Sawyerr, Adewumi & Ali Benchmarking PRCGA on BBOB-2013 Noiseless Testbed
IntroductionProjection-based Real Coded Genetic Algorithm
Experimental ProcedureExperimental Results
ConclusionThank you
ProjectionThe PRCGA Algorithm
Outline
1 IntroductionProblem StatementGenetic Algorithms
2 Projection-based Real Coded Genetic AlgorithmProjectionThe PRCGA Algorithm
3 Experimental ProcedureExperimental Settings
4 Experimental ResultsEmpirical ResultsDiscussion
5 Thank you
Sawyerr, Adewumi & Ali Benchmarking PRCGA on BBOB-2013 Noiseless Testbed
IntroductionProjection-based Real Coded Genetic Algorithm
Experimental ProcedureExperimental Results
ConclusionThank you
ProjectionThe PRCGA Algorithm
Orthogonal Projection of a vector x on a vector y
For any two n dimensional vectors, the projection of a vector xon another vector y generates a vector, defined by:
y =xT yyT y
y =xT y
‖y‖2 y =
(
‖x‖ cos(θ)‖y‖
y)
. (2)
Note that the projected vector y (the offspring) will be in thesame direction as y unless π
2 < θ < 3π2 in which case the angle,
θ, between the two vectors is such that cos(θ) < 0. As a result,the projected vector is in the opposite direction (the reflection ofy about the origin).
Sawyerr, Adewumi & Ali Benchmarking PRCGA on BBOB-2013 Noiseless Testbed
IntroductionProjection-based Real Coded Genetic Algorithm
Experimental ProcedureExperimental Results
ConclusionThank you
ProjectionThe PRCGA Algorithm
Orthogonal Projection of a vector x on a vector y
Figure: Projection of vector x on vector ySawyerr, Adewumi & Ali Benchmarking PRCGA on BBOB-2013 Noiseless Testbed
IntroductionProjection-based Real Coded Genetic Algorithm
Experimental ProcedureExperimental Results
ConclusionThank you
ProjectionThe PRCGA Algorithm
Outline
1 IntroductionProblem StatementGenetic Algorithms
2 Projection-based Real Coded Genetic AlgorithmProjectionThe PRCGA Algorithm
3 Experimental ProcedureExperimental Settings
4 Experimental ResultsEmpirical ResultsDiscussion
5 Thank you
Sawyerr, Adewumi & Ali Benchmarking PRCGA on BBOB-2013 Noiseless Testbed
IntroductionProjection-based Real Coded Genetic Algorithm
Experimental ProcedureExperimental Results
ConclusionThank you
ProjectionThe PRCGA Algorithm
The PRCGA Algorithm
PRCGA was first introduced as RCGA-P in [6, 7].The incorporated projection operator showed promisingexploratory search capability in some search problems.PRCGA is an enhanced version of RCGA-P.
Inputs
Fitness function f .
Parameters.
Outputs
The Best solution xbest .
Fitnss value of xbest , f (xbest).
Sawyerr, Adewumi & Ali Benchmarking PRCGA on BBOB-2013 Noiseless Testbed
IntroductionProjection-based Real Coded Genetic Algorithm
Experimental ProcedureExperimental Results
ConclusionThank you
ProjectionThe PRCGA Algorithm
The PRCGA Algorithm
1 Initialize Pt=0,Pt ={
x1,t , x2,t , . . . , xN,t}
from S2 f (xi,t) = evaluate(Pt ), {1 ≤ i ≤ N}3 While not stopping condition, do steps 4 - 124 ζt = σ(f (Pt)), if ζt ≤ ǫ do step 5 else step 65 Pt = perturb(Pt )6 Pt = tournamentSelection(Pt )7 Ct = blend-αCrossover(Pt , pc)8 Mt = non-uniformMutation(Ct , pm)9 Φt = projection(Mt )
10 f (xi,t) = evaluate(Φt )11 Pt+1 = replace(Pt ,Φt )12 t = t + 113 end while
Sawyerr, Adewumi & Ali Benchmarking PRCGA on BBOB-2013 Noiseless Testbed
IntroductionProjection-based Real Coded Genetic Algorithm
Experimental ProcedureExperimental Results
ConclusionThank you
Experimental Settings
Outline
1 IntroductionProblem StatementGenetic Algorithms
2 Projection-based Real Coded Genetic AlgorithmProjectionThe PRCGA Algorithm
3 Experimental ProcedureExperimental Settings
4 Experimental ResultsEmpirical ResultsDiscussion
5 Thank you
Sawyerr, Adewumi & Ali Benchmarking PRCGA on BBOB-2013 Noiseless Testbed
IntroductionProjection-based Real Coded Genetic Algorithm
Experimental ProcedureExperimental Results
ConclusionThank you
Experimental Settings
Computer System and Software
Computer System Configuration
HP Probook 6545b with AMD Turion(tm) II Ultra Dual-Coremobile M620 CPU processor.
CPU Speed: 2.5GHz.
RAM: 2.75GB
Software
Microsoft Windows 7 Professional service pack 1.
MATLAB 7.10 (R2010a).
COmparing Continuous Optimisers (COCO) software.
Post-Processing Script in Python
Sawyerr, Adewumi & Ali Benchmarking PRCGA on BBOB-2013 Noiseless Testbed
IntroductionProjection-based Real Coded Genetic Algorithm
Experimental ProcedureExperimental Results
ConclusionThank you
Experimental Settings
Experimental setup
The experimental setup was carried out according to [3] onthe benchmark functions provided in [2, 4].Two independent restart strategies were employed
Checks for stagnation [1].Maximum number of generations reached without ftarget .
For each restart strategy, the genetic run is initiated with aninitial population P0 which is ∼ Unif ([−4, 4]D).
Sawyerr, Adewumi & Ali Benchmarking PRCGA on BBOB-2013 Noiseless Testbed
IntroductionProjection-based Real Coded Genetic Algorithm
Experimental ProcedureExperimental Results
ConclusionThank you
Experimental Settings
Parameter Settings
Parameters
Population Size = min(100, 100 × D),where D = dimension.
Maximum Number of Evaluation = 105 × D.
Tournament size = 3.
Crossover rate pc = 0.8.
Mutation rate pm = 0.15.
Non-uniformity factor for Mutation β = 15.
Crafting effort CrE = 0.
Sawyerr, Adewumi & Ali Benchmarking PRCGA on BBOB-2013 Noiseless Testbed
IntroductionProjection-based Real Coded Genetic Algorithm
Experimental ProcedureExperimental Results
ConclusionThank you
Experimental Settings
CPU Timing Experiment
The CPU timing experiment was conducted using the sameindependent restart strategies on the function f8 for a durationof 30 seconds.
Time per function evaluation
Dimension 2 3 5 10 20 40Time (×10−5) 7.1 7.5 6.9 6.9 7.1 8.0
Sawyerr, Adewumi & Ali Benchmarking PRCGA on BBOB-2013 Noiseless Testbed
IntroductionProjection-based Real Coded Genetic Algorithm
Experimental ProcedureExperimental Results
ConclusionThank you
Empirical ResultsDiscussion
Outline
1 IntroductionProblem StatementGenetic Algorithms
2 Projection-based Real Coded Genetic AlgorithmProjectionThe PRCGA Algorithm
3 Experimental ProcedureExperimental Settings
4 Experimental ResultsEmpirical ResultsDiscussion
5 Thank you
Sawyerr, Adewumi & Ali Benchmarking PRCGA on BBOB-2013 Noiseless Testbed
IntroductionProjection-based Real Coded Genetic Algorithm
Experimental ProcedureExperimental Results
ConclusionThank you
Empirical ResultsDiscussion
Ellipsoid separable
2 3 5 10 20 40
0
1
2
3
4
5
410
42 Ellipsoid separable
Sawyerr, Adewumi & Ali Benchmarking PRCGA on BBOB-2013 Noiseless Testbed
IntroductionProjection-based Real Coded Genetic Algorithm
Experimental ProcedureExperimental Results
ConclusionThank you
Empirical ResultsDiscussion
Rastrigin separable
2 3 5 10 20 40
0
1
2
3
4
59
1253 Rastrigin separable
Sawyerr, Adewumi & Ali Benchmarking PRCGA on BBOB-2013 Noiseless Testbed
IntroductionProjection-based Real Coded Genetic Algorithm
Experimental ProcedureExperimental Results
ConclusionThank you
Empirical ResultsDiscussion
Skew Rastrigin-Bueche separable
2 3 5 10 20 40
0
1
2
3
4
58
1214
114 Skew Rastrigin-Bueche separ
Sawyerr, Adewumi & Ali Benchmarking PRCGA on BBOB-2013 Noiseless Testbed
IntroductionProjection-based Real Coded Genetic Algorithm
Experimental ProcedureExperimental Results
ConclusionThank you
Empirical ResultsDiscussion
Outline
1 IntroductionProblem StatementGenetic Algorithms
2 Projection-based Real Coded Genetic AlgorithmProjectionThe PRCGA Algorithm
3 Experimental ProcedureExperimental Settings
4 Experimental ResultsEmpirical ResultsDiscussion
5 Thank you
Sawyerr, Adewumi & Ali Benchmarking PRCGA on BBOB-2013 Noiseless Testbed
IntroductionProjection-based Real Coded Genetic Algorithm
Experimental ProcedureExperimental Results
ConclusionThank you
Empirical ResultsDiscussion
Discussion
Separable Functions
PRCGA performed well on separable functions f1 − f4.
PRCGA also solved Gallagher’s Gaussian 101-me PeaksFunction f21, a multi-modal function with weak globalstructure.
PRCGA showed some encouraging performance in solvingproblems f6 − f7 in dimensions 2 − 10.
Functions with high conditioning and unimodal
Functions f10 − f14 prove to be difficult for PRCGA to solveto the required level of accuracy.
Sawyerr, Adewumi & Ali Benchmarking PRCGA on BBOB-2013 Noiseless Testbed
IntroductionProjection-based Real Coded Genetic Algorithm
Experimental ProcedureExperimental Results
ConclusionThank you
Empirical ResultsDiscussion
Discussion
Comparison of PRCGA with Previous GAs
DBRCGA [1] outperformed PRCGA.
PRCGA performed better than the RCGA in [8].
PRCGA performed better than the simpleGA in [5].
Sawyerr, Adewumi & Ali Benchmarking PRCGA on BBOB-2013 Noiseless Testbed
IntroductionProjection-based Real Coded Genetic Algorithm
Experimental ProcedureExperimental Results
ConclusionThank you
Conclusion
The benchmarking of PRCGA on noiseless BBOB functiontestbed shows the strengths and weaknesses of thealgorithm.
The performance of PRCGA shows that in its current formit cannot compete with state-of-the-art evolutionaryalgorithms.
Sawyerr, Adewumi & Ali Benchmarking PRCGA on BBOB-2013 Noiseless Testbed
IntroductionProjection-based Real Coded Genetic Algorithm
Experimental ProcedureExperimental Results
ConclusionThank you
Thank You!!!
Sawyerr, Adewumi & Ali Benchmarking PRCGA on BBOB-2013 Noiseless Testbed
Appendix For Further Reading
For Further Reading I
Y.-C. Chuang and C.-T. Chen.Black-box optimization benchmarking for noiseless functiontestbed using a direction-based rcga.In GECCO (Companion), pages 167–174, 2012.
S. Finck, N. Hansen, R. Ros, and A. Auger.Real-parameter black-box optimization benchmarking2009: Presentation of the noiseless functions.Technical Report 2009/20, Research Center PPE, 2009.Updated February 2010.
Sawyerr, Adewumi & Ali Benchmarking PRCGA on BBOB-2013 Noiseless Testbed
Appendix For Further Reading
For Further Reading II
N. Hansen, A. Auger, S. Finck, and R. Ros.Real-parameter black-box optimization benchmarking2012: Experimental setup.Technical report, INRIA, 2012.
N. Hansen, S. Finck, R. Ros, and A. Auger.Real-parameter black-box optimization benchmarking2009: Noiseless functions definitions.Technical Report RR-6829, INRIA, 2009.Updated February 2010.
Sawyerr, Adewumi & Ali Benchmarking PRCGA on BBOB-2013 Noiseless Testbed
Appendix For Further Reading
For Further Reading III
M. Nicolau.Application of a simple binary genetic algorithm to anoiseless testbed benchmark.In GECCO (Companion), pages 2473–2478, 2009.
B. A. Sawyerr.Hybrid real coded genetic algorithms with pattern searchand projection.PhD thesis, University of Lagos, Lagos, Nigeria, 2010.
Sawyerr, Adewumi & Ali Benchmarking PRCGA on BBOB-2013 Noiseless Testbed
Appendix For Further Reading
For Further Reading IV
B. A. Sawyerr, M. M. Ali, and A. O. Adewumi.A comparative study of some real coded genetic algorithmsfor unconstrained global optimization.Optimization Methods and Software, 26(6):945–970, 2011.
T.-D. Tran and G.-G. Jin.Real-coded genetic algorithm benchmarked on noiselessblack-box optimization testbed.In GECCO (Companion), pages 1731–1738, 2010.
Sawyerr, Adewumi & Ali Benchmarking PRCGA on BBOB-2013 Noiseless Testbed