![Improving µDE
A modified Hill-Descend Operator:
The variables are perturbed one-by-one
For each coordinate i, xnew [i] = xbest[i] − ρ
(ρ exploratory radius)
If xnew does not outperform xbest, a half step in
the opposite direction xnew [i] = xbest[i] + ρ
2 is
performed
If an improvement occurs, xnew is the new starting
point, otherwise ρ is halved](https://image.slidesharecdn.com/microdea-170417152347/85/Micro-Differential-Evolution-with-Extra-Moves-alonf-the-Axes-6-320.jpg)
![Numerical Results
0 1e+6 2e+6 3e+6 4e+6 5e+6
10
15
10
10
10
5
10
0
10
5
10
10
Fitness function call
Fitnessvalue[Logarithmicscale]
µDEA
µDE
MDE pBX
SADE
JADE
Figure : Performance trend for f1 of CEC2010 in 1000 dimensions.](https://image.slidesharecdn.com/microdea-170417152347/85/Micro-Differential-Evolution-with-Extra-Moves-alonf-the-Axes-11-320.jpg)
Micro Differential Evolution with Extra Moves alonf the Axes
- 1. Micro-Differential Evolution with Extra Moves Along the Axes Fabio Caraffini, Ferrante Neri and Ilpo Poikolainen1 De Montfort University United Kingdom 18.04.2013 (SSCI2013, Singapore) 1 University of Jyv¨askyl¨a
- 2. Outline Background: DE vs µDE µDEA: a Micro-Differential Evolution with Extra Moves Along the Axes Numerical Results Conclusions and Future Developments
- 3. Differential Evolution (DE) DE 2 is a population-based algorithm composed by: Mutation (linear combination of individuals) Crossover (EAs like) Selection (SIAs like) In this study we made use of: DE/rand/1/exp 2 Storn and Price (1995)
- 4. DE vs µDE How to tune the population size? large population size: Increases landscape exploration minimising the probability of premature convergence in a local optimum Is preferable in noisy problems Improves algorithm stability “micro” population size: Has been widely used in various engineering applications3 Has a modest memory footprint Converges quickly 3 S. Rahnamayan and H. R. Tizhoosh, ”Image thresholding using micro opposition-based differential evolution”
- 5. Improving µDE idea: supply an alternative kind of perturbation to the one performed by the µPopulation Exploration: starting from the best individual, we perform wide movements along the coordinate axes covering the whole search space Exploitation: the perturbation step decreases during the optimisation process
- 6. Improving µDE A modified Hill-Descend Operator: The variables are perturbed one-by-one For each coordinate i, xnew [i] = xbest[i] − ρ (ρ exploratory radius) If xnew does not outperform xbest, a half step in the opposite direction xnew [i] = xbest[i] + ρ 2 is performed If an improvement occurs, xnew is the new starting point, otherwise ρ is halved
- 7. µDEA Parameters setting Population Size = 5 Scale Factor F = 0.7 Exp Xover Inheritance Factor αe = 0.5 Activation Probability η = 0.25 Extra Moves Iterations = 20 Initial Exploratory Radius ρ = 40% of the width of the decision space
- 8. Numerical Results We considered a set of 76 problems: The CEC2005 benchmark 30 dimensions (25 test problems) The BBOB2010 benchmark 100 dimensions (24 test problems) The CEC2008 benchmark 1000 dimensions (7 test problems) The CEC2010 benchmark 1000 dimensions (20 test problems) We compared µDEA against µDE, JADE4, SADE5 and MDE-pBX6by means of average value and standard deviation over 100 runs, the Wilcoxon Rank-Sum test and the Holm-Bonferroni procedure. 4 Zhang, Sanderson. JADE: Adaptive Differential Evolution With Optional External Archive. 5 Qin, Suganthan. Self-adaptive differential evolution algorithm for numerical optimization. 6 Islam, Das, Ghosh, Roy, Suganthan: An Adaptive Differential Evolution Algorithm With Novel Mutation and Crossover Strategies for Global Numerical Optimization.
- 9. Numerical Results Table : Average Fitness ± Standard Deviation and Wilcoxon Rank-Sum Test (reference =µDEA) for µDEA against µDE on CEC2005 in 30 dimensions. µDEA µDE f1 −4.50e + 02 ± 2.18e − 13 −3.98e + 02 ± 5.14e + 02 + f2 −4.50e + 02 ± 2.43e − 12 −4.29e + 02 ± 1.28e + 02 + f3 1.83e + 05 ± 1.05e + 05 1.66e + 07 ± 5.61e + 06 + f4 6.74e + 04 ± 1.60e + 04 7.59e + 02 ± 1.22e + 03 - f5 7.20e + 03 ± 2.22e + 03 9.49e + 03 ± 2.07e + 03 + f6 8.52e + 02 ± 1.03e + 03 2.79e + 07 ± 1.94e + 08 = f7 −1.80e + 02 ± 1.35e − 02 2.99e + 11 ± 1.20e + 12 + f8 −1.20e + 02 ± 4.75e − 03 −1.19e + 02 ± 5.98e − 02 + f9 −1.17e + 02 ± 1.16e + 00 −1.16e + 02 ± 1.78e + 00 = f10 2.62e + 02 ± 2.05e + 01 2.56e + 02 ± 1.89e + 01 - f11 1.18e + 02 ± 3.55e + 00 1.21e + 02 ± 2.50e + 00 + f12 1.49e + 03 ± 2.90e + 03 1.55e + 04 ± 6.55e + 03 + f13 −1.22e + 02 ± 1.45e + 00 −1.27e + 02 ± 1.20e + 00 - f14 −2.86e + 02 ± 2.78e − 01 −2.87e + 02 ± 2.60e − 01 - f15 1.45e + 03 ± 2.89e + 00 1.45e + 03 ± 4.25e + 00 - f16 1.59e + 03 ± 1.56e + 01 1.58e + 03 ± 1.20e + 01 = f17 1.74e + 03 ± 1.81e + 01 1.61e + 03 ± 1.13e + 01 - f18 9.10e + 02 ± 5.26e − 12 9.10e + 02 ± 5.41e − 02 + f19 9.10e + 02 ± 5.82e − 12 9.10e + 02 ± 1.64e − 01 + f20 9.10e + 02 ± 5.61e − 12 9.10e + 02 ± 4.18e − 01 + f21 1.72e + 03 ± 1.09e + 01 1.72e + 03 ± 8.91e + 00 + f22 2.60e + 03 ± 5.80e + 01 2.54e + 03 ± 4.93e + 01 - f23 1.73e + 03 ± 9.39e + 00 1.72e + 03 ± 8.43e + 00 - f24 1.71e + 03 ± 1.44e + 01 1.71e + 03 ± 9.66e + 00 = f 1.88e + 03 ± 3.40e + 02 1.91e + 03 ± 1.37e + 02 =
- 10. Numerical Results Table : Average Fitness ± Standard Deviation and Wilcoxon Rank-Sum Test (reference = µDEA) for µDEA against µDE on CEC2010 in 1000 dimensions. µDEA µDE f1 4.44e − 18 ± 7.20e − 19 4.45e + 07 ± 1.87e + 08 + f2 5.71e + 03 ± 3.66e + 02 5.26e + 02 ± 4.45e + 01 - f3 2.47e − 02 ± 9.79e − 02 2.88e + 00 ± 8.15e − 01 + f4 2.07e + 13 ± 3.99e + 12 3.12e + 13 ± 7.40e + 12 + f5 4.49e + 08 ± 1.16e + 08 6.32e + 08 ± 8.94e + 07 + f6 1.90e + 07 ± 3.01e + 06 2.04e + 07 ± 2.15e + 05 + f7 1.92e + 10 ± 4.99e + 09 1.83e + 10 ± 3.99e + 09 = f8 2.36e + 10 ± 1.22e + 10 2.70e + 11 ± 1.71e + 12 + f9 1.71e + 08 ± 7.24e + 06 4.55e + 08 ± 2.53e + 08 + f10 7.23e + 03 ± 2.88e + 02 6.95e + 03 ± 2.85e + 02 - f11 1.48e + 02 ± 4.56e + 01 2.08e + 02 ± 2.18e + 00 + f12 8.39e + 04 ± 1.48e + 05 3.79e + 05 ± 2.10e + 04 + f13 2.26e + 05 ± 9.13e + 04 9.57e + 07 ± 4.98e + 08 = f14 1.33e + 08 ± 2.53e + 08 9.38e + 08 ± 4.51e + 07 + f15 7.31e + 03 ± 3.08e + 02 1.37e + 04 ± 3.75e + 02 + f16 2.23e + 02 ± 1.08e + 02 4.11e + 02 ± 2.25e + 00 + f17 1.14e + 05 ± 2.39e + 05 8.07e + 05 ± 2.43e + 04 + f18 3.57e + 04 ± 1.21e + 04 3.71e + 08 ± 2.08e + 09 + f19 5.47e + 05 ± 3.43e + 04 2.82e + 05 ± 2.68e + 04 - f20 1.49e + 04 ± 1.10e + 03 1.99e + 08 ± 7.66e + 08 +
- 11. Numerical Results 0 1e+6 2e+6 3e+6 4e+6 5e+6 10 15 10 10 10 5 10 0 10 5 10 10 Fitness function call Fitnessvalue[Logarithmicscale] µDEA µDE MDE pBX SADE JADE Figure : Performance trend for f1 of CEC2010 in 1000 dimensions.
- 12. Numerical Results Table : Holm-Bonferroni test on the Fitness, reference algrithm = µDEA (Rank = 3.24e+00 ) j Optimizer Rank zj pj δ/j Hypothesis 1 SADE 3.68e+00 2.14e+00 9.84e-01 5.00e-02 Accepted 2 MDE-pBX 3.08e+00 -7.54e-01 2.25e-01 2.50e-02 Accepted 3 µDE 2.53e+00 -3.39e+00 3.46e-04 1.67e-02 Rejected 4 JADE 2.46e+00 -3.71e+00 1.05e-04 1.25e-02 Rejected
- 13. Conclusions and Future Developments µDEA improves upon µDE and is competitive against complex state-of-the-art DE variants µDEA shows excellent performances over large scale problems If well coordinated, even two simple components can provide a high performance over a huge set of different problems Its modest memory footprint and computational overhead makes µDEA suitable for real-time embedded applications (robotic, system control, wireless sensor networks...). It will be interesting to apply this algorithm to real world applications in the future.
- 14. Thank you for your attention! Any questions?