- μDEA is a micro-differential evolution algorithm with extra moves along the axes to improve exploration. It supplements the perturbation performed by a small population. - It was tested on 76 benchmark problems up to 1000 dimensions and compared to μDE, JADE, SADE, and MDE-pBX. - Results showed μDEA performed equal to or better than the other algorithms on most problems, as evidenced by average fitness values and Wilcoxon rank-sum tests. Its simplicity and low computational overhead make it suitable for real-time applications.

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?