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are again a bit worse. On rosenbrock with noise intermediate recombination is as
good as iSAR, SAR is almost comparable. The variant dSAR produces outliers,
which deteriorate the results. On the ridge functions the situation changes con-
siderably. Dominant crossover becomes the best recombination operator, while
dSAR achieves a similar approximation quality. The latter selects dominant
crossover self-adaptively. Intermediate recombination, SAR and iSAR climb the
ridge comparatively slowly. On the constrained problem 2.40 the choice of the re-
combination operator does not have a major influence. No variant is able to find
the optimum. To summarize, intermediate recombination, SAR and iSAR show
a similar behavior on the test functions. The next statistical test shows that no
significant difference between SAR and intermediate can be reported. Hence, we
cannot recommend any variant. The discrete switch between intermediate and
dominant crossover dSAR is comparatively weak. Dominant recombination and
dSAR are appropriate for ridge functions.
Wilcoxon Rank-Sum Test
sphere rastrigin roseNoise sharp ridge 2.40
p w , int vs. SAR 0.6837 0.3879
0.2072
0.1683
0.5737
A Wilcoxon rank-sum test is conducted to examine statistical differences between
intermediate recombination and SAR. The high p w -values ( p w > 0 . 05) show
that there is no statistical indication for the superiority of neither intermediate
recombination nor SAR. We observe relatively small values on rosenbrock with
noise ( p w
0 . 1683). But this does not
allow a significant statement, neither about a superiority of SAR on rosenbrock
with noise nor of intermediate recombination on the sharp ridge.
0 . 2072) and on the sharp ridge ( p w
6.5
Crossover Point Optimization
As the self-adaptation of crossover points does not lead to any significant im-
provement, we propose to find optimal crossover points with an optimization
step. This method is supposed to show the benefit of well chosen crossover points.
In the following heuristic in each step k randomly chosen crossover settings are
tested and the best is used for the final recombination procedure. We call this
crossover modification Xover opt and show the results in table 6.5. In these ex-
periments we use the settings of the previous section. Xover opt has been tested
on the sphere function, rastrigin, rosenbrock with noise, sharp ridge and the
constrained problem 2.40. In these experiments, k = 10 randomly generated ν
are tested. Xover opt searches within the convex hull 3 of the parental individuals.
On rastrigin, on rosenbrock with noise and on problem 2.40, no significant
improvement can be achieved, although k = 10 different recombination settings
have been tested. Hence, the influence of the strategy parameter vector ν is
3 The convex hull for a set of points X is the minimal convex set containing X .
 
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