Biased Mutation for Evolution Strategies - Self-Adaptive Heuristics for Evolutionary Computation - page 64

Table 4.6. Experimental population ratios analysis of the BMO on the rosenbrock
function. The (15,100)-BMO achieves the best approximation quality. No improvement
canbegatheredwithanincreaseofpopulationsizes.

(5,100) (15,100) (15,300) (15,500) (50,100) (50,300) (50,500)
best 6.61E-11 7.97E-10 9.55E-07 1.61E-05 0.00179 0.00169 0.00635
median 1.36E-09 1.22E-08 3.69E-06 0.00014 0.00352 0.00266 0.01373

worst 2.97E-05 4.72E-08 0.00011 0.00088 4.02246 0.11635 1.53753

mean 1.24E-06 1.47E-08 9.19E-06 0.00019 0.32561 0.00773 0.07479

dev 5.93E-06 1.14E-08 2.30E-05 0.00019 1.10751 0.02271 0.30477

Table 4.7. Experimental results of the DMO with various population ratios on the
rosenbrock function. The (5,100)- and the (15,100)-DMO show better approximation
capabilities than the other variants.

(5,100) (15,100) (15,300) (15,500) (50,100) (50,300) (50,500)

best 1.43E-05 1.79E-05 0.00050 0.00416 0.02784 0.03450 0.08350

median 3.20E-05 2.78E-05 0.00085 0.00634 0.04307 0.04294 0.09682

worst 6.10E-05 0.00027 4.06074 0.92337 332.30653 103.10865 23.87899

mean 3.49E-05 4.14E-05 0.45365 0.04305 14.26908 4.87227 1.86968

dev 1.17E-05 5.15E-05 1.16190 0.18340 66.29393 20.68901 5.66927

We can summarize that a relatively high selection pressure can be recom-
mended to achieve high quality results if many runs of the algorithm are pos-
sible. The probability for outstanding results among many runs is high. In this
case the best solution can be saved. If a high reliability is necessary, we recom-
mend the standard setting (15,100). Whether these results are also feasible for
the DMO, the following table 4.7 tries to answer. The results roughly confirm
the outcome of the previous experiments: the (5,100)- and the (15,100)-BMO
achieve the best approximation of the optimum. But no outstanding best result
can be obtained with the (5,100)-BMO. A decrease of the selection pressure or
an increase of μ results in a deterioration as we can see from the experiments of
the settings (15,300) to (15,500). On the other side an increase of λ is an increase
of the selection pressure and improves the results.

Noisy Fitness Functions

Noise in the fitness is a frequent condition of real-world optimization problems.
Here, we test the biased mutation on the rosenbrock function with noise. The
obtained results can be found in table 4.8. The experiments show that the bi-
ased mutation is robust against noise on the rosenbrock function. We observe
a significant better behavior of the biased mutation variants in comparison to
the standard ES, see the Wilcoxon rank-sum test below. Hence, the BMO can
be recommended for fitness landscapes where noise is present. The same holds
for the DMO. Although the mean of the cBMO is worse, the median is better