Biased Mutation for Evolution Strategies - Self-Adaptive Heuristics for Evolutionary Computation - page 60

Table 4.1. Experimental results of the BMO variants on the function sphere. The
BMO achieves a better mean than the variants sBMO, cBMO and DMO. Its results
are comparable to the results of the ES. But the unimodal monotone function does not
require the bias. This is why the unbiased ES with only one step size (sES) achieves a
much better accuracy.

ES BMO sBMO cBMO DMO

best 2.36E-150 5.83E-149 2.12E-152 5.41E-145 1.80E-146

median 3.65E-146 3.82E-146 4.89E-145 2.75E-141 4.50E-144

worst 3.60E-141 7.51E-143 9.22E-138 5.25E-138 5.89E-140

mean 2.07E-142 5.33E-144 3.95E-139 2.42E-139 2.49E-141

dev 1.07E-146 9.2699E-144 4.6107E-138 7.8575E-143 7.6016E-144

Table 4.2. Experimental results of the BMO variants on the function double sum. ES,
cBMO and DMO show the same approximation quality.

ES BMO cBMO DMO

best 3.57E-47 1.61E-38 8.06E-40 5.15E-45

median 4.94E-39 1.71E-30 1.54E-36 6.59E-41

worst 8.55E-36 1.53E-24 7.68E-34 1.58E-34

mean 7.96E-37 1.02E-25 6.34E-35 6.39E-36

dev 2.33E-36 3.588E-25 1.68E-34 3.16E-35

In table 4.2 the outcome of the experiments on the double sum function is
presented. It turns out that the BMO achieved slightly worse results than the
standard ES, cBMO and DMO. But the approximation of the optimum of the
cBMO and the DMO are as good as the ES. As no considerable improvements on
the unimodal functions could have been observed we now test the meta heuristics
on multimodal functions.
First, we compare the behavior of the BMO variants on the rosenbrock func-
tion. Table 4.3 and figure 4.6 show the results we accomplished. It turns out that
the BMO and the variant cBMO as well as the DMO deliver better results than
the ES. All parameters best , median , worst , mean and dev are smaller for the

Table 4.3. Experimental results of the BMO variants on the function rosenbrock.
BMO and its variants show much better results than the standard ES mutation.

ES BMO sBMO cBMO DMO

best 0.00027 7.97E-10 0.02016 7.83E-11 1.79E-05

median 0.02734 1.22E-08 0.02435 3.45E-08 2.78E-05

worst 0.19001 4.72E-08 0.98524 6.03E-05 0.00027

mean 0.03412 1.47E-08 0,07384 2.48E-06 4.14E-05

dev 0.03450 1.14E-08 0.19425 1.20E-05 5.15E-05