Biased Mutation for Evolution Strategies - Self-Adaptive Heuristics for Evolutionary Computation - page 69

Experimental settings
Population model

(15,100)

1
√

1
√ 2 √ N

Mutation types ES, BMO, cBMO, DMO, γ =0 . 1, τ 0 = ,τ 1 =

N

Crossover type intermediate, ρ =2

Selection type comma

Initialization [-100,100]

Constraint Handling death penalty, BMO

Termination 1000 generations (g04), 2000 generations (2.40)

Runs 25

Comparison of Variants

At first, we compare the algorithms on the constrained function 2.40, see
table 4.12. It turns out that the BMO and the DMO are both able to approxi-
mate the optimum, while the standard ES with death penalty fails. The cBMO
is worse than BMO, DMO and even almost 4 ES. The measured standard devia-
tion of the DMO on 2.40 is smaller than the accuracy of our data structure allows.
Hence, the DMO allows the best approximation of the optimum.

Table 4.12. Experimental results of the BMO variants on the constrained function
2.40. BMO and DMO are able to find the optimum in every run while the ES with
death penalty fails.

ES BMO cBMO DMO

best -4985.69 -5000.0 -4982.02 -5000.0

median -4948.04 -5000.0 -4883.70 -5000.0

worst -4676.76 -4999.99 -4683.89 -5000.0

mean -4897.72 -5000.0 -4865.01 -5000.0

dev 96.69 0.0 91.99 0.0

Figure 4.8 compares the algorithms on problem 2.40 considering typical runs
on a logarithmic scale. The standard ES with DP early suffers from premature
step size reduction while BMO and DMO show logarithmically linear approxi-
mation of the optimum. Furthermore, the figure shows that the DMO shows a
faster convergence than the BMO.

Wilcoxon Rank-Sum Test

BMO cBMO DMO
vs. ES 1.421E-09 0.09519 1.421E-09

The question about the significance of the experiments answers a Wilcoxon rank-
sum test. It reveals that the BMO and the DMO are both better than the ES
with DP (1.4E-09 << 0.05), but the cBMO is not (0.09 > 0.05). Obviously, a cube

4 See next Wilcoxon rank-sum test, p w =0 . 09 > 0 . 05.