Biased Mutation for Evolution Strategies - Self-Adaptive Heuristics for Evolutionary Computation - page 68

Table 4.11. Experimental population ratio analysis of the BMO on the function sharp
ridge. A high selection pressure with a small number of parents like the (5,100)-BMO
achieves the fastest approximation along the ridge.

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

best -2.05E+304 -1.93E+234 -8.44E+107 -1.56E+70 -1.83E+122 -1.28E+74 -4.60E+51

median -8.47E+291 -8.49E+229 -2.17E+101 -3.10E+63 -4.26E+116 -1.08E+71 -8.99E+48

worst -1.07E+244 -6.22E+206 -1.72E+86 -1.03E+54 -3.16E+108 -4.55E+66 -4.00E+45

mean -8.30E+302 -1.09E+233 -3.55E+106 -7.18E+68 -1.31E+121 -1.31E+73 -3.85E+50

Selection Pressure and Population Ratios

The question of the influence of the population ratios answers table 4.11, where
we test various population ratios of the BMO on the sharp ridge. It turns out that
an increase of the selection pressure results in an increase of optimization speed.
The (5,100)-ES achieves the best results. With this setting the optimization
speed is the fastest.
A decrease of the number of offspring λ keeping μ constant leads to an im-
provement. A decrease of μ keeping λ constant also results in a better approx-
imation quality. Consequently, diversity in the population does not seem to be
the most important aspect optimizing the ridge functions, as the fastest approx-
imation can be achieved with small population sizes.

4.5.3 Handling Constraints with Biased Mutation

For many optimization problems the search space is constrained due to a variety
of practical conditions. For EAs with a self-adaptive step size mechanism like
ES, it is not easy to find an optimum, which lies on the boundary of the feasible
search space due to premature step size reduction. This results in premature
fitness stagnation before approximating the optimum, also see [75] and chapter
7. For big step sizes the constrained search space cuts off a big area of success
to reproduce better mutations than the parent. So the step sizes reduce self-
adaptively before reaching the area of the optimum. Our experiments revealed
the benefits of biased mutation for constraint handling. The feature of unbi-
asedness of mutation operators postulated by Beyer [16] is not advantageous for
constrained search domains. The experiments reveal that the biased mutation,
i.e. the BMO, the BMO variants and the DMO exhibit useful features, which
prevent premature step size reduction while using the simple constraint handling
technique death penalty 3 . Intuitively speaking, the reason for this behavior is the
following: the distribution function is shifted along the constraint boundary so
that the success area is not cut off.

3 Death penalty discards infeasible solutions and produces new individuals until the
whole offspring population is feasible.