Self-Adaptive Crossover - Self-Adaptive Heuristics for Evolutionary Computation - page 94

As the components of Σ are usually integers, i.e. discrete crossover points,
rounding procedures discretize Λ i afterwards.
For our experiments we used the best inheritance scheme. In our analysis no
better results could be achieved with the other two variants.

Strategy Variable Mutation

Self-adaptation is only possible if the strategy variables are modified by the ge-
netic operators. The mutation of the crossover points depends on the underlying
representation. If encoded binary we propose to use classic bit-flip mutation re-
sulting in a geometric distribution. For continuous search domains we propose
to use Gaussian mutation, i.e. the addition of a random Gaussian distributed
value. We use meta-EP mutation with step size σ

Λ = Λ + σ (0 , 1) with Λ ∈ ·N [1 ,l − 1] . (6.3)

The step size σ has to be rounded and adjusted according to the length of the
representation. As an alternative log-normal mutation may be used:

Λ = Λ e σ·N (0 , 1) with Λ ∈ · [1 ,l − 1] (6.4)

Log-normal mutation is used for the step size mutation of ES, see chapter 4.

6.2 Self-Adaptive n-Point Crossover

The idea of self-adaptive n-point crossover is to divide the genome of the parents
into n+1 parts and combine them with the aim of identifying useful building
blocks. If they exist, it will make sense to set the crossover points to the borders
of the blocks in order to combine them, at least not to destroy the blocks. In the
following we introduce the self-adaptive operators SA-1-point, SA-n-point and
self-adaptive multi parent genetic algorithms (MPGAs).

6.2.1 SA-1-Point

The variant 1-point crossover works as follows: A crossover point Λ is chosen
randomly with uniform distribution. The left part of the bit-string from the first
parent is combined with the right part of the second parent and vice versa for a
second solution. For the self-adaptive extension of 1-point crossover (SA-1-point)
the strategy part of the chromosome is extended by an integer value Σ =( Λ )
with Λ ∈ [1 ,l − 1] determining the crossover point. Given two parents

p 1 =( p 1 ,...,p l ,Λ 1 )

(6.5)

and

p 2 =( p 1 ,...,p l ,Λ 2 )

(6.6)