![](bg8.png)
with
G (p)
k
=
∑
j∈Ω
p
j
· M
b⊕k, j⊕k
(51)
and
b = argmax{ f
i
: i ∈ Ω∧ p
i
> 0} . (52)
Future research will focus on the analytical derivation
of these weights ρ
ω
i
,ω
j
and the determination of the
minimum cost tributary T
ι
leading to the steady state
distribution ω
ι
of the simple genetic algorithm with
α-selection.
6 CONCLUSIONS
The intrinsic system model for the simple genetic al-
gorithm with α-selection simplifies the analysis of the
dynamical system model of genetic algorithms. It
is defined by the mixing matrix M and enables the
derivation of the unique fixed point ω. The simpli-
fications are gained because the fitness function f is
hidden from the mathematical formulation by making
use of the α-individual b. Since b enters the dynam-
ical system model via a permutation σ
b
according to
σ
b
· M
∗
· σ
b
the intrinsic system model can be formu-
lated with the help of the matrix M
∗
.
The intrinsic system model provides a means to
analyze the genetic algorithm’s exploitation and ex-
ploration of the search space Ω irrespective of the
fitness function f. This model is compatible with
the equivalence relation imposed by schemata as
shown in (Neubauer, 2008a) by explicitly deriving
the coarse-grained system model for a given schemata
family. Experimental results for the simple genetic
algorithm with α-selection, uniform crossover and
bitwise mutation presented in this paper show close
agreement to the theoretical predictions with respect
to the rapid convergence of the permuted population
vector σ
b
p to the unique fixed point ω obtained from
the intrinsic system model.
It is further conjectured that the structure of the
dynamical system model of the simple genetic algo-
rithm with α-selection and its intrinsic system model
simplify the determination of the steady state distri-
bution ω
ι
based on the fixed point graph and the min-
imum cost tributary T
ι
. The analysis of the fixed
point graph and the analytical derivation of its weights
ρ
ω
i
,ω
j
will be the focus of future research.
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