Mitchell, T. J. (2010) An exploration of evolutionary computation ...

each individual ( ) and adapted with the step-size parameters . For further reading and
implementation details of this generalised self-adaptation mechanism, see (Schwefel,
1981).
When the endogenous strategy parameters ( , and ) are adapted along with the object-
variables, optimisation takes place simultaneously in both the object and strategy
parameter spaces. This process ensures that high performing solutions are selected for
reproduction along with their corresponding strategy parameters, which may go on to yield
even stronger solutions throughout subsequent generations.
Strategy Parameter Adaptation
By selecting optimal values for the strategy parameters controlling the mutation strength,
the maximum rate of progress can be maintained. The problem then arises as to how the
strategy parameters may be continuously adapted throughout the course of evolution. For
the ES there are two standard approaches for step-size adaptation: the rule and self-
adaptation.
The Rule
By studying the dynamics of the ES when applied to two differing objective
functions, Rechenberg observed that the maximum rate of progress corresponds to a
particular value for the probability of a successful mutation (Rechenberg, 1973, as cited in
Beyer and Schwefel, 2002). As the mutation step-size tends to zero, the probability of
success becomes very high; conversely, as the step-size tends to infinity, the probability of
success becomes very low. In order to maintain an optimal rate of progress, the step-size
parameter should be adjusted to maintain a probability of success within these two
extremes; a range that has become known as the evolution window. This observation led to
the derivation of a general rule for the probability of success: mutation step-size adaptation
by the rule. Successful mutations are measured over several generations (often equal
to the dimensionality of the problem) and if the probability of a successful mutation is
found to be below , the mutation step-size is decreased. A recommended factor for the
multiplicative/multiplicative inverse adaptation of the step-size parameter by the
rule is 0.85 (Schwefel, 1995).
However, there are certain limitations that apply when adapting the mutation step-size
using the rule:
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