Biased Mutation for Evolution Strategies - Self-Adaptive Heuristics for Evolutionary Computation - page 73

step sizes. Figure 4.10 tries to explain this behavior. We assume that mutations
of x and x + b lie on the edges of the circles. The yellow circular arc is the success
area of mutations without the bias. The bias b can point into directions, which
maintain a certain distance d b ≥ d to the constraint boundary. The circle around

x + b represents BMO mutations. The red circular arc marks the success area
of the BMO with shift b . This success area is much bigger and will be preferred
in the self-adaptive process. Hence, the bias b can adapt to directions, which
increase the success rate p s without decreasing the step sizes.

4.6 Excursus: Self-Adaptive Mutation Operator Selection

The mutation operators are appropriate for different kinds of problems and
search space characteristics. Which is the appropriate mutation operator for
a given problem? We have seen various mutation operators for real-parameter
optimization. In a series of experiments we tested the self-adaptive selection of
the following mutation operators:

• isotropic mutation with σ i =1,

• uncorrelated mutation with σ i = N ,

• BMO,

• cBMO,

• DMO.

Each individual got an additional strategy parameter mut type determining the
mutation operator to apply in the current generation. At the beginning mut type
is randomly initialized. In the course of the evolutionary process this strategy
parameter is mutated randomly with uniform distribution. The other strategy
parameters like step size or bias were inherited as usual or randomly initialized
if not available from particular operator changes.
We summarize the results of this experimental series. On the same test func-
tion no particular operator was preferred. Indeed, each possible operator was
chosen randomly at the beginning, randomly changed during the first iterations
and did not or very seldom change during the rest of the run. An explanation
of this result is the following: Whenever a mutation operator with a useful set
of strategy parameters (i.e. step size or bias) was randomly generated at the be-
ginning, it created more successful offspring than an operator with random and
in most cases worse initialization. Hence, the main result of these experiments
is that the fast adaptation of strategy variables of random operators prevents
potentially better operators from showing their capabilities. They were not able
to adapt their strategy variables. We think it would be possible to overcome this
problem by evolving all operators simultaneously for short time periods under
similar conditions and select the mutation operators after this evaluation period.
But this modification is supposed to be inecient because of the additional fit-
ness evolutions during simultaneous periods.