Self-Adaptive Crossover - Self-Adaptive Heuristics for Evolutionary Computation - page 91

[39]. Second and this argument is closely connected to the previous one, the link
between crossover strategy parameters and their impact on the fitness is smaller
than necessary for successful self-adaptation [8].

6.1.1 The Role of Crossover - Building Blocks or Genetic Repair?

The assumption of the BBH by Goldberg [47] and Holland [59] is that differ-
ent good building blocks from a parental population are mixed together during
crossover in order to combine good properties of the parents and inherit these
to the offspring. Mitchell et al. [95] report that finding test functions to support
this hypothesis is surprisingly dicult. Jansen and Wegener [62] constructed an
artificial test function where one-point crossover improves the time complexity.
But it is still unexplained, whether crossover helps generally on non-artificial
test functions.
In contrast to the BBH, the GR hypothesis claims that recombination does not
let the different , but the common features flow into the offspring [14]. According
to this theory recombination extracts the similarities from the parents. Beyer
[91] assumes that similar corresponding components in the parental genomes
carry a higher probability of being beneficial. He states that the best a crossover
operator can do is to conserve these components. Beyer [16] states that recom-
bination itself usually has no benefit without mutation, but is only useful with
a high population variance. This high variance can be a result of a high mu-
tation strength or random initialization. The latter is the source for necessary
variance of GA implementations that do not use any mutations. In other words:
according to the GR hypothesis mutations are absolutely necessary for the ex-
ploration aspect of the search. Selection guides the search by choosing the useful
mutations. Up this stage BBH and GR agree. But recombination only reduces
the statistically uncorrelated parts of the mutations. This assumes that these
similarities are statistically the beneficial ones. From his hypothesis Beyer [16]
proposed the following principles for the design of crossover operators:

• Problem-specificness. Recombination should be constructed problem-specific

to make the extraction of the statistical similarities possible.

• Absence of biases. the probability for each individual to contribute to the

offspring should be equal.

• The quality of the statistical estimation increases with the number of par-

ents ρ participating. From this point of view, recombination conflicts with
selection and postulates ρ = λ = μ , i.e. no selection pressure. Of course, in
practice μ has to be chosen problem specific.

• Permanence. Recombination should always be used, i.e. p c =1 . 0. According

to Beyer [16] the empirical results for smaller values are due to improperly
chosen mutation strengths.

Beyer states that these criteria do not identify recombination operators com-
pletely. His results are theoretically valid on the sphere function. On other, more
complex functions, the principles may not be feasible possibly.