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5.8 Genetic Algorithms 209

Crossover

Two parent chromosomes may pass unchanged to the next generation or else with
a certain probability P,, called the crossover rate, they will generate new
chromosomes that result from swapping genes at specific sites or, for real-valued
genetics, interpolating the respective gene values. Some interesting crossover types
are indicated next for the MLP2:2:1 case, where we simplified the notation by
using letters for the weights.

Single crossover:

Double crossover:

Linear interpolation crossover:

The idea behind crossover is to explore alternative solutions in the weight space,
maintaining some good "qualities" of the parent generation. Usually there is also a
fixed number of parent chromosomes with high fitness factor that pass unchanged
to the next generation, the so-called elitism.

Mutation
On the generated chromosomes it is possible to add some mutation effect by
adding a small amount of noise, with probability P,, the mutation rate. The
possible benefit of adding random noise to the weights during training was already
explained in section 5.5.2.

The stopping conditions for the process of generating new populations can be
the same as for the MLP trained with back-propagation (see section 5.5.2).
Genetic algorithms rely, therefore, on random crossover and mutations to
perform a wide search in the weight space, which hopefully will converge to an
optimal solution. The explanation of how a genetically trained MLP will have the
ability to reach an optimal solution is essentially based on the schema theorem,
presented by Holland (1975). Given the insight provided by this theorem, we
review here its main ingredients for the case of binary chromosomes. A schema is