Evolution of biological systems:
- Adaptation of species to external conditions: optimization
- Adaptation strategy itself has evolved over time: sexual reproduction
More sex:
Consider some oscillatory function of a single variable,
having one global minimum within the range of definition,
. Find with
.
Start with a population of randomly chosen numbers
(individuals),
. ( is kept constant.)
- ``Gene'': bit string of
- ``Fitness'':
low
= high fitness and vice versa
- Relative fitness (probability of reproduction):
. This is a probability density,
and
is its cumulative distribution function.
Draw individuals in accordance with their reproduction
probability (Transformation method!).
The new population
is fitter than the original one. However, thus
far we have remained at the level of primitive selective
reproduction without mutation or sexual crossover.
Pick pairs of individuals at random and submit their genetic strings
are to crossover:
() Draw a position within the bit strings; () swap the bits following
between the two strings. The number of such pairings, the
``crossover rate'', should be around . The resulting
set
is called the offspring population.
Finally, mutation comes into play: within each string
every single bit is reversed with a probability
.
The resulting population is regarded as the next generation,
, and we are back at step 2.