Evolution and Optimum Seeking

142 Evolution Strategies for Numerical Optimization
5.2.3 The Step Length Control
How should one proceed in order to still achieve the maximum rate of progress, i.e.,
2
to maintain the optimum variances i i = 1(1)n, for the case of the multimembered
evolution scheme? For the (1+1) strategy this aim was met by the1=5 success rule,
which was based on the probability of success at maximum convergence rate of the sphere
and corridor model functions. Such control from outside the actual mutation-selection
game does not correspond to the biological paradigm. It should rather be assumed that
the step lengths, or more precisely the variances, have adapted and are still adapting to
circumstances arising in the course of natural evolution. Although the environmentally
induced rate of mutation cannot be interfered with directly, the existence of mutator
genes and repair enzymes strongly suggests that the consequences of such environmental
in uences are always reduced to the appropriate level. In the multimembered evolution
strategy the fact that the observed rates of mutation are also small, indeed that they
must be small to be optimal, comes out of the universal rate of progress and standard
deviation introduced above, which require to be inversely proportional to the number
of variables, as in the (1+1) strategy.
If we wish to imitate organic evolution, we can proceed as follows. Besides the variables
xEi i= 1(1)n, a set of parameters Ei i = 1(1)n, is assigned to a parent E. These
describe the variances of the random changes. Each descendant N` of the parent E should
di er from it both in x`i and `i. The changes in the variances should also be random
and small, and the most probable case should be that there is no change at all. Whether
a descendant can become a parent of the next generation depends on its vitality, thus
only on its x`i. Whichvalues of the variables it represents depends, however, not only
on the xEi of the parent, but also on the standard deviations `i, whichaect the size of
the changes zi = x`i ; xEi. In this way the \step lengths" also play an indirect r^ole in
the selection mechanism.
The highest possible probability that a descendant is better than the parent is normally
wemax =0:5
It is attained in the inclined plane case, for example, and for other model functions in the
limit of in nitely small step lengths. In order to prevent that a reduction of the i always
gives rise to a selection advantage, must be at least 2. But the optimal step lengths
can only take e ect if
> 1
weopt
This means that on average at least one descendant represents an improvement of the
value of the objective function. The number of descendants per parent thus plays a
decisive r^ole in the multimembered scheme, just as does the check on the success ratio in
the two membered evolution scheme. For comparison let us tabulate here the opt of the
(1 , ) strategy and weopt of the (1+1) strategy for the three model functions considered.
The values of weopt are taken from the work of Rechenberg (1973).