Evolution and Optimum Seeking

Numerical Comparison of Strategies 215
Forbidden
region
α
σ1
σ2
P 1
Circles : lines of equal probability of a step
P 2
Negative
gradient
direction
Figure 6.17: The situation at active constraints
σ2
To the
minimum
Lines of
constant F(x)
success is very much narrowed down by constraints. While the individuals are not yet at
the edge of the feasible region, those descendants whose step lengths have become smaller
have a higher probability of survival. Thus here too the entire population eventually
concentrates itself in a smaller and smaller area at the edge of the feasible region.
The theory of the rate of progress in the corridor model did not foresee this kind
of di culty, indeed it gives an optimal success rate, almost the same as in the sphere
model, simply because the gradient vector of the objective function always runs parallel
to the boundaries. In this case the search weaves backwards and forwards between the
center and side of the corridor. The reduced probability of success at multidimensional
edges is compensated by the fact that with a uniform probability of occupation over the
cross section of the corridor, the space that counts as near to the edges represents a very
small fraction of the total. Provided that the success rate is obtained over long enough
periods the 1=5 success rule does not lead to permanent reduction of the variances but to
a constant near optimal step size (it really uctuates) that depends only on the width of
the corridor and the number of variables.
The situation is happier than in Figure 6.17 if the constraints are given explicitly as
xi ai or xi bi
For anyonevariable, the region of success at a boundary is reduced by one half. If at some
position m variables are each bounded on one side, then on average it costs 2 m mutations
before one lands within the feasible region. Here again, the 1=5 success rule for m >2
will continuously reduce the variances until they reach their minimum value. Depending
on the route chosen by the search process the limiting values of the variances, which are
individually adjustable for each variable, will be reached at di erent times. Their relative
values thereby alter, and with the new combination of step lengths the convergence can
be faster.
The extra exibility of the multimembered evolution strategy with recombination,
in which thevariances of the changes in the variables are individually adaptable during