Evolution and Optimum Seeking

A Multimembered Evolution Strategy 145
We shall not go into further details since another kind of individual step length control
will o er itself later, i.e., recombination.
At this point afurtherword should be said about the alternative (1+ )or(1, )
strategies. Let us assume that by virtue of a jump landing far from the expectation value,
a descendant has made a very large and useful step towards the optimum, thus becoming
a parent of the next generation. While the variance allocated to it was eminently suitable
for the preceding situation, it is not suited to the new one, being in general much too
big. The probability that one of the new descendants will be successful is thereby low.
Because the (1+ ) strategy permits no worsening of the objective function value, the
parent survives{and may do so for many generations. This increases the probability ofa
successful mutation still having a poorly adapted step length. In the (1 , ) strategy such
a stray member will indeed also turn up in a generation, but it will be in e ect revoked in
the following generation. The descendant that regresses the least survives and is therefore
probably the one that most reduces the variance. The scheme thus has better adaptation
properties with regard to the step length. In fact this phenomenon can be observed in the
simulation. Since we have seen that for 5 the maximum rate of progress is practically
independent of whether or not the parent survives, we should favor a ( , ) strategy, at
least when = is not chosen to be very small, e.g., less than 5 or 6.
5.2.4 The Convergence Criterion for > 1 Parents
In Section 5.2.2 wewere really looking for the rate of progress of a ( , )evolution method.
Because of the analytical di culties, however, we had to fall back onthe = 1 case,
with only one parent. We shall now proceed again on the assumption that > 1. In
each generation state vectors xE and associated step lengths are stored, which should
always be the best of the mutants of the previous generation. We naturally require
more storage space for doing this on the computer, but on the other hand we havemore suitable values at our disposal for each variable. Supposing that the topology of the
objective function is complicated or even \pathological," and an individual reaches a point
that is unfavorable to further progress, we still have su cient alternative starting points,
which mayeven be much more favorable. According to the usefulness of their parameter
sets, some parents place more mutants in the prime group of descendants than others.
In general the best individuals of a generation will di er with respect to their variable
vectors and objective function values as long as the optimum has not been reached. This
provides us with a simple convergence criterion.
From the population of
function value:
parents Ek k = 1(1) ,welet Fb be the best objective
Fb = min fF (x
k (g)
k )k= 1(1) g
and Fw the worst
Fw = max
k
Then for ending the search we require that either
fF (x(g)
k )k= 1(1) g
Fw ; Fb "c