Evolution and Optimum Seeking

110 Evolution Strategies for Numerical Optimization
5.1.2 The Step Length Control
In experimental optimization, the appropriate step lengths can frequently be predicted.
The values of the variables usually have to be determined exactly at only a few points.
Thus constant values of the variances are often all that is required to complete an extreme
value search. It is a matter of fact that in most experimental applications of the simple
evolution strategy xed (and discrete) distributions of mutations have been used.
By contrast, in mathematically formulated problems that are to be solved on a digital
computer, the variables often run over muchofthenumber range of the computer, which
corresponds to many powers of 10. In a numerical optimum search the step lengths
must be continuously modi ed if the algorithm is to be e cient{a problem reminiscent
of steering safely between Scylla and Charybdis for if the step length is too small the
search takes an unnecessarily large number of iterations if it is too large, on the other
hand, the optimum can only be crudely approached and the search caneven get stuck
far from the optimum, for example, if the route to the minimum passes along a narrow
valley. Thus in all optimization strategies the step length control is the most important
part of the algorithm after the recursion formula, and it is furthermore closely linked to
the convergence behavior.
The corresponding remarks hold for the evolution strategy, with the following di erence:
In place of a predetermined step length for a parameter of the objective function
there is the variance of the random changes in this parameter, and instead of the statement
that an improvement will or will not be made in a given direction with a speci ed
step length, there can only be a statement of probability of the success or failure for a
chosen variance.
In his theoretical investigations of the two membered evolution strategy, Rechenberg
discovered using two basically di erent model objective functions (sphere model = Problem
1.1, corridor model = Problem 3.8 of the problem catalogue see Appendix A) that
the maximal rate of convergence corresponds to a particular value for the probability ofa
success, i.e., an improvement in the objective function value. He was thus led to formulate
the following rule for controlling the size of the random changes:
The 1=5 success rule:
From time to time during the optimum search obtain the frequency of successes,
i.e., the ratio of the number of successes to the total number of trials
(mutations). If the ratio is greater than 1=5, increase the variance,ifitisless
than 1=5, decrease the variance.
In many problems this rule proves to be extremely e ective inmaintaining approximately
the highest possible rate of progress towards the optimum. While in the rightangled
corridor model the variances are adjusted once and for all in accordance with this
rule and subsequently remain constant, in the sphere model they must steadily become
smaller. The question then arises as to how often the success criterion should be tested
and by what factor the variances are most e ectively reduced or increased.
This question will be answered with reference to the sphere model introduced by
Rechenberg, since this is the simplest non-linear model objective function and requires