Evolution and Optimum Seeking

116 Evolution Strategies for Numerical Optimization
normally distributed random numbers with the expectation values zero and the variances
unity. The formulae are
Z
and
0 q
1 = ;2lnY1 sin (2 Y2)
Z 0 q (5.8)
2 = ;2lnY1 cos (2 Y2)
where the Yi are the uniformly distributed and the Z0 i (0 1)-normally distributed random
numbers respectively. Toobtain a distribution with a variance di erent from unity, the
Z0 i must simply be multiplied by the desired standard deviation i (the \step length"):
Zi = i Z 0 i
The transformation rules are contained in a function procedure separate from the actual
subroutine. To make use of both Equations (5.8) a switch withtwo settings is de ned,
the condition of which must be preset in the subroutine once and for all. In spite of
Neave's (1973) objection to the use of these rules with uniformly distributed random
numbers that have been generated by amultiplicative congruence method, no signi cant
di erences could be observed in the behavior of the evolution strategy when other random
generators were used. On the other hand the trapezium method of Ahrens and Dieter
(1972) is considerably faster.
Most algorithms for parameter optimization include a second termination rule, independent
of the actual convergence criterion. They end the search after no more than a
speci ed number of iterations, in order to avoid an in nite series of iterations in case the
convergence criterion should fail. Such a rule is e ectively a bound on the computation
time. The program libraries of computers usually contain a procedure with which the
CPU time used by the program can be determined. Thus instead of giving a maximum
number of iterations one could specify a maximum computation time as a termination
criterion. In the present program the latter option is adopted. After every n iterations
the elapsed CPU time is checked. As soon as the limit is reached the search ends and
output of the results can be initiated from the main program.
The 1=5 success rule assumes that there is always some combination of variances
i > 0 with which, on average, at least one improvement can be expected within ve
mutations. In Figure 5.3 two contour diagrams are shown for which the above condition
cannot always be met. At some points the probability of a success cannot exceed 1=5 : for
example, at points where the objective function has discontinuous rst partial derivatives
or at the edge of the allowed region. Especially in the latter case, the selection principle
progressively forces the sequence of iteration points closer up to the boundary and the
step lengths are continuously reduced in size, without the optimum being approached
with comparable accuracy.
Even in the corridor model (Problem 3.8 of Appendix A, Sect. A.3) di culties can
arise. In this case the rate of progress and probability of success depend on the current
position relative to the edges of the corridor. Whereas the maximum probability of success
in the middle of the corridor is 1=2, at the corners it is only 2 ;n . If one happens to be in the
neighborhood of the edge of the corridor for several mutations, the probability of success