Evolution and Optimum Seeking

114 Evolution Strategies for Numerical Optimization
Gradient methods, which seek a point withvanishing rst derivatives, frequently also
apply this necessary condition for the existence of an extremum as a termination criterion.
Alternatively the search can be continued until 4F = F (x (k;1) ) ; F (x (k) ), the change
in the objective function value in one iteration, goes to zero or to below a prescribed
limit. But this requirement can also be ful lled far from the minimum if the valley in
which the deepest point issought happens to be very at in shape. In this case the
step length control of the two membered evolution strategy ensures that the variances
become larger, and thus the function value di erences between two successful trials also
on average become larger. This is guaranteed even if the function values are equal (within
computational accuracy), since a change in the variables is always then registered as a
success. One thus has only to take care that 4F is summed over a number of results
in order to derive a termination criterion. Just as lower bounds are de ned for the step
lengths, an absolute and a relative bound can be speci ed here:
Termination rule:
End the search if
or
where
and
1
"d
"c > 0
1+"d > 1
F (x (g;4g)
E ) ; F (x (g)
E ) "c
h (g;4g)
F (x E ) ; F (x (g)
E ) i
)
4g 20 n
jF (x (g)
E )j
according to the computational accuracy
The condition 4g 20 n is designed to ensure that in the extreme case the standard
deviations are reduced or increased within the test period by at least the factor
(0:85) 20 ' (25) 1 , in accordance with the 1=5 success rule. This will prevent the search
being terminated only because the variances are forced to change suddenly. It is clear
from Equation (5.4) that the more variables are involved in the problem, the slower is
the rate of progress. Hence it does not make sense to test the convergence criterion very
frequently. A recommended procedure is to make a test every 20 n mutations. Only one
additional function value then needs to be stored.
Another reason can be adduced for linking the termination of the search to the function
value changes. While every success in an optimum search means, in the end, an economic
pro t, every iteration costs computer time and thus money. If the costs exceed the pro t,
the optimization may well provide useful information, but it is certainly not on the whole
of any economic value. Thus someone who only wishes to optimize from an economic
point of view can, by a suitable choice of values for the accuracy parameters, restrain the
search process as soon as it starts running into a loss.