Evolution and Optimum Seeking

206 Comparison of Direct Search Strategies for Parameter Optimization
The assessment is tricky if a method does not converge for a particular problem but
terminates the search following its own criteria without getting anywhere near the solution.
Any strategy that fails frequently in this way cannot be recommended for use in practice
even if it is especially fast in other cases. In a practical problem, unlike a test problem,
the correct solution is not, of course, known in advance. One therefore has to be able
to rely on the results given by a strategy if they cannot be checked by another method.
Hence, reliability is just as important a criterion for assessing optimization methods as
speed.
The second part of the strategy comparison is therefore designed to test the robustness
of the optimization methods. The scale for assessing this is the number of problems that
are solved by agiven method. Since in this respect it is the complexity rather than size
of the problem that is signi cant, the number of variables ranges only from one to six.
All numerical iteration methods in practice can only approximate a solution with a
nite accuracy. In order to be able either to accept the end result of an optimum search
as adequate, or to reject it as inadequate, a border must be de ned explicitly, on one side
of which the solution is exact enough and on the other side of which it is unsatisfactory.
It is the structure of the objective function that is the decisive factor determining the
accuracy that can be achieved (Hyslop, 1972). With this in mind the border values for
the purpose of ranking the test results were obtained by the following scheme. Starting
from the known exact or best solution
x =(x 1 x 2 ::: x n ) T
the variables were individually altered by the amounts
4xi =
(
for x i =0
x i for x i 6= 0
in all combinations. For example for n = 2 one obtains eight di erent test values of the
objective function (see Fig. 6.16). In the general case there are 3 n ; 1 di erent values.
The greatest deviation 4F ( ) from the optimal value F (x ) de nes the border between
results that approach the objective su ciently closely and results that do not. To obtain
anumber of grades of merit, four di erent test increments j j = 1(1)4 were selected:
1 =10 ;38
2 =10 ;8
3 =10 ;4
4 =10 ;2
A problem is deemed to have been solved \exactly" at ~x if
F (~x) F (x )+4F ( 1)
is attained. On the other hand, if at the end of the search
F (~x) >F(x )+4F ( 4)