Evolution and Optimum Seeking

Numerical Comparison of Strategies 207
x 2
x
1
Optimum
Test position
Figure 6.16: Eight di erent testvalues of the objective function in case of n =2
the strategy employed has failed. Three intermediate classes of approximation are de ned
in the obvious way.
The maximum possible accuracy was required of all strategies. The corresponding
free parameters of the strategies that enter the termination criteria have already been
de ned in Table 6.2. In contrast to the rst test, no additional common termination rule
was employed.
A total of 50 problems were to be solved. The mathematical formulations of the
problems are given in Appendix A, Section A.2. Some of them are only distinguished by
the chosen initial conditions, others by the applied constraints. Nine out of 14 strategies
or versions of basic strategies are not suited to solving constrained problems, at least not
directly. Methods involving transformations of the variables and penalty function methods
were not employed. An exception is the method of Rosenbrock, which only alters the
objective function near the boundaries and can be applied in one pass otherwise penalty
functions require a sequence of partial optimizations to be executed. The second series of
tests therefore comprises one set of 28 unconstrained problems for all 14 strategies and a
second set of 22 constrained problems for 5 of the strategies. The results are displayed
together in Tables 6.5 to 6.8. The approximation to the objective that has been achieved
in each case is indicated by a corresponding symbol, using the classes of accuracy de ned
above.
Any interesting features in the solution of individual problems are documented in the
Appendix A, Section A.2, in some cases together with a brief analysis. Thus at this point
it is only necessary to make some general observations about the reliability of the search
methods for the totality of problems.
Unconstrained Problems
The results of the three versions of the coordinate strategies are very similar and generally
unsatisfactory. A third of all the problems cannot be solved with them at all, or only
very inaccurately. Exact solutions ( =10 ;38 ) are the exception and only in less than a