Evolution and Optimum Seeking

232 Comparison of Direct Search Strategies for Parameter Optimization
departs from the general trend of the others simply because no orthogonalizations were
performed in this case. But the di erence is not dramatic, because the cost of testing the
constraints is of the same order of magnitude as that of rotating the coordinates. The
complex method takes computation times that initially increase somewhat more rapidly
than O(n 3 ). This corresponds to a greater than linearly increasing number of objective
function evaluations. As we have already seen in other problems, the increase becomes
even steeper as the number of parameters increases. With n =95variables, the required
distance was only partially covered within the maximum computation time.
Problem 3.9 represents a modi cation of Problem 3.8 with respect to the constraints.
In place of the (2 n ; 2) linear constraints, the corridor is bounded by a single non-linear
boundary condition. The cost of testing the feasibility of an iteration point is thereby
greatly reduced. The number of mutations or generations of the evolution strategies is
higher than in Problem 3.8 but still increases as O(n) the computation times in contrast
to Problem 3.8 only increase as O(n 2 ). The Rosenbrock method also has no di culty with
this problem, although the necessary rotations of the coordinate system make the times
of order O(n 3 ). The complex method could only solve Problem 3.9 for n =3upwards
of n = 10 it no longer converged.
The last problem, Problem 3.10, which also has inequality constraints, turned out
to be extremely di cult for all the search methods in the test. The main problem is
one of scaling. Convergence in the neighborhood of the minimum can be achieved if, and
practically only if, the step lengths in the coordinate directions are individually adjustable.
They have to di er from each other by several powers of 10. For n = 30, no strategy
managed to solve the problem within the maximum allowed computation time. The
complex method sometimes failed to end the search within this time for n = 10. The
intermediate results achieved after 8 hours are presented in Appendix A, Section A.3. All
of the evolution strategies do better than the methods of Rosenbrock and Box.
The result that the two membered evolution strategy came closer to the objective
than the multimembered evolution without recombination was not completely unexpected,
because considerably fewer generations than mutations can occur within the allowed time.
What is more surprising is that the (10 , 100) strategy with recombination does almost as
well as the two membered version. Here once again, the degree of freedom gained by the
possibilities of recombination shows itself to advantage. The variances of the mutation
step lengths do adjust themselves individually quite di erently according to the situation
and thus permit much faster convergence than with equal variances for all variables.
The other evolution strategies only come as close as they do to the solution because
the variances reach their relative lower bounds at di erent times, whereby di erences in
their sizes are introduced. This scaling process is, however, very much slower than the
continuous process of adaptation brought aboutby the recombination mechanism.
6.4 Core storage required
Up to now, only the time has been considered as a measure of the computational cost.
There is, however, another important characteristic that a ects the applicability of optimization
strategies, namely the core storage required. (Today nobody would use this