Evolution and Optimum Seeking

216 Comparison of Direct Search Strategies for Parameter Optimization
the whole of the optimization process, is a very clear advantage in solving constrained
problems. Suitable combinations of variances are set up in this case before the smallest
possible step lengths are reached. Thus the total computation time is reduced and the
nal accuracy is better. The recombination option also appears to have a bene cial e ect
at boundaries that are not explicit it clearly improves the chance that descendants, even
with a larger step size, will be successful near the boundary. Inany case the population
clusters more slowly together than when there is no recombination.
Global Convergence Properties
Among the 50 test problems there are 8 having at least a second local minimum besides
the global one. In the reliability test, the accuracy achieved was only assessed with respect
to the particular optimum that was being approximated. What now is the capability of
each strategy for locating global minima? Several problems were speci cally designed to
investigate this question by having very many local optima, namely Problems 2.3, 2.26,
2.30, and 2.44. In Table 6.9 this aspect of the test results is evaluated.
Except for one problem (Problem 2.32), whose global minimum was found by all the
strategies under test, the method of Rosenbrock onlyconverged to local optima. The
complex method and the (1+1) evolution strategy were only better in one case: namely,
in Problem 2.45 they both approached the global minimum.
Table 6.9: Results of all strategies in the second comparison test:
global convergence properties
Problem F G L H D D P D S R C E G R
I O A O S S O F I O O V R E
B L G J C C W P M S M O U K
No. O D R E G P E S P E P L P O
2.3 L1 L1 L3 L1 L7 L7 L1 L3 L1 L6 L1 Lm G G
2.36 L1 L1 L1 L1 L1 L1 L1 L1 L1 L1 L1 L1 G G
2.30 L4 L1 Lm G G
2.32 G G G G G
2.44 L1 L1 L1 G G
2.45 L G G G G
2.47 L3 L1 L2 G G
2.48 L2 Lm Lm GL GL
Meaning of symbols:
L Search converges to local minimum.
L3 Search converges to the 3rd local minimum (in order of decreasing objective
function values).
Lm Search converges to various local minima depending on the random numbers.
G Search converges to global minimum.
GL Search converges to local or global minimum depending on the random numbers.