Evolution and Optimum Seeking

212 Comparison of Direct Search Strategies for Parameter Optimization
Table 6.6: Summary of the results from Table 6.5
Strategy Total number of problems No solution Fatal No normal
solved in the accuracy class or >10 ;2 computation termination
10 ;38 10 ;8 10 ;4 10 ;2 errors
FIBO 3 9 18 19 9 0 0
GOLD 4 9 18 19 9 0 0
LAGR 2 7 17 21 7 0 0
HOJE 6 21 26 26 2 0 0
DSCG 11 23 24 26 2 2 2
DSCP 12 24 26 26 2 2 2
POWE 4 20 21 21 7 8 4
DFPS 5 16 18 22 6 12 0
SIMP 7 18 24 26 2 0 9
ROSE 11 23 26 26 2 0 3
COMP 5 17 24 26 2 0 0
EVOL y 17 20 24 28 0 0 0
GRUP y 18 22 27 28 0 0 0
REKO y 23 24 28 28 0 0 2
Table 6.6 presents again a summary of the number of unconstrained problems that
were solved with given accuracy by the search methods under test, together with the number
of unsolved problems, the number of cases of fatal execution errors, and the number
of cases in which the termination criteria failed.
Constrained Problems
Tables 6.7 and 6.8 show the results of 5 strategies in the 22 constrained problems. Execution
errors such as exceeding the number range or dividing by zero did not occur in
any case. Neither were there any di culties in the termination of the searches.
The method of Rosenbrock can only be applied if the starting point of the search lies
within the allowed or feasible region. For this reason the initial values of the variables in
seven problems had to be altered. All other methods very quickly found a feasible solution
to start with. As in the unconstrained problems, the strategies that depend on random
numbers were each run three times with di erent sequences of random numbers. The
best of the three results was accepted for evaluation. The results of the complex method
and the two membered evolution turned out to be very variable in quality, whereas the
multimembered versions of the strategy, especially with recombination, proved to be less
in uenced by the particular random numbers. Two problems (Problems 2.40 and 2.41)
caused great di culty to all the search methods. These are simple linear programs that
can be solved rapidly and exactly by, for example, the simplex method of Dantzig. In
y Search terminated twice in each case due to too slow convergence