Evolution and Optimum Seeking

336 Appendix A
Figure A.9: Graphical representation of Problem 2.13 for n =2:
a11 = ;2 a12 = 27 a21 = ;70 a22 = ;48
b11 = ;76 b12 = ;51 b21 = 63 b22 = ;50
1 = ;3:0882 2 = 2:0559
F (x) ==238:864 581:372 1403:11 3283:14 7153:45
13635:3 21479:6 27961:4 31831:7 33711:8 34533:5=
nX
j=1
(aij sin xj + bij cos xj) =Ai for i = 1(1)n
The solution is again approximated in the least squares sense.
Minimum:
x i = i for i = 1(1)n F (x )=0
Because the trigonometric functions are multivalued there are in nitely many equivalent
minima (real solutions of the system of equations), of which upto2 n lie in the interval
Start:
i ; xi i + for i = 1(1)n
x (0)
i = i + i for i = 1(1)n
where i are random numbers in the range [; =10 =10]. To provide the same conditions
for all the search methods the same sequence of random numbers was used in each case,
and hence
F (x (0) ) ' 1182
Because of the proximity of the starting point to the one solution, x i = i for i = 1(1)n,
all the strategies approached this minimum only.