Evolution and Optimum Seeking

346 Appendix A
on the interval 0 y 1 with the boundary condition z(y =0)=0. The function
sought, z(y), is to be approximated by a polynomial
~z(c y) =
nX
j=1
cj y j;1
In the present case only the rst six terms are considered. Suitable values of the polynomial
coe cients cj j = 1(1)6, are to be determined. The deviation from the exact
solution of the di erential equation is measured in the Gaussian sense as the sum of the
squares of the errors at m = 30 argument values yi, uniformly distributed in the range
[0,1]
F 1(c) =
0
mX
@
i=1
@~z(c y)
; ~z
@y yi 2 (c y)
yi
The boundary condition is treated as a second simultaneous equation by means of a
similarly constructed term:
F 2(c) = ~z 2 (c y) y=0
By inserting the polynomial and rede ning the parameters ci as variables xi we obtain
the objective function F (x) =F 1(x)+F 2(x), the minimum of which isanapproximate
solution of the parameterized functional problem.
Minimum:
Start:
x ' (;0:0158 1:012 ;0:2329 1:260 ;1:513 0:9928) F (x ) ' 0:002288
; 1
x (0) =(0 0 0 0 0 0) F (x (0) )=30
Judging by the number of objective function evaluations all the search methods found
this a di cult problem to solve. The best solution was provided by the complex strategy.
Problem 2.29 after Beale (1967)
Objective function:
Constraints:
Minimum:
Start:
F (x) =2x 2
1 +2x 2
2 + x 2
3 +2x 1 x 2 +2x 1 x 3 ; 8 x 1 ; 6 x 2 ; 4 x 3 +9
x = 4 7 4
3 9 9
Gj(x) =xj 0 for j = 1(1)3
G 4(x) =;x 1 ; x 2 ; 2 x 3 +3 0
F (x )= 1
9 only G 4 active i.e., G 4(x )=0
x (0) =(0:1 0:1 0:1) F (x (0) )=7:29
1
A
2