Evolution and Optimum Seeking

352 Appendix A
The constraints form a feasible region, which could be described as a corridor with a
square cross section (three dimensionally speaking). The axis of the corridor runs along
the diagonal in the space
x 1 = x 2 = x 3 = :::= xn
The contours of the linear objective function run perpendicular to this axis. In order
to obtain a nite minimum further constraints were added, whereby a kind of pencil
point is placed on the end of the corridor. In the absence of these additional constraints
the problem corresponds to the corridor model used by Rechenberg (1973), for which he
derived theoretically the rate of progress (a measure of the convergence rate) of the two
membered evolution strategy.
Minimum:
Start:
Problem 2.38
x i =100 for i = 1(1)n F (x )=;300 G 1 to Gn active
x (0)
i =0 for i =1(1)n F (x (0) )=0
As Problem 2.25, but with the additional constraints:
Minimum:
Start:
Gj(x) =xj ; 1 0 for j = 1(1)n n =5
x i =1 for i = 1(1)n F (x )=0 all Gj active
x (0)
i = ;10 for i = 1(1)n F (x (0) ) = 48884
The starting point is not in the feasible region.
Problem 2.39 after Rosen and Suzuki (1965)
Objective function:
Constraints:
Minimum:
F (x) =x 2
1 + x 2
2 +2x 2
3 + x 2
4 ; 5 x 1 ; 5 x 2 ; 21 x 3 +7x 4
G1(x) = ;2 x 2
1 ; x2
2 ; x2
3 ; 2 x1 + x2 + x4 +5 0
G2(x) = ;x 2
1 ; x 2
2 ; x 2
3 ; x 2
4 ; x1 + x2 ; x3 + x4 +8 0
G3(x) = ;x 2
1 ; 2 x 2
2 ; x 2
3 ; 2 x 2
4 + x1 + x4 +10 0
x =(0 1 2 ;1) F (x )=;44 G 1 active