Evolution and Optimum Seeking

Test Problems for the Second Part of the Strategy Comparison 349
Minimum:
Start:
x =(1:275 0:6348 2:0 2:0) F (x ) ' 189:1 all Gj active
x (0) =(1 2 3 4) F (x (0) )=1512
The (1+1) evolution strategy only solved the problem very inaccurately. Due to the 1=5
success rule the mutation variances vanish prematurely.
Problem 2.34 after Fletcher and Powell (1963)
Objective function:
where
or
Constraints:
Minimum:
=
F (x) = 100 h (x 3 ; 10 ) 2 +(R ; 1) 2i + x 2
3
8
><
>:
1
2
x 1 = R cos (2 )
x 2 = R sin (2 )
R =
q x 2
1 + x 2
2
arctan x2
x1 if x 2 6= 0 and x 1 > 0
1
2 if x2 =0
1
2
+ arctan x2
x1
if x 2 6= 0 and x 1 < 0
G 1(x) =;x 3 +7:5 0 G 2(x) =x 3 +2:5 0
x =(1 ' 0 0) F (x )=0 no constraint isactive
The objective function itself has a discontinuity atx 2 = 0, right at the minimum sought.
Thus x 2 should only be allowed to approach closely to zero. Because of the multivalued
trigonometric functions there are in nitely many solutions to the problem, of which only
one, however, lies within the feasible region.
Start:
Problem 2.35 after Rosenbrock (1960)
Objective function:
x (0) =(;1 0 0) F (x (0) )=2500
F (x)=;x 1 x 2 x 3