Evolution and Optimum Seeking

332 Appendix A
Several of the search procedures wereunableto ndtheminimum. They converge to a
point on the line x 1+x 2 = 4, which joins together the sharpest corners of the rhombohedral
contours. The partial derivatives of the objective function are discontinuous there in the
unit vector directions, parallel to the coordinate axes, no improvement can be made.
Besides the coordinate strategies, the methods of Hooke and Jeeves and of Powell are
thwarted by this property.
Problem 2.7 after Box (1966)
Objective function:
Minima:
F (x)=
10X
j=1
(exp (;0:1 jx 1) ; exp (;0:1 jx 2) ; x 3 [exp (;0:1 j) ; exp (;j)]) 2
x =(1 10 1) F (x )=0
x =(10 1 ;1) F (x )=0
Besides these two equivalent, strong minima there is a weak minimum along the line
x 0
1 = x 0
2 x 0
3 =0 F (x 0 )=0
Because of the
into a region:
nite computational accuracy the weak minimum is actually broadened
x 00
1 ' x 00
2 x 00
3 ' 0 F (x 00 )=0 if x1 1
Figure A.6: Graphical representation of Problem 2.7 on the plane
x3 =1F(x) ==0:03 0:3 1' 3:064 10 30=