Evolution and Optimum Seeking

Test Problems for the Second Part of the Strategy Comparison 337
Problem 2.14 after Powell (1962)
Objective function:
Minimum:
Start:
F (x) =(x 1 +10x 2) 2 +5(x 3 ; x 4) 2 +(x 2 ; 2 x 3) 4 +10(x 1 ; x 4) 4
x =(0 0 0 0) F (x )=0
x (0) =(3 ;1 0 1) F (x (0) )=215
The matrix of second partial derivatives of the objective function goes singular at the
minimum. Thus it is not surprising that a quasi-Newton method like thevariable metric
method of Davidon, Fletcher, and Powell (applied here in Stewart's derivative-free form)
got stuck a long way from the minimum. Geometrically speaking, there is a valley which
becomes extremely narrow asitapproaches the minimum. Theevolution strategies therefore
ended up by converging very slowly with a minimum step length, and the search had
to be terminated for reasons of time.
Problem 2.15
As Problem 2.14, except:
Start:
Problem 2.16 after Leon (1966a)
Objective function:
x (0) =(1 2 3 4) F (x (0) )=1512
F (x) = 100 (x 2 ; x 3
1) 2 +(x 1 ; 1) 2
Figure A.10: Graphical representation of Problem 2.16
F (x) ==0:25 4 64 250 1000 5000 10000=