Evolution and Optimum Seeking

Test Problems for the Second Part of the Strategy Comparison 329
There are in nitely many local minima, the position of which can be speci ed by a
transcendental equation: q q
jx j = 2 tan ( jx j)
For jx j 1wehave approximately
and
x ' ( (0:5+k)) 2 for k =1 2 3:::integer
F (x ) 'jx j
Whereas in reality none of the nite local minima is at the same time a global minimum,
the nite word length of the digital computer used together with the system-speci c
method of evaluating the sine function give rise to an apparent global minimum at
x = 4:44487453 10 16
F (x )=;4:44487453 10 16
Counting from the origin it is the 67 108 864th local minimum in each direction. If x is
increased above this value, the objective function value is always set to zero. (Note that
this behavior is machine dependent.)
Start:
x (0) =0 F (x (0) )=0
Most strategies located the rst or highest local minimum left or right of the starting
point (the origin). Depending on the sequence of random numbers, the two membered
evolution method found (for example) the 2nd, 9th, and 34th local minimum. Only the
(10, 100) evolution strategy almost always reached the apparent global minimum.
Problem 2.4
Objective function:
Minimum:
Start:
F (x) =
nX
i=1
Problem 2.5 after Booth (1949)
Objective function:
x 1 ; x 2
i
2
+(xi ; 1] 2
for n =5
x i =1 for i = 1(1)n F (x )=0
x (0)
i =10 for i = 1(1)n F (x (0) ) = 40905
F (x) =(x 1 +2x 2 ; 7) 2 +(2x 1 + x 2 ; 5) 2