Evolution and Optimum Seeking

Multidimensional Strategies 49
where
and
wi =
8
><
>:
ai
P
ai ; i;1
(a
j=1
T i v (k+1)
j
ai =
nX
j=i
) v (k+1)
j
d (k)
j v (k)
j for all i = 1(1)n
for i =1
for i = 2(1)n
(3.21)
A scalar d (k)
i represents the distance covered in direction v (k)
i in the kth iteration.
Thus v (k+1)
1 points in the overall successful direction of the step k. It is expected that a
particularly large search step can be taken in this direction at the next iteration. The
requirement ofwaiting for at least one success in each direction has the e ect that no
direction is lost, and the v (k)
i always span the full n-dimensional Euclidean space. The
termination rule or convergence criterion is determined by the lengths of the vectors
a (k)
1
and a (k)
2 . Before each orthonormalization there is a test whether ka (k)
1 k < " and
ka (k)
2 k > 0:3 ka (k)
1 k. When this condition is satis ed in six consecutive iterations, the
search is ended. The second condition is designed to ensure that a premature termination
of the search does not occur just because the distances covered have become small. More
signi cantly, the requirement is also that the main success direction changes su ciently
rapidly something that Rosenbrock regards as a sure sign of the proximity of a minimum.
As the strategy comparison will show (see Chap. 6), this requirement is often too strong.
It even hinders the ending of the procedure in many cases.
In his original publication Rosenbrock has already given detailed rules as to how
inequality constraints can be treated. His procedure for doing this can be viewed as
a partial penalty function method, since the objective function is only altered in the
neighborhood of the boundaries. Immediately after each variation of the variables, the
objective function value is tested. If the comparison is unfavorable, a failure is registered
as in the unconstrained case. For equality oranimprovement, however, if the iteration
point lies near a boundary of the region, the success criterion changes. For example, for
constraints of the form Gj(x) 0 for all j = 1(1)m, the extended objective function
~F (x) takes the form (this is one of several suggestions of Rosenbrock):
in which
and
'j(x) =
~F(x) =F (x)+
8
><
>:
mX
j=1
0
3 ; 4 2 +2 3
1
'j(x)(fj ; F (x))
=1; 1 Gj(x)
if Gj(x)
if 0