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construct of as many orthogonal dimensions as there are genes in the gene
subset with a given length.
No starting guess is necessary (or even possible), and the operator only
has to define basic parameters such as the number of genes in a gene sub-
set and the minimal and maximal gene IDs for each gene variable (i.e. the
boundary conditions). After the process is started, the operator can
observe the maximal and minimal values (within which the global opti-
mum resides) for the various gene variables approaching each other. At
the end of the run, there will be no gene variable whose search space
(maximal to minimal) is greater than the specified value (as little as one
gene). By using SDL global optimization strategies, an operator can be
assured that he/she has found the best solution physically possible, inde-
pendent of his/her so-called best guess.
5.3.4.
Mathematical form of SDL optimization
Consider a multi-dimensional continuous function
f
(
x
) with multiple
global minima and local minima on subset
G
of
R
n
.
5.3.4.1.
Definition of local minima
For a given point
x
*
∈
G, if there exists a
δ
-neighborhood of
x
*,
O
(
x
*,
δ
),
such that for
xOx
Œ
(*, ),
d
and
fx
(*)
£
fx
(),
(1)
then
x
* is called a local minimal point of
f
(
x
).
5.3.4.2.
Definition of global minima
If for every
x
G
the inequality (1) is correct, then
x
* is called a global
minimum of
f
(
x
) on
G
, and the global minima of
f
(
x
) on
G
form a global
minimum set.
∈
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