Evolution and Optimum Seeking

24 Hill climbing Strategies
and Vogl (1966), Klerer and Korn (1967), Abadie (1967, 1970), Fletcher (1969a), Rosen,
Mangasarian, and Ritter (1970), Geo rion (1972), Murray (1972a), Lootsma (1972a),
Szego (1972), and Sebastian and Tammer (1990).
Formulated as a minimum problem without constraints, the task can be stated as
follows:
minfF
(x) j x 2 IR
x n g (3.1)
The column vector x (at the extreme position) is required
x =
2
6
4
x 1
x 2
.
xn 3
7
5 =(x 1 x 2 :::x n )T
and the associated extreme value F = F (x ) of the objective function F (x), in this case
the minimum. The expression x 2 IR n means that the variables are allowed to take all
real values x can thus be represented by any point inann-dimensional Euclidean space
IR n . Di erent types of minima are distinguished: strong and weak, local and global.
For a local minimum the following relationship holds:
for
and
0 kx ; x k =
F (x ) F (x) (3.2)
vu
u
t nX
i=1
x 2 IR n
(xi ; x i ) 2 "
This means that in the neighborhood of x de ned by the size of " there is no vector
x for which F (x) is smaller than F (x ). If the equality sign in Equation (3.2) only
applies when x = x , the minimum is called strong, otherwise it is weak. An objective
function that only displays one minimum (or maximum) is referred to as unimodal. In
many cases, however, F (x) has several local minima (and maxima), which maybeof
di erent heights. The smallest, absolute or global minimum (minimum minimorum) ofa
multimodal objective function satis es the stronger condition
F (x ) F (x) for all x 2 IR n
This is always the preferred object of the search.
If there are also constraints, in the form of inequalities
or equalities
(3.3)
Gj(x) 0 for all j = 1(1)m (3.4)
Hk(x) =0 for all k = 1(1)` (3.5)