Evolution and Optimum Seeking

One Dimensional Strategies 25
then IR n in Equations (3.1) to (3.3) must either be replaced by the hopefully non-empty
subset M 2 IR n to represent the feasible region in IR n de ned by Equation (3.4), or by
IR n;` , the subspace of lower dimensionality spanned by the variables that now depend
on each other according to Equation (3.5). If solutions at in nity are excluded, then
the theorem of Weierstrass holds (see for example Rothe, 1959): \In a closed compact
region a x b every function which iscontinuous there has at least one (i.e., an
absolute) minimum and maximum." This can lie inside or on the boundary. In the case
of discontinuous functions, every point of discontinuity is also a potential candidate for
the position of an extremum.
3.1 One Dimensional Strategies
The search for a minimum is especially easy if the objective function only depends on one
variable.
F(x)
a
b
c d e f g h
Figure 3.1: Special points of a function of one variable
a: local maximum at the boundary
b: local minimum at a point of discontinuity ofFx(x)
c: saddle point, or point of in ection
d-e: weak local maximum
f: local minimum
g: maximum (may be global) at a point ofdiscontinuity ofF (x)
h: global minimum at the boundary
x