Evolution and Optimum Seeking

The Two Membered Evolution Strategy 109
x 2
Line of equal
probability
density
(g)
x
N
(g)
N
x (g)
E
z (g)
x
E
(g+2)
(g) (g+1)
E = E
z (g+1)
Opt.
N (g+1)
= E (g+2)
x
1
Figure 5.1: Two membered evolution strategy
Contours
F (x) = const.
E : Parent
N : Descendant
(g) : Generation index
The geometric locus of equally likely changes in variation of the variables can be
derived immediately from the probability density function, Equation (5.2). It is an ndimensional
hyperellipsoid (n-fold variance ellipse) with the equation
nX
i=1
zi
i
2
= const:
referred to its center, which is the starting point x (g)
E . In the multidimensional case, the
random changes can be regarded as a vector ending on the surface of a hyperellipsoid
with the semi-axes i orif i = for all i = 1(1)n, in the language of two dimensions
they are distributed circumferentially. Figure 5.1 serves to illustrate two iteration steps
of the evolution strategy on a two dimensional contour diagram. Whereas in other,
fully deterministic search strategies both the direction and length of the search step are
determined in the procedure in a xed manner, or on the basis of previously gathered
information and plausible assumptions about the topology of the objective function, in
the evolution strategy the direction is purely random and the step length{except for
a small number of variables{is practically xed. This should be emphasized again to
distinguish this random method from Monte-Carlo procedures, in which the selected trial
point isalways fully independent of the previous choice and its outcome. Darwin (1874)
himself emphasized that the evolution of living things is not a purely random process. Yet
against his theory of descendancy, a polemic is still waged in which the impossibility is
demonstrated that life could arise by a purely random process (e.g., Jordan, 1970). Even
at the level of the simplest imitation of organic evolution, a suitable choice of the step
lengths or variances turns out to be of fundamental signi cance.