Evolution and Optimum Seeking

A Multimembered Evolution Strategy 119
thus consisted of two individuals. In order to reach a higher level of imitation of the
evolutionary process, the number of individuals must be increased. This is precisely the
concept behind the evolution strategy referred to in the following as multimembered. In
his basic work (Rechenberg, 1973), Rechenberg already presented a scheme for a multimembered
evolution. The one considered here is somewhat di erent. It turns out to
be particularly useful with respect to the individual control of several step lengths to be
described later. As yet, however, no detailed comparison of the two variants has been
undertaken.
It is useful to introduce at this point anomenclature for the di erent evolution strategies.
We shall call the number of parents of a generation ,andthenumber of descendants
, so that the selection takes place between + = 1+1 = 2 individuals in the two membered
strategy. Wethus characterize the simplest imitation of evolution in abbreviated
notation as the (1+1) strategy. Since the multimembered evolution scheme described by
Rechenberg allows a selection between > 1 parents and = 1 o spring it should be
called the ( +1) strategy. Accordingly a more general form, a ( + )evolution strategy,
should be formulated in such away that a basic population of parents of generation g
produces o spring. The process of selection only allows the best of all + individuals
to proceed as parents of the following generation, be they o spring of generation
g or their parents. In this model it could happen that a parent, because of its vitality,
is far superior to the other parents in the same generation, \lives" for a very long time,
and continues to produce further o spring. This is at variance to the biological fact of a
limited lifespan, or more precisely a limited capacity for reproduction. Aging phenomena
do not, as far as is known, a ect biological selection (see Savage, 1966 Osche, 1972). As
a further conceptual model, therefore, let us introduce a population in which parents
produce > o spring but the parents are not included in the selection. Rather
the parents of the following generation should be selected only from the o spring. To
preserve a constant population size, we require that each time the best of the o spring
become parents of the following generation. We will refer to this scheme in what follows
as the ( , ) strategy. As for the (1+1) strategy in Section 5.1.1, the algorithm of the
multimembered ( , ) strategy will rst be formulated in the language of biology.
Step 0: (Initialization)
A given population consists of individuals. Each ischaracterized by its
genotype consisting of n genes, which unambiguously determine the vitality,
or tness for survival.
Step 1: (Variation)
Each individual parent produces = o spring on average, so that a total of
new individuals are available. The genotype of a descendant di ers only
slightly from that of its parents. The number of genes, however, remains to
be n in the following, i.e., neither gene duplication nor gene deletion occurs.
Step 2: (Filtering)
Only the best of the o spring become parents of the following generation.
In mathematical notation, taking constraints into account, the rules are as follows: