Evolution and Optimum Seeking

154 Evolution Strategies for Numerical Optimization
practiced in ESs and EP. An important linkbetween both levels, i.e., the genetic code
as well as the so-called epigenetic apparatus, is neglected at least in the canonical GA.
For dealing with integer or real values on the level of the object variables GAs make use
of a normal Boolean representation or they use the so-called Gray code. Both, however,
present the di culty of so-called Hamming cli s. Depending on its position, a single
bit reversal thus can lead to small or very large changes on the phenotypic level. This
important fact has advantages and disadvantages. The advantage lies in the broad range
of di erent phenotypes available in a GA population at the same time, a matter a ecting
its global convergence reliability (for a thorough convergence analysis of the canonical GA
see Rudolph, 1994a). The corresponding disadvantage stems from the other side of the
same coin, i.e., the inability to focus the search e ort in a close enough vicinity of the
current positions of individuals in one generation.
There is a second reason to cling to binary representations of object variables within
GAs, i.e., Holland's schema theorem (Holland, 1975, 1992). This theorem tries to assure
exponential penetration of the population by individuals with above average tness under
proportional selection, with su ciently higher reproduction rates for better individuals,
one point crossover with xed crossover probability, and small, xed mutation rates.
If, at some time, especially when starting the search, the population contains the
globally optimal solution, this will persist in the case where there are zero probabilities
for mutation and recombination. Mutation, according to the theorem, is an always destructive
force and thus called a subordinate operator. It only serves to introduce missing
or reintroduce lost correct bits into nite populations. Recombination (here, one point
crossover) mayormay not be destructive, depending on whether the crossover point happens
to lie within a so-called building block, i.e., a short substring of the bit string that
contributes to above-average tness of one of the mating individuals, or not. Building
blocks are especially important in case of decomposable objective functions (for a more
detailed description see Goldberg, 1989).
GAs in their original form do not permit the handling of implicit inequality or equality
constraints. On the other hand, explicit upper and lower bounds have tobeprovided for
the range of the object variables:
ui xi vi for all i = 1(1)n
in order to have a basis for the binary decoding and encoding process, e.g.,
xi = ui + vi ; ui
2 l ; 1
lX
j=1
aij 2 j;1
where aij for j = 1(1)l represents the bit string segment of length l for encoding the ith
element of the object variable vector x.
Instead of this Boolean mapping one also may choose the Gray code, which has the
property that neighboring values for the xi di er in one bit position only. Looking for
the probability distribution p( xi) of phenotypic changes xi from one generation to the
next at a given position x (0)
i and a given mutation probability pm shows that changing
the code from Boolean to Gray only shifts, but never avoids, the so-called Hamming