Evolution and Optimum Seeking

158 Evolution Strategies for Numerical Optimization
without applying the selection operator. Otherwise it may happen simply by chance that
one or the other descendant is not di erent from one of its parents.
In contrast to ESs, the number of o spring always is equal to the number of parents
( = ). There is no surplus of descendants to cope with lethal mutations and
recombinations. ESs need that kind of surplus for handling constraints, at least. In
the non-preserving case of its comma-version, a multimembered ES also needs a surplus
( > ) for the selection process. The ; worst o spring are handled as if they do
not survive to the adult reproductive state the best, however, have the same reproduction
probability ps =1= ,which does not depend on their individual phenotypes or
corresponding objective function values. Thus,onaverage, every parent has = descendants.
This is depicted on the left-hand side of Figure 5.17, where the average number
of descendants of the two bestof = 10 descendants (evenly distributed on the tness
scale just for simpli cation purposes) is just = = 5 for a (2,10) ES, and zero for all
others.
Within a GA it largely depends on the scaling function (f), how many o spring are
produced on average by their ancestors. The right-hand part of Figure 5.17 presents two
possible situations. Crosses (+) belong to a steep, triangles (4) to a at reproduction
probability curve (average number of o spring) over the tness of the individuals. In
the former case it typically happens that, just like in ESs, only the best individuals
produce o spring (here the best parent has 6, the second best 3, the third best only 1,
and all others zero o spring). One would call this strong selection. Weak selection, on
the contrary, characterizes the other case (only the worst parent has no o spring, the
best one just 2, and all others 1). It will strongly depend on the actual topology how one
should choose the proportionality factor and it mayeven be necessary to change it during
one optimum seeking process.
Self-adaptation of internal strategy parameters is possible within the framework of
GAs, too. Back (1992a,b, 1993, 1994a,b) has demonstrated this with respect to the
mutation rate. For that purpose he adopts the selection mechanism of the multimembered
ES.
Last but not least, the question remains whether a stochastic or a deterministic approach
to modelling selection is more appropriate. The argument that a stochastic model
is closer to reality, is not su cient for the purpose at hand: optimization and adaptation.
5.3.5 Further Remarks
Of course, one would like to incorporate at least one close-to-canonical GA version into the
comparative test series with all the other optimization procedures. But there are problems
with that kind of endeavor. First, GAs do not permit general inequality constraints.
This does not matter too much, since there are other algorithms that are not applicable
directly in such cases, too. Next, GAs must be provided with lower and upper bounds for
all parameters, which of course have tobechosen to contain the solution, probably in or
near the middle of the hypercube de ned by the explicit bounds. The GA thus would be
provided with information that is not available for the other algorithms.
For all other methods the starting point is of great importance, not only because it