Evolution and Optimum Seeking

Genetic Algorithms 151
too is the case of the starting point x (0) lying outside the feasible region.
Whereas the subroutine GRUP with option REKO has been taken into account in
the test series of Chapter 6, this is not so for the third version KORR, which was created
later (Schwefel, 1974). Still, more often than any multimembered version, the (1+1)
strategy has been used in practice. Nonetheless it has proved its usefulness in several
applications: for example, in conjunction with a linearization method for minimizing
quadratic functions in surface tting problems (Plaschko andWagner, 1973). In this case
the evolution process provides useful approximate values that enable the deterministic
method to converge. It should also serve to locate the global minimum of the multimodal
objective function. Another practically oriented multiparameter case was to nd the
optimum weight disposition of lightweight rigidly jointed frameworks (Ho er, Ley ner,
and Wiedemann, 1973 Ley ner, 1974). Here again the evolution strategy is combined
with another method, this time the simplex method of linear programming. Each strategy
is applied in turn until the possible improvements remaining at a step are very small.
The usefulness of this procedure is demonstrated by checking against known solutions.
A third example is provided by Hartmann (1974), who seeks the optimal geometry of
a statically loaded shell support. He parameterizes the functional optimization problem
by assuming that the shape of the cross section of the cylindrical shell is described by a
suitable polynomial. Its coe cients are to be determined such that the largest absolute
value of the transverse moment is as small as possible. For various cases of loading,
Hartmann nds optimal shell geometries di ering considerably from the shape of circular
cylinders, with sometimes almost vanishingly small transverse moments. More examples
are mentioned in Chapter 7.
5.3 Genetic Algorithms
At almost the same time that evolution strategies (ESs) were developed and used at the
Technical University of Berlin, two other lines of evolutionary algorithms (EAs) emerged
in the U.S.A., all independently of each other. One of them, evolutionary programming
(EP), was mentioned at the end of Chapter 4 and goes back to the workofL.J.Fogel
(1962 see also Fogel, Owens, and Walsh, 1965, 1966a,b). For a long time, activity on this
front seemed to have become quiet. However, in 1992 a series of yearly conferences was
started by D.B.Fogel and others (Fogel and Atmar, 1992, 1993 Sebald and Fogel, 1994)
to disseminate recent results on the theory and applications of EP. Since EP uses concepts
that are rather similar to either ESs or genetic algorithms (GAs) (Fogel, 1991, 1992), it
will not be described in detail here, nor will it be compared to ESs on the basis of test
results. This was done in a paper presented at the second EP conference (Back, Rudolph,
and Schwefel, 1993). Similarly, contributions to comparing ESs and GAs in detail may
be found in Ho meister and Back (1990, 1991, 1992 see also Back, Ho meister, and
Schwefel, 1991 Back andSchwefel, 1993).
The third line of EAs mentioned above, genetic algorithms, has become rather popular
today and di ers from the others in several aspects. This approach will be explained in
the following according to its classical (also called canonical) form.
Even to attentive scientists, GAs did not become apparent before 1975 when the rst