Evolution and Optimum Seeking

Particular Problems and Methods of Solution 7
The algorithm of Rechenberg's evolution strategy will be treated in detail in Chapter 5.
In the experimental eld it has often been applied successfully: for example, to the solution
of multiparameter shaping problems (Rechenberg, 1964 Schwefel, 1968 Klockgether and
Schwefel, 1970). All variables are simultaneously changed by a small random amount.
The changes are (binomially or) normally distributed. The expected value of the random
vector is zero (for all components). Failures leave the starting condition unchanged,
only successes are adopted. Stochastic disturbances or perturbations, brought about by
errors of measurement, do not a ect the reliability but in uence the speed of convergence
according to their magnitude. Rechenberg (1973) gives rules for the optimal choice of a
common variance of the probability density distribution of the random changes for both
the unperturbed and the perturbed cases.
The EVOP method of G. E. P. Boxchanges only two or three parameters at a time{if
possible those which have the strongest in uence. A square or cube is constructed with
an initial condition at its midpoint its 2 2 = 4 or 2 3 = 8 corners represent the points in
a cycle of trials. These deterministically established states are tested sequentially, several
times if perturbations are acting. The state with the best result becomes the midpoint
of the next pattern of points. Under some conditions, one also changes the scaling of
the variables or exchanges one or more parameters for others. Details of this altogether
heuristic way of proceeding are described by Box and Draper (1969, 1987). The method
has mainly been applied to the dynamic optimization of chemical processes. Experiments
are performed on the real system, sometimes over a period of several years.
The counterpart to experimental optimization is mathematical optimization. The
functional relation between the criterion of merit or quality and the variables is known,
at least approximately to put it another way, a more or less simpli ed mathematical
model of the object, process or system is available. In place of experiments there appears
the manipulation of variables and the objective function. It is sometimes easy to set up
a mathematical model, for example if the laws governing the behavior of the physical
processes involved are known. If, however, these are largely unresearched, as is often the
case for ecological or economic processes, the work of model building can far exceed that
of the subsequent optimization.
Depending on what deliberate in uence one can have on the process, one is either
restricted to the collection of available data or one can uncover the relationships between
independent and dependent variables by judiciously planning and interpreting tests. Such
methods (Cochran and Cox, 1950 Kempthorne, 1952 Davies, 1954 Cox, 1958 Fisher,
1966 Vajda, 1967 Yates, 1967 John, 1971) were rst applied only to agricultural problems,
but later spread into industry. Since the analyst is intent on building the best
possible model with the fewest possible tests, such an analysis itself constitutes an optimization
problem, just as does the synthesis that follows it. Wald (1966) therefore
recommends proceeding sequentially, that is to construct a model as a hypothesis from
initial experiments or given a priori information, and then to improve it in a stepwise
fashion by a further series of tests, or, alternatively, to sometimes reject the model completely.
The tting of model parameters to the measured data can be considered as an
optimization problem insofar as the expected error or maximum risk is to be minimized.
This is a special case of optimization, called calculus of observations ,whichinvolves sta-