Evolution and Optimum Seeking

118 Evolution Strategies for Numerical Optimization
property (see Rechenberg, 1973) of the evolution strategy can only be proved under the
condition of constant step lengths. With the introduction of the success rule, it is lost,
or to be more precise: the probability of nding the global minimum among several
local minima decreases continuously as a local minimum is approached with continuous
reduction in the step lengths. Rapid convergence and reliable global convergence behavior
are two contradictory requirements. They cannot be reconciled if one has absolutely no
knowledge of the topology of the objective function. The 1=5 success rule is aimed at high
convergence rates. If several local optima are expected, it is thus advisable to keep the
(0)
variances large and constant, or at least to start with large i and perhaps to require a
lower success probability than 1/5. This measure naturally costs extra computation time.
Once one is sure of having located a point near the global extremum, the accuracy can be
improved subsequently in a follow-up computation. For more sophisticated investigations
of the global convergence see Born (1978), Rappl (1984), Scheel (1985), Back, Rudolph,
and Schwefel (1993), and Beyer (1993).
5.2 A Multimembered Evolution Strategy
While the simple, two membered evolution strategy is successful in application to many
optimization problems, it is not a satisfactory method of solving certain types of problem.
As we have seen,by following the 1=5 success rule, the step lengths can be permanently
reduced in size without thereby improving the rate of progress. This phenomenon occurs
frequently if constraints become active during the search, and greatly reduce the size of
the success scoring region. A possible remedy would be to alter the probability distribution
of the random steps in such away astokeep the success probability su ciently
large. To do so the standard deviations i would have to be individually adjustable.
The contour surfaces of equal probability could then be stretched or contracted along
the coordinate axes into ellipsoids. Further possibilities for adjustment would arise if the
random components were allowed to depend on each other. For an arbitrary quadratic
problem the rate of convergence of the sphere model could even be achieved if the random
changes of the individual variables were correlated so as to make the regression line of
the random vector run parallel to the concentric ellipsoids F (x) =const:, which now lie
at some angle in the space. To put this into practice, information about the topology
of the objective function would have to be gathered and analyzed during the optimum
search. This would start to turn the evolution strategy into something resembling one
of the familiar deterministic optimization methods, as Marti (1980) and recently again
Ostermeier (1992) have done this is contrary to the line pursued here, which is to apply
biological evolution principles to the numerical solution of optimization problems. Following
Rechenberg's hypothesis, construction of an improved strategy should therefore be
attempted by taking into account further evolution principles.
5.2.1 The Basic Algorithm
When the ground rules of the two membered evolution strategy were formulated in the
language of biology, reference was to one parent and one o spring the basic population