Evolution and Optimum Seeking

134 Evolution Strategies for Numerical Optimization
Both the (1 , ) and (1+ )schemes were run on the computer for the sphere model,
with n = 100rE = 100, and variable . In each case' was evaluated over 10 000
generations. The resulting data are shown in terms of ' and in Figure 5.9. In
comparison with the approximate theory, deviations are apparent mainly for > opt .
The skewness of the probability distribution w(s 0 ) and the error in the estimate of the
integral R D(y) dy have only a weak e ect in the region of greatest interest, where the
rate of progress is maximum. Furthermore, the results of the simulation fall closer to
the approximate theory if n is taken to be greater than 100 however, the computation
time then becomes excessive. For large values of the possible survival of the parent
only becomes noticeable when the variance is too large to allow rapid convergence. The
greatest di erences, as expected, appear for =1.
On the whole we see that the theory worked out here gives at least a qualitative
account ofthebehavior of the (1 , ) strategy. Amuch more elegant method yielding an
even better approximation may be found in Back, Rudolph, and Schwefel (1993), or Beyer
(1993, 1994a,b).
5.2.2.3 The Corridor Model
As a third and last model objective function, we will now consider the right-angled corridor.
The contours of F (x) in the two dimensional picture (Fig. 5.11) are straight and
parallel, but not necessarily equidistant.
F (x) =c0 +
For the sake of simplifying the calculation we will again give the coordinate system a
particular position and orientation with c1 = ;1 ci = 0 for all i = 2 3:::n: The
right-angled corridor (Problem 2.37, see Appendix A, Sect. A.2){we are using here three
dimensional concepts for the essentially n-dimensional case{is de ned by constraints of
the form
Gj(x) =jxjj b for j = 2(1)n
It has the width 2 b for all coordinate directions xi i = 2(1)n hence the cross section
(2 b) n;1 . As a starting point, the position xE of the parent E, wechoose the origin with
respect to x1 = 0. The useful part of a random step is just its component z1 in the
x1 direction, which is the negative gradient direction. The formulae for w(s` = s 0 ) and
p(s`