Evolution and Optimum Seeking

214 Comparison of Direct Search Strategies for Parameter Optimization
Table 6.8: Summary of the results from Table 6.7
Total number of problems
solved with accuracy class No solution
Strategy 10 ;38 10 ;8 10 ;4 10 ;2 or >10 ;2
ROSE 4 4 20 20 2
COMP 6 11 18 18 4
EVOL 10 12 14 19 3
GRUP 10 13 17 20 2
REKO 16 17 19 20 2
such edges takes considerable e ort and time. In Figure 6.17 the situation is illustrated
for the case of two variables and one constraint.
The contours of the objective function run at a narrow angle to the boundary of
the region. Foramutation to count as successful it must fall within the feasible region
as well as improve the objective function value. For simplicity let us assume that all the
mutations fall on the circumference of a circle about the current starting point. In the case
of many variables this point ofviewisvery reasonable (see Chap. 5, Sect. 5.1). To start
with the center of the circle (P 1) will still lie some way from the boundary. If the angle
between the contours of the objective function and the edge of the feasible region is small
and the step size, or variance of the mutation step size, is large then only a small fraction
of the mutations will be successful (thickly drawn part of the circle 1). The 1=5 success
rule ensures that this fraction is raised to 20%, which if the angle is small enough can
only be achieved by reducing the variance to 2. The search point P is driven closer and
closer to the boundary and eventually lies on it (P 2). Since there is no longer any nite
step size that can provide a su ciently large success rate, the variance is permanently
reduced to the minimum value speci ed in the program. Depending on the particular
problem structure and the chosen values of the parameters in the convergence criteria the
search is either slowly continued or it is terminated before reaching the optimum. The
more constraints become active during the search, the smaller is the probability that the
objective will be closely approached. In fact, even in problems with only two variables
and one constraint (Problem 2.46) the angle between the contours and the edge of the
feasible region can become vanishingly small in the neighborhood of the minimum.
Similar situations to the one depicted in Figure 6.17 can even arise in unconstrained
problems if the objective function displays discontinuities in its rst partial derivatives.
Examples of this kind of behavior are provided by Problems 2.6 and 2.21. If only a
few variables are involved there is still a good chance of reaching the objective. Other
search methods, especially those which execute line searches, are generally defeated by
such points of discontinuity.
The multimembered evolution strategy, although it works without a rigid step length
adaptation, also loses its otherwise reliable convergence characteristics when the region of