Evolution and Optimum Seeking

The Two Membered Evolution Strategy 117
x 2
To the
optimum
x
1
Circle : line of equal probability density
Bold segment : fraction where success can be scored
x 2
Figure 5.3: Failure of the 1/5 success rule
Forbidden
region
Optimum
x
1
calculated by the above rule will be very di erent from that associated with the same
step length if an average over the corridor cross section were taken. If now, on the basis
of this low estimate of the success probability, the step length is further reduced, there
is a corresponding decrease in the probability of escaping from the edge of the corridor.
It would therefore be desirable in this special case to average the probability of success
over a longer time period. Opposed to this, however, is the requirement from the sphere
model that the step lengths should be adjusted to the topology as directly as possible.
The present subroutine o ers several means of dealing with the problem. For example,
the lower bounds on the variances (variables EA, EB in the subprogram EVOL) can be
chosen to be relatively large, or the number of mutations (the variable LS) after which
the convergence criterion is tested can be altered by the user. The user has besides a free
choice with regard to the required probability of success (variable LR) and the multiplier
of the variance (variable SN). The rate of change of the step lengths, given by the factor
0:85 per n mutations, was xed on the basis of the sphere model. It is not ideal for all
types of problems but rather in the nature of a lower bound. If it seems reasonable to
operate with constant variances, the parameter in question should be set equal to one.
An indication of a suitable choice for the initial step lengths (variable array SM) can be
obtained from Equation (5.4). Since r increases as the root of the number of parameters,
one is led to set
(0)
i
= 4xi
pn
in which 4xi is a rough measure of the expected distance from the optimum. This does
not actually give the optimal step length because r is a kind of local scale of curvature of
the contours of the objective function. However, it does no harm to start with variances
that are too large they will quickly be reduced to a suitable size by the1=5 success rule.
During this transition phase there is still a chance of escaping from the neighborhood
of a merely local optimum but very little chance afterwards. The global convergence