Evolution and Optimum Seeking

The Two Membered Evolution Strategy 113
the quantity eventually becomes zero after a nite number of divisions. Every subsequent
multiplication leaves the value as zero. If it happens to one of the standard deviations i,
the a ected variable xi remains constant thereafter. The optimization continues only in a
subspace of IR n .To guard against this it must be required that i > 0 for all i = 1(1)n.
The random changes should furthermore be su ciently large that at least the last stored
place of a variable is altered. There are therefore two requirements:
Lower limits for the \step lengths":
and
where
"a > 0
1+"b > 1
(g)
i "a for all i =1(1)n
(g)
i "b x (g)
i for all i = 1(1)n
)
according to the computational accuracy
It is thereby ensured that the random variations are always active and the region of the
search stays spanned in all dimensions.
5.1.3 The Convergence Criterion
In experimental optimization it is usually decided heuristically when to terminate the
series of trials: for example, when the trial results indicate that no further signi cant
improvement can be gained. One always has an overall view of how the experiment is
running. In numerical optimization, if the calculations are made by computer, one must
build into the program a rule saying when the iteration sequence is to be terminated. For
this purpose objective, quantitative criteria are needed that refer to the data available at
any time. Sometimes, although not always, one will be concerned to obtain a solution as
exactly as possible, i.e., accurate to the last stored digit. This requirement can relate to
the variables or to the objective function. Remember that the optimum may beaweak
one.
Towards the minimum, the step lengths and distances covered normally become
smaller and smaller. A frequently used convergence criterion consists of ending the search
when the changes in the variables become zero (in which case no further improvementin
the objective function is made), or when the step lengths have become zero. As a rule one
sets the lower bound not to zero but to a su ciently small, nite value. This procedure
has however one disadvantage that can be serious. Small step lengths occur not only if
the minimum is nearby, but also if the search ismoving through a narrow valley. The
optimization may then be practically halted long before the extreme value being sought is
reached. In Equations (5.4) and (5.5), r can equally well be thought of as the local radius
of curvature. Neither ', the distance covered, nor , the step length, are a measure of the
closeness to the optimum. Rather they convey information about the complexity ofthe
minimum problem: the number of variables and the narrowness of valleys encountered.
The requirement >"or kx (g) ; x (g;1) k >"for the continuation of the search isthus
no guarantee of su cient convergence.