Evolution and Optimum Seeking

One Dimensional Strategies 29
The boxing-in method has also been proposed occasionally as a one dimensional optimization
strategy (Rosenbrock, 1960 Berman, 1966) in its own right. In order not to
waste too many trials far from the target when the accuracy requirement isvery high, it
is useful to start with relatively large steps. Each time a loop ends with a failure the step
length is reduced by a factor less than 0:5, e.g., 0:25. If the above rules for increasing
and reducing the step lengths are combined, a very exible procedure is obtained. Dixon
(1972a) calls it the success/failure routine. If a starting interval [a (0) b (0) ] is already at
hand, however, there are signi cantly better strategies for successively reducing the size
of the interval.
3.1.2.2 Interval Division Methods
If an equidistant division method is applied repeatedly, the interval of uncertainty is
reduced at each stepby the same factor ,andthus for k steps by k . This exponential
progression is considerably stronger than the linear dependence of the value of on the
number of trials per step. Thus as few simultaneous trials as possible would be used. A
comparison of two schemes, with two and three simultaneous trials, shows that except in
the rst loop, only two new objective function values must be obtained at a time in both
cases, since of three trial points in one step, one coincides with a point of the previous
step. The total number of trials required with sequential application of the equidistant
three point scheme is
1+
2 log b;a
"
log 2