Evolution and Optimum Seeking

One Dimensional Strategies 31
F(x)
a
a
a d
d
c
d
c
b
Figure 3.2: Interval division in the Fibonacci search
c
b
b
x
Step k
Step k+1
Step k+2
so that each time only one new value of the objective function needs to be obtained.
Figure 3.2 illustrates two steps of the procedure. The process is continued until k =
N ; 2. At the next division, because f 2 = 2 f 1d (k) and c (k) coincide. A further
interval reduction can only be achieved by slightly displacing one of the test points. The
displacement must be at least big enough for the two objective function values to still
be distinguishable. Then the remaining interval after N trials is of length
`N = 1
fN
(b (0) ; a (0) )+
As ! 0 the e ectiveness tends to f ;1
N . Johnson (1956) and Kiefer (1957) show
that this value is optimal in the sense of the -minimax concept, according to which the
Fibonacci search is the best of all sequential interval division procedures. However, by
taking account ofthe displacement, not only at the last but at all the steps, Oliver and
Wilde (1964) give a recursion formula that for the same number of trials yields a slightly
smaller residual interval. Avriel and Wilde (1966a) provide a proof of optimality. Ifone
has a priori information about the structure of the objective function it can be exploited
to advantage (Gal, 1971) in order to reduce further the number of trials. Overholt (1967a,
1973) suggests that in general there is no a priori information available to x suitably,
and it is therefore better to omit the nal division using a displacement rule and to choose
N one bigger from the start. In order to obtain the minimum with accuracy ">0one