Evolution and Optimum Seeking

One Dimensional Strategies 37
F
P
F(x)
a (k)
Minimum P(x)
P(x)
d (k) b (k)
(k+1) (k+1) (k+1)
a b c
Figure 3.3: Lagrangian quadratic interpolation
the objective function and the trial function. In the most favorable case the objective
function is also quadratic. Then one iteration is su cient. This is why it can be advantageous
to use an interpolation method rather than an interval division method such as
the optimal Fibonacci search. Dijkhuis (1971) describes a variant of the basic procedure
in which four argument values are taken. The two inner ones and each of the outer ones
in turn are used for two separate quadratic interpolations. The weighted mean of the two
results yields a new iteration point. This procedure is claimed to increase the reliability
of the minimum search for non-quadratic objective functions.
3.1.2.3.4 Hermitian Interpolation. If one chooses, instead of a parabola, a third
order polynomial as a test function, more information is needed to make it agree with the
objective function. Beveridge and Schechter (1970) describe such acubic interpolation
procedure. In place of four argument values and associated objective function values, two
points a (k) and b (k) are enough, if, in addition to the values of the objective function, values
of its slope, i.e., the rst order di erentials, are available. This Hermitian interpolation is
mainly used in conjunction with gradient or quasi-Newton methods, because in any case
they require the partial derivatives of the objective function, or they approximate them
using nite di erence methods.
The interpolation formula is:
c (k) = a (k) +(b (k) ; a (k) )
w ; Fx(a (k) ) ; z
2 w + Fx(b (k) ) ; Fx(a (k) )
c (k)
x