Evolution and Optimum Seeking

Multidimensional Strategies 73
otherwise set x (0) = y (3) v (0)
1
v (0)
i
= v (k)
i
= y (3) ; y (1)
for i = 2(1)n, k = 0, and go to step 2.
Figure 3.9 illustrates a few iterations for a hypothetical two parameter function. Each
of the rst loops consists of n +1 = 3 line searches and leads to the adoption of a new
search direction. If the objective function had been of second order, the minimum would
certainly have been found by the last line search of the second loop. In the third and
fourth loops it has been assumed that the trial steps have led to a decision not to exchange
directions, thus the old direction vectors, numbered v 3 and v 4 are retained. Further loops,
e.g., according to step 9, are omitted.
The quality of the line searches has a strong in uence on the construction of the
conjugate directions. Powell uses a sequence of Lagrangian quadratic interpolations. It is
terminated as soon as the required accuracy is reached. For the rst minimization within
an iteration three points and Equation (3.16) are used. The argument values taken in
direction vi are: x (the starting point), x + si vi, and either x +2si vi or x ; si vi,
according to whether F (x + si vi)