Evolution and Optimum Seeking

Multidimensional Strategies 71
speci cation of the gradient directions makes it seem inadvisable to apply nite di erence
methods to approximate the slopes of the objective function. This is taken into account
by some procedures that attempt to construct conjugate directions without knowledge
of the derivatives. The oldest of these was devised by Smith (1962). On the basis of
numerical tests by Fletcher (1965), however, the version of Powell (1964) has proved to
be better. It will be brie y presented here. It is arguable whether it should be counted
as a gradient strategy. Its intermediate position between direct search methods that only
use function values, and Newton methods that make use of second order properties of the
objective function (if only implicitly), nevertheless makes it come close to this category.
The strategy of conjugate directions is based on the observation that a line through the
minimum of a quadratic objective function cuts all contours at the same angle. Powell's
idea is then to construct such special directions by a sequence of line searches. The
unit vectors are taken as initial directions for the rst n line searches. After these, a
minimization is carried out in the direction of the overall result. Then the rst of the old
direction vectors is eliminated, the indices of the remainder are reduced by one and the
direction that was generated and used last is put in the place freed by the nth vector. As
shown by Powell, after n cycles, each ofn +1 line searches, a set of conjugate directions
is obtained provided the objective function is quadratic and the line searches are carried
out exactly.
Zangwill (1967) indicates how this scheme might fail. If no success is obtained in
one of the search directions, i.e., the distance covered becomes zero, then the direction
vectors are linearly dependent and no longer span the complete parameter space. The
same phenomenon can be provoked by computational inaccuracy. To prevent this, Powell
has modi ed the basic algorithm. First of all, he designs the scheme of exchanging
directions to be more exible, actually by maximizing the determinant of the normalized
direction vectors. It can be shown that, assuming a quadratic objective function, it is
most favorable to eliminate the direction in which the largest distance was covered (see
Dixon, 1972a). Powell would also sometimes leave the set of directions unchanged. This
depends on how thevalue of the determinant would change under exchange of the search
directions. The objective function is here tested at the position given by doubling the
distance covered in the cycle just completed. Powell makes the termination of the search
depend on all variables having changed by less than 0:1 " within an iteration, where "
represents the required accuracy. Besides this rst convergence criterion, he o ers a second,
stricter one, according to which the state reached at the end of the normal procedure
is slightly displaced and the minimization repeated until the termination conditions are
again ful lled. This is followed by a line search in the direction of the di erence vector
between the last two endpoints. The optimization is only nally ended when the result
agrees with those previously obtained to within the allowed deviation of 0:1 " for each
component.
The algorithm of Powell runs as follows:
Step 0: (Initialization)
Specify a starting point x (0)
and accuracy requirements "i > 0 for all i =1(1)n.