Evolution and Optimum Seeking

76 Hill climbing Strategies
s (k) = 1 may lead to a point withaworse value of the objective function. The
search diverges, e.g., when r 2 F (x (k) ) is not positive-de nite.
It can happen that r 2 F (x (k) ) is singular or almost singular. The Hessian matrix
cannot be inverted.
Furthermore, it depends on the starting point x (0) whether a minimum, a maximum,
or a saddle point is approached, or the whole iteration diverges. The strategy itself does
not distinguish the stationary points with regard to type.
If the method does converge, then the convergence is better than of linear order
(Goldstein, 1965). Under certain, very strict conditions on the structure of the objective
function and its derivatives even second order convergence can be achieved (e.g., Polak,
1971) that is, the number of exact signi cant gures in the approximation to the minimum
solution doubles from iteration to iteration. This phenomenon is exhibited in the solution
of some test problems, particularly in the neighborhood of the desired extremum.
All the variations of the basic procedure to be described are aimed at increasing the
reliability of the Newton iteration, without sacri cing the high convergence rate. A distinction
is made here between quasi-Newton strategies, which donotevaluate the Hessian
matrix explicitly, andmodi ed Newton methods, for which rst and second derivatives
must be provided at each point. The only strategy presently known which makes use of
higher than second order properties of the objective function is due to Biggs (1971, 1973).
The simplest modi cation of the Newton-Raphson scheme consists of determining the
step length s (k) by a line search in the Newton direction v (k) (Equation (3.28)) until the
relative optimum is reached (e.g., Dixon, 1972a):
F (x (k) + s (k) v (k) ) = min
s fF (x (k) + sv (k) )g (3.30)
To save computational operations, the second partial derivatives can be redetermined
less frequently and used for several iterations. Care must always be taken, however, that
v (k) always points \downhill," i.e., the angle between v (k) and ;rF (x (k) ) is less than
90 0 . The Hessian matrix must also be positive-de nite. If the eigenvalues of the matrix
are calculated when it is inverted, their signs show whether this condition is ful lled.
If a negative eigenvalue appears, Pearson (1969) suggests proceeding in the direction of
the associated eigenvector until a point isreached with positive-de nite r 2 F (x). Greenstadt
(1967a) simply replaces negative eigenvalues by their absolute value and vanishing
eigenvalues by unity. Other proposals have been made to keep the Hessian matrix positivede
nite by addition of a correction matrix (Goldfeld, Quandt, and Trotter, 1966, 1968
Shanno, 1970a) or to include simple gradient steps in the iteration scheme (Dixon and
Biggs, 1972). Further modi cations, which operate on the matrix inversion procedure
itself, have beensuggestedby Goldstein and Price (1967), Fiacco and McCormick (1968),
and Matthews and Davies (1971). A good survey has been given by Murray (1972b).
Very few algorithms exist that determine the rst and second partial derivatives numerically
from trial step operations (Whitley, 1962 see also Wasscher, 1963c Wegge,
1966). The inevitable approximation errors too easily cancel out the advantages of the
Newton directions.