Page 158 - Compact Numerical Methods For Computers

Optimisation and nonlinear equations 147

pointed out, however, starting points can be close to the desired solution without
guaranteeing convergence to that solution. They found that certain problems in
combination with certain methods have what they termed magnetic zeros to which
the method in use converged almost regardless of the starting parameters emp-
loyed. However, I did not discover this ‘magnetism’ when attempting to solve the
cubic-parabola problem of Brown and Gearhart using a version of algorithm 23.
In cases where one root appears to be magnetic, the only course of action once
several deflation methods have been tried is to reformulate the problem so the
desired solution dominates. This may be asking the impossible!
Another approach to ‘global’ minimisation is to use a pseudo-random-number
generator to generate points in the domain of the function (see Bremmerman
(1970) for discussion of such a procedure including a FORTRAN program). Such
methods are primarily heuristic and are designed to sample the surface defined by
the function. They are probably more efficient than an n-dimensional grid search,
especially if used to generate starting points for more sophisticated minimisation
algorithms. However, they cannot be presumed to be reliable, and there is a lack
of elegance in the need for the shot-gun quality of the pseudo-random-number
generator. It is my opinion that wherever possible the properties of the function
should be examined to gain insight into the nature of a global minimum, and
whatever information is available about the problem should be used to increase
the chance that the desired solution is found. Good starting values can greatly
reduce the cost of finding a solution and greatly enhance the likelihood that the
desired solution will be found.