Evolution and Optimum Seeking

Multidimensional Strategies 39
the second variable. Both end results are then used to reject one of the values of the
rst variable that were held constant, and to reduce the size of the interval with respect
to this parameter. By analogy, a three dimensional minimization consists of a recursive
sequence of two dimensional Fibonacci searches. If the number of function calls to reduce
the uncertainty interval [aibi] su ciently with respect to the variable xi is Ni, then the
total number N also obeys Equation (3.19). The advantage compared to the grid method
is simply that Ni depends logarithmically on the ratio of initial interval size to accuracy
(see Equation (3.11)). Aside from the fact that each variable must be suitably xed in
advance, and that the unimodality requirement of the objective function only guarantees
that local minima are approached, there is furthermore no guarantee that a desired
accuracy will be reached within a nite number of objective function calls (Kaupe, 1964).
Other elimination procedures have been extended in a similar way tothemultivariable
case, such as, for example, the dichotomous search (Wilde, 1965) and a sequential boxingin
method (Berman, 1969). In each case the e ort rises exponentially with the number
of variables. Another elimination concept for the multidimensional case, the method of
contour tangents, is due to Wilde (1963) (see also Beamer and Wilde, 1969). It requires,
however, the determination of gradient vectors. Newman (1965) indicates how to proceed
in the two dimensional case, and also for discrete values of the variables (lattice search).
He requires that F (x) beconvex and unimodal. Then the cost should only increase
linearly with the number of variables. For n 3, however, no applications of the contour
tangent method are as yet known.
Transferring interpolation methods to the n-dimensional case means transforming the
original minimum problem into a series of problems, in the form of a set of equations to
be solved. As non-linear equations can only be solved iteratively, this procedure is limited
to the special case of linear interpolation with quadratic objective functions. Practical
algorithms based on the regula falsi iteration can be found in Schmidt and Schwetlick
(1968) and Schwetlick (1970). The procedure is not widely used as a minimization method
(Schmidt and Vetters, 1970). The slopes of the objective function that it requires are
implicitly calculated from function values. The secant method described by Wolfe (1959b)
for solving a system of non-linear equations also works without derivatives of the functions.
From n +1 current argument values, it extracts the required information about the
structure of the n equations.
Just as the transition from simultaneous to sequential one dimensional search methods
reduces the e ort required at the expense of global convergence, so each further
acceleration in the multidimensional case is bought by a reduction in reliability. High
convergence rates are achieved by gathering more information and interpreting it in the
form of a model of the objective function. If assumptions and reality agree, then this
procedure is successful if they do not agree, then extrapolations lead to worse predictions
and possibly even to abandoning an optimization strategy. Figure 3.4 shows the contour
diagram of a smooth two parameter objective function.
All the strategies to be described assume a degree of smoothness in the objective
function. They do not converge with certainty to the global minimum but at best to one
of the local minima, or sometimes only to a saddle point.