Actual numerical results, are of course, unimportant. Instead, a simple
implementation of the principle described in the last section will be
given. It was found that the idea described above is so simple that
it can even be implemented in Pascal. See the appendix for the
corresponding source code. Of course, the implementation is totally
inefficient, as it has to be the case for a prototype code.
The input parameters as well as the rules have to be given in a file
with name `rules`. The output is standard LaTeX code. This is, at
the moment, the only important output format, since actual calculations with
the generated functions within a computer code are not an issue when one
tries to find efficient algorithms. In our example, the input file looks
like this:

5 2 18 @ + @ @ - \left( @ \right) @ @ \sqrt{\left| @ \right|} \frac{1}{d+\left(@\right)^2} \cos\left(@\right) \sin\left(@\right) @ + c \alpha @ x^{\top} a \Vert x \Vert_2 \Vert x \Vert_1 \Vert x \Vert_{\infty} \Vert x \Vert_p \sum_{i=1}^m w_i \left( @ - a_i \right) \max_{i=1}^m w_i \left( @ - a_i \right) \left| @ \right|^p \sqrt[p]{\left| @ \right|}

As it can be seen, we employ the recursive rules exactly five times,
we consider problems with two unknowns, and we have specified eighteen
rules. The first rule given takes the form

while, e. g., the last one has the form

i. e. the place holder @ is replaced by .
Of course, more or completely different rules can be specified
at will. Note also that several rules are able to stop the recursive
process prematurely. Moreover, as it can be seen in the input file displayed
above, the rules used here are rather locationally oriented, i. e. they mimic
problems typically encountered in location science. One of the first
applications of the program produced the following results:

{

f(x_1, x_2) :=\alpha \sin\left( \Vert x \Vert_{\infty} + x_{1} + c\right)

f(x_1, x_2) :=\sqrt[p]{\left| \sqrt{\left| \max_{i=1}^m w_i \left( x_{2} + c^{\top} a - a_i \right) \right|} \right|}

f(x_1, x_2) :=\sum_{i=1}^m w_i \left( \sqrt{\left| \left| \sum_{i=1}^m w_i \left( x_{1} - a_i \right) \right|^p \right|} - a_i \right)\Vert x \Vert_2

f(x_1, x_2) :=\sum_{i=1}^m w_i \left( \sqrt{\left| x_{2} \sum_{i=1}^m w_i \left( x_{1} + x_{2} - a_i \right) \right|} - a_i \right)}

These outputs translate readily to

and

The author firmly believes that none of these functions has ever been considered in location science. It it perhaps surprising that a comparatively simple strategy as the one proposed here can make such a significant contribution to research. After such a generation process, all that remains is to apply a standard optimization strategy and to invent an application with a corresponding model for the objective function. Then, a new paper has been finished.

The present implementation uses randomly chosen rules and recursion symbols. As a consequence, the results are somewhat arbitrary. Of course, it is relatively simple to replace this random element by a deterministic procedure. In this way, a complete list of all problems composed with the specified rules and with a fixed recursion depth can be generated. When the rules are chosen to reflect all functions appearing in a specific application field, one could therefore generate all objective functions which might occur in this field. After the generation process, these function can then be classified according to some criterion.

Tue Dec 21 21:12:32 CET 1999