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# Numerical Results

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)
}

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.

Next: Outlook Up: A Problem Generator Previous: The Problem Generator

Joerg Fliege
Tue Dec 21 21:12:32 CET 1999