4 Further Remarks

Note: This paper was reconstructed into L A T E X. The formatting is different and there might be errors created during reconstruction. reduce the amount of redundancy generated at each step, Fourier's method becomes practical for a class of parametric linear programs. More recently, [12], [5] and [6] discuss further applications. Acknowledgement: We thank Tien Huynh for his comments. Sometimes these results may be easy to establish once one has a good grasp of the basics of affine geometry and linear programming , but why use such machinery if there is a trivial proof using this simple extension to Fourier's method? For instance it is obvious from theorem 8 that if an initial set of constraints does not contain implicit equalities, none of the sets generated by the elimination of variables will contain implicit equalities. In geometric terms this means that the projections of a full dimensional polyhedral set are also full dimensional. This remark is useful as it helps avoid unnecessary computation in a problem of redundancy removal [9]. In a language like CLP(<), constraints are added incrementally to the solver. There arises the problem of determining the implicit equalities in a set S fcg, formed by adding an inequality c to a set S known not to contain implicit equalities. To avoid doing unnecessary computation one remarks that the new set contains implicit equalities iff c is an implicit equality and, furthermore, if c is an implicit equality then there is at least one implicit equality in S. So one first tests c for equality: if c is not an implicit equality the new set does not contain any; if c is an implicit equality we must search S for others, but with a guarantee that some will be found. This useful property is easy to establish, for instance using the mathematics of the simplex algorithm or using geometric arguments [8], but here the statement is clearly a very trivial consequence of the extension to Fourier's method. In [11] it was proved that if in a system of equalities and inequalities all implicit equalities are replaced by equalities, the resulting system contains at least one redundant equality. The proof is existential and rests on some non-trivial topological lemmas. In [9] it is shown as a direct consequence of theorem 8 that in fact all the equalities that replace implicit equalities are redundant, which leads to a characterization …