Stochastic Programming - Index of

274 STOCHASTIC PROGRAMMING
direction, is found to be in pos W , we can halve the number of points in A.
In some cases we may therefore reduce the testing to one single problem.
It is of importance to understand that the reduction in the size of A has
two positive aspects. First, if we do not have (or do not know that we have)
relatively complete recourse, the test for feasibility, and therefore generation of
feasibility cuts, becomes much easier. But equally important is the fact that
it tells us something about our problem. If a column from H is in pos W ,
we have found a direction in which we can move as far as we want without
running into feasibility problems. This will, in a real setting, say something
about the random effect we have modelled using that column.
5.4 Bibliographical Notes
Preprocessing and similar procedures have been used in contexts totally
different from ours. This is natural, since questions of model formulations and
infeasibilities are equally important in all areas of mathematical programming.
For further reading, consult e.g. Roodman [7], Greenberg [3, 4, 5] or Chinneck
and Dravnieks [1].
An advanced algorithm for finding frames can be found in Wets and
Witzgall [12]. Later developments include the work of Rosen et al. [8] and Dulá
et al. [2]. The algorithm for finding a support was described by Tschernikov[9],
and later also by Wets [11]. For computational tests using the procedure see
Wallace and Wets [10]. Similar procedures for networks will be discussed in
Chapter 6.
For an overview of methods for extreme point enumeration see e.g.
MattheissandRubin[6].
Exercises
1. Let W be the coefficient matrix for the following set of linear equations:
x + 1 2 y − z + s 1 =0,
2x + z + s 2 =0,
x, y, z, s 1 , s 2 ≥ 0.
(a) Find a frame of pos W .
(b) Draw a picture of pos W , and find the generators of pol pos W by
simple geometric arguments.
(c) Find the generators of pol pos W by using procedure support in
Figure 5. Make sure you draw the cones pos W and pol pos W after
each iteration of the algorithm, so that you see how it proceeds.