Stochastic Programming - Index of

288 STOCHASTIC PROGRAMMING
a solution, the column is not part of the frame, and can be removed. An
important property of this procedure is that to determine if a column can be
discarded, we have to use all other columns in the test. This is a major reason
why procedure framebylp is so slow when the number of columns gets very
large.
So, a generator w ∗ of the cone pos W ∗ has the property that a right-hand
side h must satisfy h T w ∗ ≤ 0 to be feasible. In the uncapacitated network
case we saw that a right-hand side β had to satisfy b(Y ) T β ≤ 0torepresenta
feasible problem. Therefore the index vector b(Y ) corresponds exactly to the
column w ∗ . And calling procedure framebylp to remove those columns that
are not in the frame of the cone pos W ∗ corresponds to using Proposition 6.5.
Therefore the index vector of a node set from Proposition 6.5 corresponds to
the columns in W ∗ .
Computationally there are major differences, though. First, to find a
candidate for W ∗ , we had to start out with W , and use procedure support,
which is an iterative procedure. The network inequalities, on the other hand,
are produced more directly by looking at all subsets of nodes. But the
most important difference is that, while the use of procedure framebylp,
as just explained, requires all columns to be available in order to determine if
one should be discarded, Proposition 6.5 is totally local. We can pick up
an inequality and determine if it is needed without looking at any other
inequalities. With possibly millions of candidates, this difference is crucial.
We did not develop the LP case for explicit bounds on variables. If such
bounds exist, they can, however, be put in as explicit constraints. If so, a
column w ∗ from W ∗ corresponds to the index vector
(
b(Y )
−a(Q + )
)
.
6.3 Generating Relatively Complete Recourse
Let us now discuss how the results obtained in the previous section can help
us, and how they can be used in a setting that deserves the term preprocessing.
Let us first repeat some of our terminology, in order to see how this fits in
with our discussions in the LP setting.
A two-stage stochastic linear programming problem where the second-stage
problem is a directed capacitated network flow problem can be formulated as
follows:
[
min x c T x + Q(x) ]
s.t. Ax = b, x ≥ 0,
where
Q(x) = ∑ Q(x, ξ j )p j