Stochastic Programming - Index of

266 STOCHASTIC PROGRAMMING
pos W
pol pos W
Figure 3 Finding the generators of pol pos W .
There is another important aspect of the polar cone pos W ∗ that we have
not yet discussed. It is indicated in Figure 3 by showing that the generators
are pairwise normals. However, that is slightly misleading, so we have to turn
to a three-dimensional figure to understand it better. We shall also need the
term facet. Let a cone pos W have dimension k. Then every cone K positively
spanned by k−1 generators from pos W , such that K belongs to the boundary
of pos W , is called a facet. Consider Figure 4.
What we note in Figure 4 is that the generators are not pairwise normals,
but that the facets of one cone have generators of the other as normals. This
goes in both directions. Therefore, when we state that h ∈ pos W if and only
if h T y ≤ 0 for all generators of pol pos W , we are in fact saying that either h
represents a feasible problem because it is a linear combination of columns in
W or because it satisfies the inequality implied by the facets of pos W . In still
other words, the point of finding W ∗ is not so much to describe a new cone,
but to replace the description of pos W in terms of generators with another
in terms of inequalities.
This is useful if the number of facets is not too large. Generally speaking,
performing an inner product of the form b T y is very cheap. In parallel
processing, an inner product can be pipelined on a vector processor and the
different inner products can be done in parallel. And, of course, as soon as we
find one positive inner product, we can stop—the given recourse problem is
infeasible.
Readers familiar with extreme point enumeration will see that going from