Stochastic Programming - Index of

NETWORK PROBLEMS 295
0
2
1
1
(0,4)
(0,3)
2
(0,3)
1
3
2
3
3
4 6
4
2
(0,3)
(0,6)
8
1
(0,4)
4 7 5
5 -1
2
1
(0,7)
2
(0,1)
Slack
3
Figure 12 Example network with arc capacities and external flows
corresponding to the Jensen lower bound.
number of points. If there are k random variables, we must work with 2 k
points. This means that with more than about 10 random variables we are
not in business. Thus, since there are 11 random variable in the example, we
must solve 2 11 problems to find the upper bound. We have not done that here.
In what follows, we shall demonstrate how to obtain a piecewise linear upper
bound that does not exhibit this exponential characterization. A weakness of
this bound is that it may be +∞ even if the problem is feasible. That may
not happen to the Edmundson–Madansky upper bound. We shall continue to
use the network in Figure 11 to illustrate the ideas.
6.5.1 Piecewise Linear Upper Bounds
Let us illustrate the method in a simplified setting. Define φ(ξ,η) by
φ(ξ,η) =min{q T y | W ′ y = b + ξ, 0 ≤ y ≤ c + η},
y
where all elements of the random vectors ˜ξ = (˜ξ 1 T , ˜ξ 2 T ,...) T and ˜η =
(˜η 1 T, ˜ηT 2 ,...)T are mutually independent. Furthermore, let the supports be
given by Ξ(˜ξ) =[A, B] andΞ(˜η) =[0,C]. The matrix W ′ is the node–arc
incidence matrix for a network, with one row removed. That row represents
the slack node. The external flow in the slack node equals the negative sum of
the external flows in the other nodes. The goal is to create an upper bounding
function U(ξ,η) that is piecewise linear, separable and convex in ξ, aswellas
easily integrable in η:
U(ξ,η) =φ(E ˜ξ,0) + H(η)+ ∑ i
{
d
+
i (ξ i − E ˜ξ i ) if ξ i ≥ E ˜ξ i ,
d − i (E ˜ξ i − ξ i ) if ξ i