Stochastic Programming - Index of

200 STOCHASTIC PROGRAMMING
The minimal value for the pair UL:UU is therefore 16.643.
If we were to pick the pair with the largest minimum of α and β, we should
pick the pair UL:UU, over which it is ξ 2 that varies. In such a case we have
tried to find that part of the function that is the most nonlinear. When we
look at Figure 21, we see that as ξ 2 increases (with ξ 1 = 20), the optimal
solution moves from F to E and then to D, where it stays when ξ 2 comes
above the y coordinate in D. It is perhaps not so surprising that this is the
most serious nonlinearity in φ.
If we try to find the random variable with the highest average nonlinearity,
by summing the errors over those pairs for which the given random variable
varies, we find that for ˜ξ 1 the sum is 9.5+15.143 = 24.643, and for ˜ξ 2 it is
8+16.643, which also equals 24.643. In other words, we have no conclusion.
The next approach we suggested was to look at the dual variables as in
(5.1). The right-hand side structure is very simple in our example, so it is
easy to find the connections. We define two random variables: ˜π 1 for the row
constraining x, and˜π 2 for the row constraining y. With the simple kind of
uniform distributions we have assumed, each of the four values for ˜π 1 and ˜π 2
will have probability 0.25. Using Table 1, we see that the possible values for
˜π 1 are 0, 1 and 3 (with 0 appearing twice), while for ˜π 2 they are 0, 2 and 3.5
(also with 0 appearing twice). There are different ideas we can follow.
1. We can find out how the dual variables vary between the extreme points.
The largest individual change is that ˜π 2 fallsfrom3.5to0aswegofromUL
to UU. This should again confirm that ˜ξ 2 is a candidate for partitioning.
2. We can calculate E˜π =(1, 11 8
), and the individual variances to 1.5 and
2.17. If we choose based on variance, we pick ˜ξ 2 .
3. We also argued earlier that the size of the support was of some importance.
A way of accommodating that is to multiply all outcomes with the length
of the support. (That way, all dual variables are, in a sense, a measure of
change per total support.) That should make the dual variables comparable.
The calculations are left to the reader. We now end up with ˜π 1 having the
largest variance. (And if we now look at the biggest change in dual variable
over pairs of neighboring extreme points, ˜ξ 1 will be the one to partition.)
No conclusions should be made based on these numbers in terms of what
is a good heuristic. We have presented these numbers to illustrate the
computations and to indicate how it is possible to make arguments about
partitioning. Before we conclude, let us consider the “look-ahead” strategy
(5.2). In this case there are two possibilities: either we split at ξ 1 =10orwe
split at ξ 2 = 5. If we check what we need to compute in this case, we will
find that some calculations are required in addition to those in Table 1, and