Stochastic Programming - Index of

RECOURSE PROBLEMS 201
Table 2
Function values needed for the “look-ahead”strategy.
(5,5) 15 (10,7.5) 25
(15,5) 25 (10,2.5) 15
(10,10) 27.143 (0,5) 10
(20,5) 25 (10,0) 10
φ(E ˜ξ) =φ(10, 5) = 20, which we have already found. The additional numbers
are presented in Table 2.
Based on this, we can find the total error after splitting to be about 4.5
both for ˜ξ 1 and for ˜ξ 2 . Therefore, based on “look-ahead”, we cannot decide
what to do.
✷
3.5.2 Using the L-shaped Method within Approximation Schemes
We have now investigated how to bound Q(x) forafixedx. Wehavedone
that by combining upper and lower bounding procedures with partitioning of
the support of ˜ξ. On the other hand, we have earlier discussed (exact) solution
procedures, such as the L-shaped decomposition method (Section 3.2) and the
scenario aggregation (Section 2.6). These methods take a full event/scenario
tree as input and solve this (at least in principle) to optimality. We shall now
see how these methods can be combined.
The starting point is a set-up like Figure 18. We set up an initial partition
of the support, possibly containing only one cell. We then find all conditional
expectations (in the example there are five), and give each of them a
probability equal to that of being in their cell, and we view this as our
“true” distribution. The L-shaped method is then applied. Let ξ i denote the
conditional expectation of ˜ξ, giventhat˜ξ is contained in the ith cell. Then
the partition gives us the support {ξ 1 ,...ξ l }.Wethensolve
where
min c T x + L(x)
s.t. Ax = b,
x ≥ 0,
L(x) =
⎫
⎬
⎭
l∑
p j Q(x, ξ j ),
j=1
(5.3)