Stochastic Programming - Index of

220 STOCHASTIC PROGRAMMING
Remember, however, that this is not the true value of Q(x k )—just an estimate.
In other words, we have now observed two major differences from the
exact L-shaped method (page 171). First, we operate on a sample rather
than on all outcomes, and, secondly, what we calculate is an estimate of
a lower bound on Q(x k ) rather than Q(x k ) itself. Hence, since we have a
lower bound, what we are doing is more similar to what we did when we
used the L-shaped decomposition method within approximation schemes, (see
page 204). However, the reason for the lower bound is somewhat different. In
the bounding version of L-shaped, the lower bound was based on conditional
expectations, whereas here it is based on inexact optimization. On the other
hand, we have earlier pointed out that the Jensen lower bound has three
different interpretations, one of which is to use conditional expectations (as
in procedure Bounding L-shaped) and another that is inexact optimization
(as in SD). So what is actually the principal difference
For the three interpretations of the Jensen bound to be equivalent, the
limited set of bases must come from solving the recourse problem in the points
of conditional expectations. That is not the case in SD. Here the points are
random (according to the sample ξ j ). Using a limited number of bases still
produces a lower bound, but not the Jensen lower bound.
Therefore SD and the bounding version of L-shaped are really quite
different. The reason for the lower bound is different, and the objective value
in SD is only a lower bound in terms of expectations (due to sampling). One
method picks the limited number of points in a very careful way, the other
at random. One method has an exact stopping criteria (error bound), the
other has a statistically based stopping rule. So, more than anything else,
they are alternative approaches. If one cannot solve the exact problem, one
either resorts to bounds or to sample-based methods.
In the L-shaped method we demonstrated how to find optimality cuts. We
can now find a cut corresponding to x k (which is not binding and might even
not be a lower bound, although it represents an estimate of a lower bound).
As for the L-shaped method, we shall replace Q(x) in the objective by θ, and
then add constraints. The cut generated in iteration k is given by
θ ≥ 1 k
k∑
(πj k )T [ξ j − T (ξ j )x] =α k k +(βk k )T x.
j=1
The double set of indices on α and β indicate that the cut was generated in
iteration k (the subscript) and that it has been updated in iteration k (the
superscript).
In contrast to the L-shaped decomposition method, we must now also look
at the old cuts. The reason is that, although we expect these cuts to be loose
(since we use inexact optimization), they may in fact be far too tight (since
they are based on a sample). Also, being old, they are based on a sample that