Stochastic Programming - Index of

RECOURSE PROBLEMS 217
3.7.3 Optimality Cuts
The creation of optimality cuts is the same in both cases, since in the integer
case we create such cuts only for feasible (integer) solutions.
3.7.4 Stopping Criteria
The stopping criteria are basically the same, except that what halts the whole
procedure in the continuous case just fathoms a node in the integer case.
3.8 Stochastic Decomposition
Throughout this book we are trying to reserve superscripts on variables and
parameters for outcomes/realizations, and subscripts for time and components
of vectors. This creates difficulties in this section. Since whatever we do will
be wrong compared with our general rules, we have chosen to use the indexing
of the original authors of papers on stochastic decomposition.
The L-shaped decomposition method, outlined in Section 3.2, is a
deterministic method. By that, we mean that if the algorithm is repeated with
the same input data, it will give the same results each time. In contrast to this,
we have what are called stochastic methods. These are methods that ideally
will not give the same results in two runs, even with the same input data. We
say “ideally” because it is impossible in the real world to create truly random
numbers, and hence, in practice, it is possible to repeat a run. Furthermore,
these methods have stopping criteria that are statistical in nature. Normally,
they converge with probability 1.
The reason for calling these methods random is that they are guided by
some random effects, for example samples. In this section we are presenting
the method called stochastic decomposition (SD). The approach, as we present
it, requires relatively complete recourse.
We have until now described the part of the right-hand side in the
recourse problem that does not depend on x by h 0 + Hξ. This was done
to combine two different effects, namely to allow certain right-hand side
elements to be dependent, but at the same time to be allowed to work on
independent random variables. SD does not require independence, and hence
we shall replace h 0 + Hξ by just ξ, since we no longer make any assumptions
about independence between components of ξ. Wedoassume,however,that
q(ξ) ≡ q 0 , so all randomness is in the right-hand side. The problem to solve
is therefore the following:
min{φ(x) ≡ c T x + Q(x)}
s.t. Ax = b,
x ≥ 0,