Stochastic Programming - Index of

214 STOCHASTIC PROGRAMMING
problem
min c T x
s.t. Ax = b,
x ≥ 0,
i.e. (7.3) without the last set of constraints added. Then, if the resulting ˆx
makes the last set of constraints in (7.3) feasible for all ξ, we are done. If not,
an implied feasibility cut is added.
An integer program, on the other hand, could be written as
min c T x
s.t. Ax = b,
x i ∈{a i ,...,b i } for all x i .
⎫
⎬
⎭
(7.4)
A cutting-plane procedure for (7.4) will solve the problem with the constraints
a ≤ x ≤ b so that the integrality requirement is relaxed. Then, if the resulting
ˆx is integral in all its elements, we are done. If not, an integrality cut is added.
This cut will, if possible, be a facet of the solution space with all extreme points
integer.
By now, realizing that integrality cuts are also feasibility cuts, the
connection should be clear. Integrality cuts in integer programming are just
a special type of feasibility cuts.
For the bounding version of the L-shaped decomposition method we
combined bounding (with partitioning of the support) with cuts. In the same
way, we can combine branching and cuts in the branch-and-cut algorithm for
integer programs (still deterministic). The idea is fairly simple (but requires
a lot of details to be efficient). For all waiting nodes, before or after we
have solved the relaxed LP, we add an appropriate number of cuts, before
we (re)solve the LP. How many cuts we add will often depend on how well we
know the facets of the (integer) solution space. This new LP will have a smaller
(continuous) solution space, and is therefore likely to give a better result—
either in terms of a nonintegral optimal solution with a higher objective value
(increasing the probability of bounding), or in terms of an integer solution.
So, finally, we have reached the ultimate question. How can all of this be
used to solve integer stochastic programs Given the simplification that we
have integrality only in the first-stage problem, the procedure is given in
Figure 27. In the procedure we operate with a set of waiting nodes P. These
are nodes in the cut-and-branch tree that are not yet fathomed or bounded.
The procedure feascut was presented earlier in Figure 9, whereas the new
procedure intcut is outlined in Figure 28. Let us try to compare the L-shaped
integer programming method with the continuous one presented in Figure 10.