Wireless Network Design: Optimization Models and Solution ...

4 An Introduction to Integer and Large-Scale Linear Optimization 91
ˆπ T (A k ˆx)− ˆγk < 0, then add ˆx to Ω k . If any new columns are added (corresponding to
extreme points or extreme directions) for any k ∈ K, then return to Step 1. Otherwise,
the current solution to the restricted master problem is optimal, and the procedure
stops.
For the multicommodity flow problem, we have a restricted master problem of
the form (4.27), and a subproblem that takes the form of a simple network flow
problem:
min ∑ (c
(i, j)∈A
k i j − ˆπi js k i j)xi j
(4.29a)
s.t. ∑ x ji − ∑ xi j = d
i∈FS( j) i∈RS( j)
k j ∀ j ∈ V, k ∈ K (4.29b)
x ≥ 0. (4.29c)
Note that the cost for pushing flow across arc (i, j) is given by the per-unit flow
cost ci j, plus the shadow price ˆπi j on the capacity of arc (i, j) multiplied by the rate
of capacity usage s k i j xi j of commodity k on arc (i, j). That is, the costs include the
direct cost of sending a unit of flow on (i, j), plus the costs associated with utilizing
capacity on arc (i, j). (Note that the π-values here will be nonpositive, because those
duals are associated with ≤ constraints in a minimization problem; therefore, all coefficients
of the x-variables in the objective function are nonnegative.) Furthermore,
because there are no negative coefficients in the objective function, we will never
generate any extreme directions for the restricted master problem in this application.
(We refer interested readers to classical surveys [2, 15] of multicommodity flow algorithms
as a starting point for a more thorough treatment of these problems, and to
[17] for more recent work in the area.)
Remark 4.4. The structure of (4.29) suggests another important facet of these decomposition
procedures. Note that not only are the subproblems (4.29) separable,
but they are also solvable by much more efficient means than linear programming.
(See [1] for a discussion of low-polynomial time algorithms for network flow problems.)
Hence, even if the number of separable problems is small, one may still opt
to use decomposition in order to exploit specially-structured subproblems that are
quickly solvable.
Remark 4.5. In this subsection, we have compared in some detail how Benders decomposition
compares to Dantzig-Wolfe decomposition. In fact, taking the dual of a
problem (4.25) (after some simple adjustments) yields a problem of the form (4.14).
It is thus not surprising that applying Benders decomposition to the dual of (4.25) is
equivalent to solving (4.25) directly by Dantzig-Wolfe decomposition.