Wireless Network Design: Optimization Models and Solution ...

88 J. Cole Smith and Sibel B. Sonuc
(i, j) consumes sk i j units of its total capacity. (For instance, if capacity is given by
the bandwidth of some particular communication link, video communication will
consume bandwidth at a different rate than voice communication.) The objective of
the problem is to satisfy each commodity’s demands and exhaust each commodity’s
supplies at a minimum cost, while obeying capacity limits on the arcs.
Define flow variables xk i j as the flow of commodity k ∈ K over arc (i, j) ∈ A. Then
the multicommodity network flow problem is formulated as follows.
min ∑
∑
k∈K (i, j)∈A
c k i jx k i j
(4.24a)
s.t. ∑ s
k∈K
k i jx k i j ≤ qi j ∀(i, j) ∈ A (4.24b)
∑ x
i∈FS( j)
k ji − ∑ x
i∈RS( j)
k i j = d k j ∀ j ∈ V, k ∈ K (4.24c)
x k i j ≥ 0 ∀(i, j) ∈ A, k ∈ K. (4.24d)
The objective (4.24a) minimizes the commodity flow costs over the network. Constraints
(4.24b) enforce the arc capacity restrictions. Constraints (4.24c) enforce the
flow balance constraints (at each node, and for each commodity), and (4.24d) enforce
nonnegativity restrictions on the flow variables.
Here, there is no set of complicating variables. However, if the capacity constraints
(4.24b) were removed from the problem, then we could solve (4.24) as |K|
separable network flow problems. Hence, we say that (4.24b) are complicating constraints.
One common way of solving problems like this is via Dantzig-Wolfe decomposition,
which employs the Representation Theorem (Theorem 4.2) as a key
component. An alternative method for solving these problems is via Lagrangian
optimization, which we discuss in Section 4.3.3.
Our generic model for this subsection contains |K| sets of variables, x k for k ∈ K,
which would be separable if the complicating constraints were removed from the
problem.
min ∑ (c
k∈K
k ) T x k
(4.25a)
s.t. ∑ A
k∈K
k x k = b (4.25b)
H k x k = g k
∀k ∈ K (4.25c)
x k ≥ 0 ∀k ∈ K, (4.25d)
where Ak ∈ Rm0 ×nk and Hk ∈ Rmk ×nk , ∀k ∈ K, and all other vectors have conforming
dimensions.
In the Benders decomposition method, we decomposed the dual feasible region
into its extreme points and extreme directions, and included a constraint in a relaxed
master problem corresponding to each extreme point and extreme direction, added
to the formulation as necessary. In the Dantzig-Wolfe decomposition method, we
will decompose the feasible regions given by X k = {xk ≥ 0 : Hkxk = gk } into its