Wireless Network Design: Optimization Models and Solution ...

232 Dag Haugland and Di Yuan
but not all destinations, some arc in the corresponding cut set must be used for transmission.
Expressed in terms of y, this means that yi j = 1 for some i ∈ S and some
j ∈ V for which there exists some k ∈ ¯S = V \ S such that pi j ≥ pik. Note that, since
i can send directly to k if it is assigned power pi j, node j does not have to be found
in ¯S.
The number of constraints in the cut-based models is exponential in |V |, and
therefore constraint generation schemes to solve them were proposed in [10]. The
strongest cut-based model was shown to be exactly as strong as MET-F2; however,
their overall performance turned out to be inferior to that of MET-F2. Since the
cut-based models are non-compact, they will not be further discussed in the chapter.
10.3.2 Models Based on Incremental Power
The power of the nodes can be represented in an incremental way. This amounts to
introducing the following variables:
�
1 if the power of node i is at least pi j,
y ′ i j =
0 otherwise.
At each node i ∈ V , a necessary condition for feasibility is y ′ (ik) ≤ y′ (i,k−1) ,k =
2,...,N, that is, if the power of node i is greater than or equal to that of a link,
then the y ′ -variables of all links requiring less power must be equal to one as
well. Introducing a notation convention p (i0) = 0,i ∈ V , the objective function
is ∑i∈V ∑ N−1
k=1 (p (ik) − p (i,k−1))y ′ (ik)
. By utilizing these results, we can reformulate
MET-F1 and MET-F2 using y ′ and thus treating the power levels incrementally.
The relationship between the y- and y ′ -variables is defined by the following equations.
Note that the mapping defined by the equations is unique.
y ′ (ik) =
N−1
∑
ℓ=k
y (i,N−1) = y ′ (i,N−1) ,y (il) = y ′ (iℓ) − y′ (i,ℓ+1)
y (il), k = 1,...,N − 1, (10.10)
, ℓ = 1,...,N − 2. (10.11)
As results of (10.10) and (10.11), a solution defined in the y-variables corresponds
to a solution in the y ′ -variables, and vice versa. This holds also for the LP
relaxations of the models, provided that (10.9) is present. Because of this type of
equivalence, and the observation that reformulation by incremental power does not
bring noticeable computational benefit [10], we do not discuss the models using
incremental power in any further detail.