Wireless Network Design: Optimization Models and Solution ...

5 Mathematical Programming Models for Third Generation Wireless Network Design 119
At any particular node in the branch-and-bound tree, there can be multiple variables
that could be selected to branch on, and in the generic branch-and-bound algorithm
described in Chapter 4 the choice is made arbitrarily. In fact, this happens at the
root node of the example branch-and-bound tree shown in Figure 4.7. For towerlocation/subscriber-assignment
problems, however, adopting a branching rule that
gives priority to branching on fractional y variables over branching on non-integer
x variables has been shown to speed up the branch-and-bound process [34]. Intuitively,
this branching rule implements a strategy for searching the feasible region
of first determining the tower locations and then solving for the best subscriber assignment.
Limiting test point assignments to the nearest selected tower can also be built into
the integer programming model itself via the following cuts proposed by Kalvenes
et al. [34] for KKOIP:
xmℓ ≤ dm(1 − y j) ∀m ∈ M, j ∈ Lm such that gmℓ < gm j. (5.45)
If tower j is selected, then (5.45) forces xmℓ = 0 for all any selected tower ℓ that
receives signals from test point m at lower power. That is, it only allows solutions
where every test point served is assigned to a nearest selected tower. The advantage
of adding these cuts to the model is that they can eliminate a large number
of sub-optimal points from the feasible region. However, there can be a relatively
large number of these cuts for any given problem instance, and so adding them
increases the number of constraints in the model and this in turn can lead to significantly
longer solution times. Kalvenes et al. [34] studied this trade-off in a series of
computational experiments which we summarize in Section 5.4.4. Note that when
applied to ACMIP, where xmℓ is a binary variable, the right-hand side of (5.45) is
simply 1 - y j.
By focusing only on the interference from test points assigned to tower ℓ, Amaldi
et al. [7] observe that the QoS constraint (5.7) implies
∑ dmxmℓ ≤ s ⇒ ∑ xmℓ ≤ s. (5.46)
m∈Mℓ m∈Mℓ
Hence, the QoS constraints implicitly impose a capacity limit on the number of test
points that can be assigned to a tower, and the second set of cuts proposed in [34]
are the valid inequalities
∑ xmℓ ≤ s ℓ ∈ L. (5.47)
m∈Mℓ Although these valid inequalities are implied by (5.7) and (5.22), Kalvenes et al.
[34] demonstrate that including them explicitly in the KKOIP formulation improves
solution quality.
Chapter 4 describes how the branch-and-bound process can be stopped before
it finds a provably optimal solution, and how in such cases the relative optimality
gap is used to measure the quality of the best (incumbent) solution found at that
point. The relative optimality gap can be used as a stopping criterion for branch-