Wireless Network Design: Optimization Models and Solution ...

120 Eli Olinick
and-bound so that the process is stopped when the relative optimality gap of the
incumbent solution is within a user-defined threshold. This strategy is widely used in
practice to make a trade-off between solution quality and running time; the process
terminates more quickly with larger thresholds but at the cost of leaving portions
of the feasible region unexplored and/or finding a good (tight) dual bound on the
objective function (i.e., the quality of the incumbent solution may in fact be better
than that implied by the relative optimality gap). It is up to the user to determine
an acceptable threshold for the relative optimality gap; typical values used in the
literature are 1%, 5%, and 10% [34].
5.4.4 Dealing with Numerical Instability
As noted earlier, the wide range of coefficients on the left-hand side of the QoS
constraints (5.7) or (5.22) can lead to numerical instability of CPLEX’s branch-andbound
algorithm. CPLEX, and presumably other commercially available branchand-bound
codes, scale the constraint matrix of the user-supplied problem so that
the absolute value of all the coefficients in any given constraint are between zero
and one. It then solves the scaled problem and applies the reverse scaling to the
result to obtain the solution returned to the user. Kalvenes et al. [34] observed that
although the solutions CPLEX found to the scaled problems in their test sets were
feasible (within a given tolerance), the unscaled solutions often violated some of
the QoS constraints. To correct this, they apply a post-processing procedure that
minimizes the number of subscribers that must be dropped from the scaled solution
to ensure that the unscaled solution satisfies the QoS constraints. The procedure
involves solving the following integer programming problem
min ∑
(m,ℓ)∈A
δmℓ
s.t. ∑
gmℓ
gm (m, j)∈A j
(5.48)
(xm j − δm j) ≤ s ∀ℓ ∈ LV (5.49)
δmℓ ∈ N ∀(m,ℓ) ∈ A, (5.50)
where in the unscaled solution constant xmℓ is the value of xmℓ, A = {m ∈ M,ℓ ∈ Lm :
xmℓ > 0}, and LV is set of selected towers, and δmℓ is the number of subscribers at
test point m that are dropped from tower ℓ in the scaled solution. After solving the
integer program above, the post-processing procedure sets xmℓ = xmℓ − δmℓ for all
m ∈ M and ℓ ∈ Lm. Thus, Kalvenes et al. [34] propose a two-phase solution: Phase
I applies branch-and-bound to the KKIOP model and if the solution violates any of
the QoS constraints, Phase II applies the post-processing procedure to the Phase I
solution.
Kalvenes et al. [34] present a computational study using this post-processing
procedure, the branching rule and cuts, (5.45) and (5.47), described in Section 5.4.3