Wireless Network Design: Optimization Models and Solution ...

78 J. Cole Smith and Sibel B. Sonuc
containing the condition that xi ≤ ⌊ f ⌋, and the other with xi ≥ ⌈ f ⌉. The subproblems
are solved recursively, and the best solution obtained from the two subproblems is
optimal.
Note that the recursive application of branch-and-bound implies that the branching
operations may need to be applied to subproblems at multiple levels of the recursive
algorithm. The branches created throughout this algorithm yield a branchand-bound
tree.
Before presenting the remainder of the branch-and-bound algorithm, we illustrate
the first stages of the branching process on formulation (4.10). First, observe that the
optimal solution to the LP relaxation of (4.10) is x0 = (12/7,27/7). Of course, x0 is
infeasible to (4.10), because x 0 1 and x0 2
are both fractional. Therefore, the next step
in our algorithm is to branch and either eliminate the portion of the feasible region
in which 1 < x1 < 2, or the portion in which 3 < x2 < 4. We (arbitrarily) choose to
branch on x2. As a result, we create two subproblems:
Subproblem 1: max x1 + x2
(4.11a)
s.t. constraints (4.9b) − (4.9e), (4.10c) (4.11b)
x2 ≤ 3 (4.11c)
Subproblem 2: max x1 + x2
(4.12a)
s.t. constraints (4.9b) − (4.9e), (4.10c) (4.12b)
x2 ≥ 4. (4.12c)
Figure 4.6 illustrates these two new subproblems, which we call R1 and R2, respectively
associated with subproblems 1 and 2.
Now, both regions R1 and R2 are active, meaning that they must be searched by
recursively solving both of the associated subproblem LPs. This process is depicted
in the branch-and-bound tree shown in Figure 4.7. We must solve subproblems over
Fig. 4.6 Feasible regions for
branch-and-bound algorithm
for IP (4.10)
x2
R2
R1
x2 ≥ 4
x2 ≤ 3
x1