Wireless Network Design: Optimization Models and Solution ...

4 An Introduction to Integer and Large-Scale Linear Optimization 81
also say that it is fathomed by integrality in case 3 since it is an integer solution.)
Subproblem 6 yields another integer solution x 6 = (0,5) which gets fathomed both
by bound and by integrality as well. Because there are no more active branches left,
the algorithm terminates, and we conclude that the incumbent solution x ⋆ = (2,3)
is an optimal solution for our problem.
It is important to note that the number of subproblems generated within branchand-bound
can grow exponentially in terms of the number of integer variables in the
corresponding IP. Therefore, a practical concern is that the algorithm will not terminate
within a reasonable amount of time. The execution time for branch-and-bound
can be reduced by adding certain valid inequalities that cut off fractional solutions,
which would otherwise need to be explored and eliminated by branch-and-bound.
Algorithmic considerations may attempt to improve computational time by prescribing
“node selection” rules (i.e., the order in which subproblems are branched on and
explored) and “variable selection” rules (i.e., the choice of variables to branch on
when case 4 above is encountered). The field is far too large and complex to be discussed
in any detail here, and so we refer to [20] for a comprehensive treatment of
the subject.
However, some problems appear to be too difficult to solve, either because of
the problem complexity or size, regardless of the algorithmic tricks used to solve
them. If the algorithm must be terminated early, then we can still attempt to assess
the quality of the incumbent solution. Recall that, for a maximization problem, the
incumbent solution’s objective function value is a lower bound (LB) on the optimal
IP solution. The upper bound (UB) is the largest objective function value present
at any active node of the branch-and-bound tree. We say that the optimality gap is
the difference UB − LB, and the relative optimality gap is (UB − LB)/LB × 100%.
When the relative optimality gap is sufficiently small, we may terminate the branchand-bound
procedure with the knowledge that the incumbent solution is within a
certain percentage of optimality. (The relative optimality gap is given with respect to
the incumbent solution’s objective function value. Thus, for minimization problems,
the incumbent solution’s objective function value yields an upper bound, and the
relative optimality gap replaces LB with UB in the denominator.) In the example
above, after subproblems 2 and 3 are solved, we have that LB = 5 and UB = 11/2.
Thus, the relative optimality gap is (0.5/5)×100% = 10%.
4.3 Large Scale Optimization
Many real-world optimization applications require very large mathematical programming
models for which classical optimization techniques require too much
time or memory to solve to optimality, due to a prohibitively large number of variables
and constraints. In some formulations, though, the problem would be separable
(solvable as a series of smaller optimization problems) if not for the presence of
variables that are present in multiple sets of constraints, or constraints that jointly
constrain several sets of variables. In this situation, we say that the problem con-