Wireless Network Design: Optimization Models and Solution ...

4 An Introduction to Integer and Large-Scale Linear Optimization 77
the formulation by “cutting off” a set of fractional solutions. The new formulation
is written as:
max x1 + x2
(4.10a)
s.t. (4.9b) − (4.9f) (4.10b)
2x1 + 3x2 ≤ 15 (4.10c)
A constraint that does not remove any integer-feasible solution (i.e., of the form
α T x ≤ β such that α T x i ≤ β ∀x i ∈ X, where X is the set of all feasible solutions),
such as (4.10c), is called a valid inequality. We say that the formulation (4.10) is
stronger than formulation (4.9), because the set of fractional feasible solutions to
the LP relaxation of (4.10) is a strict subset of those to the LP relaxation of (4.9).
The classic work of Nemhauser and Wolsey [20] discusses common techniques used
to generate valid inequalities for integer programming problems, often based on
specific problem structures.
One generic approach for solving an IP is to solve its LP relaxation and obtain
an optimal solution ˆx. (If no solution exists to the LP relaxation, then the IP is
also infeasible.) If ˆx is integer-valued, then it is optimal. Otherwise, add a valid
inequality that cuts off ˆx, i.e., find a valid inequality α T x ≤ β such that α T ˆx >
β. This type of valid inequality is called a cutting plane (or just a “cut”), and the
given technique is called a cutting-plane algorithm. Practically speaking, though,
pure cutting-plane algorithms are not practical, due to the fact that many cuts must
usually be generated before termination, and the cutting-plane process is potentially
not numerically stable.
Instead, most common algorithms to solve integer programs involve the branchand-bound
process. The branch-and-bound algorithm starts in the same fashion as
the cutting-plane algorithm above, obtaining optimal LP relaxation solution ˆx and
terminating if ˆx is integer-valued (or if no solution exists to the LP relaxation). Otherwise,
at least some variable xi equals to a value f that is fractional-valued (noninteger)
in the optimal LP solution ˆx. Clearly, all feasible solutions must obey either
the condition that xi ≤ ⌊ f ⌋ or xi ≥ ⌈ f ⌉. We thus branch into two subproblems, one
Fig. 4.5 Feasible region for
problem (4.10)
x2
New constraint:
2x1 + 3x2 ≤ 15
x1