Wireless Network Design: Optimization Models and Solution ...

76 J. Cole Smith and Sibel B. Sonuc
as good as that of the original IP. That is, for a maximization problem, the optimal
objective function value of the LP relaxation is at least as large as that of the IP;
and for a minimization problem, it is at least as small as that of the IP. Therefore,
for a maximization problem, the optimal LP relaxation objective function value is
an upper bound on the optimal IP objective function value. A lower bound on the
optimal (maximization) IP objective function value can be found by computing the
objective function value of any feasible solution. This is evident because an optimal
solution cannot be worse than any candidate solutions thus identified. In the subsequent
discussion, we refer to the incumbent solution as the best solution identified
for the IP thus far. (Note that for a minimization problem, the LP relaxation solution
yields a lower bound, and feasible solutions yield upper bounds, on the optimal IP
objective function value.)
Returning to our example, suppose that we add constraints x1 +x2 ≤ 5 and x1 ≤ 2
to the LP (4.9), as in Figure 4.4, so that all extreme points to the LP relaxation are
integer-valued. In general, we refer to a formulation that yields the convex hull of
integer-feasible solutions as an ideal formulation. Then solving this ideal linear program
(and identifying an extreme-point optimal solution) is equivalent to solving the
IP. This is evident because the optimal objective function value of the LP relaxation
yields an upper bound on the optimal IP objective function value, but in this case,
one also obtains a lower bound from this relaxation due to the fact that an extreme
point optimal LP solution is integer-valued as well. Noting that the lower and upper
bounds match, the LP optimal solution must be IP optimal as well.
However, obtaining an ideal formulation is extremely difficult when dealing with
large-scale problems. Not only is it computationally difficult to obtain each of the
required constraints, but there may be an exponential number of these constraints.
Instead, we can improve a given mathematical formulation to contain fewer fractional
points, while retaining all of the feasible integer points. For example, consider
the addition of constraint 2x1 + 3x2 ≤ 15 to (4.9), as depicted in Figure 4.5.
This constraint does not eliminate any of the integer feasible points, but improves
Fig. 4.4 Ideal formulation of
problem (4.9)
x2
x1 ≤ 2
x1 + x2 ≤ 5
x1