Wireless Network Design: Optimization Models and Solution ...

70 J. Cole Smith and Sibel B. Sonuc
within a polyhedron (intersection of halfspaces formed by linear constraints) for a
linear program. A point ¯x is an extreme point of X if and only if it cannot be written
as a strict convex combination of any two distinct points in X, i.e., if there is no
x ′ ∈ X, x ′′ ∈ X, and 0 < λ < 1 such that ¯x = λx ′ + (1 − λ)x ′′ . Then the following
critical theorem holds true.
Theorem 4.1. If there exists an optimal solution to a linear program, at least one
extreme point must be optimal.
Note that this theorem does not (incorrectly) assert that all optimal solutions exist
at extreme points. When alternative optimal solutions exist, then an infinite number
of non-extreme-point solutions are optimal. For instance, examining (4.1), if the objective
function is changed to “max x1 + 2x2,” then all solutions on the line segment
between (x1,x2) = (0, 5) and (8/5, 21/5) are optimal.
The caveat in Theorem 4.1 is necessary because LPs need not have optimal solutions.
If there are no feasible solutions, the problem is termed infeasible. If there
is no limit to how good the objective can become (i.e., if the objective can be made
infinitely large for maximization problems, or infinitely negative for minimization
problems), then it is unbounded. If a problem is unbounded, then there exists a direction
d, such that if ˆx is feasible, then ˆx + λd is feasible for all λ ≥ 0, and the
objective improves in the direction d (i.e., c T d > 0 for a maximization problem).
Note that the set of all directions to a feasible region, unioned with the origin, is
itself a polyhedron that lies in the nonnegative orthant of the feasible region (assuming
x ≥ 0). If this polyhedron is normalized with a constraint of the form e T x ≤ 1
(where e is a vector of 1’s), then the nontrivial extreme points are extreme directions
of the feasible region. It follows that all directions can be represented as nonnegative
linear combinations of the extreme directions. Indeed, if a linear program is
unbounded, it must be unbounded in at least one extreme direction. For the remaining
discussion, we assume that extreme directions d have been normalized so that
e T d = 1.
The following Representation Theorem gives an important alternative characterization
of polyhedra, which we use in the following section.
Theorem 4.2. Consider a polyhedron defining a feasible region X, with extreme
points x i for i = 1,...,E and extreme directions d j for j = 1,...,F. Then any point
x ∈ X can be represented by a convex combination of extreme points, plus a nonnegative
linear combination of extreme directions, i.e.,
x =
E
∑
i=1
λix i +
λ,µ ≥ 0, and
F
∑ µ jd
j=1
j , where (4.3)
E
∑ λi = 1. (4.4)
i=1
Full details of LP solution techniques are beyond the scope of this chapter, but we
present a sketch of the simplex method for linear programs here. Observe that each
extreme point can be associated with at least one set of m columns of A that form