Linear Programming Lecture Notes - Penn State Personal Web Server

These two facts hold since Gd = 0 and if Ai· is a row of A with Ai·x0 < bi (or x > 0), then
there is at least one non-zero ɛ so that Ai·(x0 ± ɛd) < bi (or x0 ± ɛd > 0) still holds and
therefore (x0 ± ɛd) ∈ P . Since we have a finite number of constraints that are non-binding,
we may choose ɛ to be the smallest value so that the previous statements hold for all of
them. Finally we can choose λ = 1/2 and see that x0 = λx + (1 − λ)ˆx and x, ˆx ∈ P . Thus
x0 cannot have been an extreme point, contradicting our assumption. This completes the
proof.
Definition 4.33. Let P be the polyhedral set from Theorem 4.31. If x0 is an extreme
point of P and more than n hyperplanes are binding at x0, then x0 is called a degenerate
extreme point.
Definition 4.34 (Face). Let P be a polyhedral set defined by
P = {x ∈ R n : Ax ≤ b}
where A ∈ R m×n and b ∈ R m . If X ⊆ P is defined by a non-empty set of binding linearly
independent hyperplanes, then X is a face of P .
That is, there is some set of linearly independent rows Ai1·, . . . Ail· with il < m so that
when G is the matrix made of these rows and g is the vector of bi1, . . . , bil then:
(4.27) X = {x ∈ R n : Gx = g and Ax ≤ b}
In this case we say that X has dimension n − l.
Remark 4.35. Based on this definition, we can easily see that an extreme point, which
is the intersection n linearly independent hyperplanes is a face of dimension zero.
Definition 4.36 (Edge and Adjacent Extreme Point). An edge of a polyhedral set P is
any face of dimension 1. Two extreme points are called adjacent if they share n − 1 binding
constraints. That is, they are connected by an edge of P .
Example 4.37. Consider the polyhedral set defined by the system of inequalities:
3x1 + x2 ≤ 120
x1 + 2x2 ≤ 160
28
16 x1 + x2 ≤ 100
x1 ≤ 35
x1 ≥ 0
x2 ≥ 0
The polyhedral set is shown in Figure 4.9. The extreme points of the polyhedral set are shown
as large diamonds and correspond to intersections of binding constraints. Note the extreme
point (16, 72) is degenerate since it occurs at the intersection of three binding constraints
3x1 + x2 ≤ 120, x1 + 2x2 ≤ 160 and 28
16 x1 + x2