Linear Programming Lecture Notes - Penn State Personal Web Server

Example 4.23. Consider the unbounded convex set shown in Figure 4.6. This set has
direction [1, 0] T . To see this note that for any positive scaling parameter λ and for any vertex
Figure 4.6. Convex Direction: Clearly every point in the convex set (shown in
blue) can be the vertex for a ray with direction [1, 0] T contained entirely in the
convex set. Thus [1, 0] T is a direction of this convex set.
point x0, we can draw an arrow pointing to the right (in the direction of [1, 0] T ) with vertex
at x0 scaled by λ that is entirely contained in the convex set.
Exercise 43. Prove the following: Let C ⊆ R n be a convex cone and let x1, x2 ∈ C. If
α, β ∈ R and α, β ≥ 0, then αx1 + βx2 ∈ C. [Hint: Use the definition of convex cone and
the definition of convexity with λ = 1/2, then multiply by 2.]
Exercise 44. Use Exercise 43 to prove that if C ⊆ R n is a convex cone, then every
element x ∈ C (except the origin) is also a direction of C.
5. Directions of Polyhedral Sets
There is a unique relationship between the defining matrix A of a polyhedral set P and
a direction of this set that is particularly useful when we assume that P is located in the
positive orthant of R n (i.e., x ≥ 0 are defining constraints of P ).
Theorem 4.24. Suppose that P ⊆ R n is a polyhedral set defined by:
(4.16) P = {x ∈ R n : Ax ≤ b, x ≥ 0}
If d is a direction of P , then the following hold:
(4.17) Ad ≤ 0, d ≥ 0, d = 0.
Proof. The fact that d = 0 is clear from the definition of direction of a convex set.
Furthermore, d is a direction if and only if
(4.18)
(4.19)
A (x + λd) ≤ b
x + λd ≥ 0
for all λ > 0 and for all x ∈ P (which is to say x ∈ R n such that Ax ≤ b and x ≥ 0). But
then
Ax + λAd ≤ b
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