Linear Programming Lecture Notes - Penn State Personal Web Server

for all λ > 0. This can only be true if Ad ≤ 0. Likewise:x + λd ≥ 0 holds for all λ > 0 if
and only if d ≥ 0. This completes the proof.
Corollary 4.25. If
(4.20) P = {x ∈ R n : Ax = b, x ≥ 0}
and d is a direction of P , then d must satisfy:
(4.21) Ad = 0, d ≥ 0, d = 0.
Exercise 45. Prove the corollary above.
Example 4.26. Consider the polyhedral set defined by the equations:
x1 − x2 ≤ 1
2x1 + x2 ≥ 6
x1 ≥ 0
x2 ≥ 0
This set is clearly unbounded as we showed in class and it has at least one direction. The
direction d = [0, 1] T pointing directly up is a direction of this set. This is illustrated in
Figure 4.7. In this example, we have:
(4.22) A =
Figure 4.7. An Unbounded Polyhedral Set: This unbounded polyhedral set has
many directions. One direction is [0, 1] T .
1
−1
−2 −1
Note, the second inequality constraint was a greater-than constraint. We reversed it to
a less-than inequality constraint −2x1 − x2 ≤ −6 by multiplying by −1. For our chosen
direction d = [0, 1] T , we can see that:
1 −1 0 −1
(4.23) Ad =
= ≤ 0
−2 −1 1 −1
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