Linear Programming Lecture Notes - Penn State Personal Web Server

[0 1]
[1 2]
System 2 has
a solution
Positive cone
[1 -1]
System 1 has
a solution
Figure 8.3. An example of Farkas’ Lemma: The vector c is inside the positive cone
formed by the rows of A, but c ′ is not.
and the vector c = 1 2 . For this matrix and this vector, does System 1 have a solution or
does System 2 have a solution? [Hint: Draw a picture illustrating the positive cone formed
by the rows of A. Draw in c. Is c in the cone or not?]
2.2. Theorems of the Alternative. Farkas’ lemma can be manipulated in many ways
to produce several equivalent statements. The collection of all such theorems are called
Theorems of the Alternative and are used extensively in optimization theory in proving
optimality conditions. We state two that will be useful to us.
Corollary 8.5. Let A ∈ R k×n and E ∈ R l×n . Let c ∈ R n be a row vector. Suppose
d ∈ R n is a column vector and w ∈ R k is a row vector and v ∈ R l is a row vector. Let:
and
M =
A
E
u = w v
[1 0]
Then exactly one of the following systems has a solution:
(1) Md ≤ 0 and cd > 0 or
(2) uM = c and u ≥ 0
Proof. Let x = −d. Then Md ≤ 0 implies Mx ≥ 0 and cd > 0 implies cx < 0. This
converts System 1 to the System 1 of Farkas’ Lemma. System 2 is already in the form found
in Farkas’ Lemma. This completes the proof.
Exercise 61. Prove the following corollary to Farkas’ Lemma:
Corollary 8.6. Let A ∈ R m×n and c ∈ R n be a row vector. Suppose d ∈ R n is a
column vector and w ∈ R m is a row vector and v ∈ R n is a row vector. Then exactly one of
the following systems of inequalities has a solution:
(1) Ad ≤ 0, d ≥ 0 and cd > 0 or
(2) wA − v = c and w, v ≥ 0
[Hint: Write System 2 from this corollary as wA − Inv = c and then re-write the system
with an augmented vector [w v] with an appropriate augmented matrix. Let M be the
augmented matrix you identified. Now write System 1 from Farkas’ Lemma using M and x.
Let d = −x and expand System 1 until you obtain System 1 for this problem.]
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