CO 355 Mathematical Optimization (Fall 2010)
Assignment 2
Due:
Thursday October 14th, in class.
Policy.
No collaboration is allowed. You may use the course notes / textbook and the lecture slides, but
please be very specific
when using citing results found there. (Don’t just say “from some claim in class
we know
....
”.) Every other resource that you might stumble upon must be properly referenced. You are
welcome to seek help from the current instructor and TAs for CO 355.
The Fundamental Theorem of Linear Programming:
We now complete the proof of this theorem.
Question 1:
(10 points)
In class we proved the following variant of Farkas’ Lemma:
Ax
≤
b
has no solution
⇐⇒
∃
y
≥
0 such that
y
T
A
= 0 and
y
T
b <
0
(1)
Using Eq. (1), prove the following variant of Farkas’ Lemma:
Mu
=
d, u
≥
0 has no solution
⇐⇒
∃
w
such that
w
T
M
≥
0 and
w
T
d <
0
.
(2)
Question 2:
(10 points)
Consider the LP max
{
c
T
x
:
Ax
≤
b
}
.
Suppose that it is feasible but its dual is
in
feasible.
Use the
variant of Farkas’ lemma in Eq. (2) to prove that the (primal) LP is unbounded.
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 Winter '10
 Harvey
 Vector Space, Englishlanguage films, Trigraph, Ellipsoid, convex combinations

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