CO 350 Assignment 3 – Winter 2010 Solutions
Due: Friday January 29 at 10:25 a.m.
Question #
Max. marks
Part marks
1
4
2
4
3
8
2
2
2
2
4
8
4
4
5
6
6(bonus)
(3)
Total
30 + (3)
1. Write the dual of the LP problem:
Max
2
x
1
+
3
x
2
+
4
x
3
+
x
4

x
1
+
2
x
2
+
x
3
+
3
x
4
=
3
x
1

x
2

x
3
+
2
x
4
≤
7
x
1
,
x
3
,
≥
0
.
Solution:
Min
3
y
1
+
7
y
2

y
1
+
y
2
≥
2
2
y
1

y
2
=
3
y
1

y
2
≥
4
3
y
1
+
2
y
2
=
1
y
2
≥
0
.
2. Prove the following version of the Duality Theorem. Let (
P
) be an LP problem in standard
equality form and suppose that (
P
) and its dual both have feasible solutions. Then (
P
) has
an optimal solution
x
*
and its dual has an optimal solution
y
*
having equal objective values.
You may use the Fundamental Theorem of LP and the results of Sections 4.1 and 4.2.
Solution:
We are given that (
P
) has a feasible solution. By Corollary 4.2 of weak duality,
since the dual has a feasible solution, we know that (
P
) is not unbounded. Therefore, by the
Fundamental Theorem, (
P
) must have an optimal solution. We can now apply Theorem 4.5
to obtain the desired conclusion.
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 Fall '10
 GUENIN
 Vector Space, Optimization, feasible solution, x∗

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