CO350
Assignment 2
Due: May 18 at 2pm.
Question 1:
Consider the following linear programs.
(
LP
a
)
min
x
1
+ 2
x
3
Subject to
2
x
1
+
x
2

3
x
3
≤
4
3
x
1
+ 2
x
2

4
x
3
= 3

3
x
1

x
2
+
x
3
≥
3
x
1
,x
2
≥
0
(
LP
b
)
min
x
1

x
3
Subject to
2
x
1
+
x
2
+ 3
x
3
≥
4
x
1
+ 2
x
2

2
x
3
≤
1

3
x
1
+
x
3
= 2
x
1
≥
0
,x
2
≤
0
(a)
Write (
LP
a
) in standard inequality form.
(b)
Write (
LP
b
) in standard equality form.
Question 2:
Consider the linear program max(
c
T
x
:
Ax
= 0
, x
≥
0), where
A
∈
R
m
×
n
and
c
∈
R
n
. Show that if
x
= 0 is not an optimal solution, then the linear program is unbounded.
Question 3:
Consider the linear program max(
c
T
x
:
Ax
=
b, x
≥
0), where
A
∈
R
m
×
n
,
b
∈
R
m
, and
c
∈
R
n
. Show that, if the linear program has two distinct optimal solutions, then
it has infinitely many optimal solutions.
Question 4:
Let
A
∈
R
m
×
n
and
b
∈
R
m
.
(a)
Show that, if there exists
y
∈
R
m
such that
y
T
A
≤
0 and
y
T
b >
0, then the system
(
Ax
=
b, x
≥
0) is infeasible.
(b)
Show that the following linear program is infeasible by using the result proved in part (a)
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 Spring '07
 S.Furino,B.Guenin
 Operations Research, Optimization, linear program, LPA, following linear program, linear program MAX

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