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Math 171A:
Mathematical Programming
Instructor: Philip E. Gill
Winter Quarter 2009
Homework Assignment #4
Due Monday February 9, 2009
Starred exercises require the use of
Matlab
.
Exercise 4.1.
Let
F
denote the feasible region
F
=
{
x
∈
R
n
:
Ax
=
b
}
,
where
A
be an
m
×
n
matrix of rank
r
and
b
is an
m
vector. Assume that
F
is not
empty. Let
A
r
denote an
r
×
n
matrix of
r
linearly independent rows of
A
and let
b
r
denote the corresponding
r
vector of righthand sides. Show that
F
is identical to the set
F
r
4
=
{
x
∈
R
n
:
A
r
x
=
b
r
}
.
Exercise 4.2.
Suppose that the constant vector
c
is such that
c
T
p
≥
0 for all
p
such that
Ap
= 0. Show that this implies that
c
T
p
= 0 for all
p
such that
Ap
= 0.
Exercise 4.3.
Consider the equalityconstrained linear program: minimize
c
T
x
subject to
Ax
=
b
, where
A
=
±

1
5
0
1
1
3

1
1
4
2
²
,
b
=
±
4

5
²
and
c
=

8

2
6
3
5
.
(a)
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 Winter '08
 staff
 Math, matlab

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