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Math 171A: Linear Programming
Lecture 14
Farkas’ Lemma and its Implications
Philip E. Gill
c
±
2011
http://ccom.ucsd.edu/~peg/math171a
Monday, February 7th, 2011
Recap: Optimality conditions for LP
LP
minimize
x
c
T
x
subject to
Ax
≥
b
with
A
an
m
×
n
matrix,
b
an
m
vector and
c
an
n
vector.
The vector
x
*
is a solution of LP
if and only if
:
(a)
Ax
*
≥
b
(b)
c
=
A
T
a
λ
*
a
for some vector
λ
*
a
≥
0, where
A
a
is the active
constraint matrix at
x
*
UCSD Center for Computational Mathematics
Slide 2/41, Monday, February 7th, 2011
Example: minimize 2
x
1
+
x
2
subject to the constraints:
constraint #1:
x
1
+
x
2
≥
1
constraint #2:
x
2
≥
0
constraint #3:
x
1
≥
0
Written in the form min
c
T
x
subject to
Ax
≥
b
, we get
c
=
2
1
!
,
A
=
1
1
0
1
1
0
,
b
=
1
0
0
UCSD Center for Computational Mathematics
Slide 3/41, Monday, February 7th, 2011
c
a
1
c
a
1
a
3
a
2
x
1
x
2
x
*
¯
x
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View Full Document At the point
x
*
=
0
1
!
,
the active set is
A
=
{
1
,
3
}
with
A
a
=
1
1
1
0
!
,
b
a
=
1
0
!
Solving for the Lagrange multipliers gives
A
T
a
λ
a
=
c
⇒
1
1
1
0
!
λ
a
=
2
1
!
⇒
λ
a
=
1
1
!
≥
0
⇒
x
*
is optimal.
UCSD Center for Computational Mathematics
Slide 5/41, Monday, February 7th, 2011
At the point
¯
x
=
1
0
!
,
the active set is
A
=
{
1
,
2
}
with
A
a
=
1
1
0
1
!
,
b
a
=
1
0
!
Solving for the Lagrange multipliers gives
A
T
a
λ
a
=
c
⇒
1
0
1
1
!
λ
a
=
2
1
!
⇒
λ
a
=
2

1
!
6≥
0
⇒
¯
x
is
not
optimal.
UCSD Center for Computational Mathematics
Slide 6/41, Monday, February 7th, 2011
Proof of Farkas’ lemma
Result (Farkas’ Lemma)
(A)
c
T
p
≥
0 for all
p
such that
A
a
p
≥
0
if and only if
(B)
c
=
A
T
a
λ
*
a
for some
λ
*
a
≥
0
UCSD Center for Computational Mathematics
Slide 7/41, Monday, February 7th, 2011
Proof: We have two statements that we must show are equivalent:
(A)
c
T
p
≥
0 for all
p
such that
A
T
a
p
≥
0
(B)
c
=
A
T
a
λ
*
a
for some
λ
*
a
≥
0
We must show that
(A)
⇒
(B)
and
(B)
⇒
(A)
UCSD Center for Computational Mathematics
Slide 8/41, Monday, February 7th, 2011
It is “EASY” to show that
(B)
⇒
(A)
If
(B)
holds, then
c
=
A
T
a
λ
*
a
for some
λ
*
a
≥
0.
Then,
c
T
p
= (
A
T
a
λ
*
a
)
T
p
= (
λ
*
a
)
T
(
A
a
p
)
≥
0 for all
p
with
A
a
p
≥
0
⇒
c
T
p
≥
0 for all
p
with
A
a
p
≥
0
⇒
(A)
holds.
UCSD Center for Computational Mathematics
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This note was uploaded on 03/19/2012 for the course MATH 171a taught by Professor Staff during the Spring '08 term at UCSD.
 Spring '08
 staff
 Linear Programming

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