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Math 171A: Linear Programming
Lecture 23
Linear Programming Duality
Philip E. Gill
c
±
2011
http://ccom.ucsd.edu/~peg/math171a
Friday, March 4th, 2011
Recap: Choice of the generic form
minimize
w
∈
R
n
d
T
w
subject to
Gw
≥
f
,
w
≥
0
Convert to allinequality form if
m
>
n
, i.e.,
G
=
Convert to standard form if
m
<
n
, i.e.,
G
=
Duality theory
can turn a
bad shape
into a
good shape
UCSD Center for Computational Mathematics
Slide 2/35, Friday, March 4th, 2011
Problem conversion involves deﬁning a
primal linear program
and
converting it into another
dual linear program
First, we deﬁne the primal problem as an LP in allinequality form:
minimize
x
∈
R
n
c
T
x
subject to
Ax
≥
b
where
A
has
m
rows.
UCSD Center for Computational Mathematics
Slide 3/35, Friday, March 4th, 2011
Primal problem in allinequality form
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View Full Document First, we deﬁne the primal problem as an LP in allinequality form:
minimize
x
∈
R
n
c
T
x
subject to
Ax
≥
b
where
A
has
m
rows.
Assume for the moment that the problem is feasible with a
bounded optimal value of
c
T
x
*
.
UCSD Center for Computational Mathematics
Slide 5/35, Friday, March 4th, 2011
The optimality conditions are:
c
=
A
T
λ
*
,
λ
*
≥
0;
λ
*
·
(
Ax
*

b
) = 0
Suppose the Lagrange multipliers are viewed as particular values of
dual variables
λ
i
. The optimality conditions
A
T
λ
*
=
c
and
λ
*
≥
0
become the values of
dual constraints
A
T
λ
=
c
and
λ
≥
0
evaluated at
λ
=
λ
*
.
UCSD Center for Computational Mathematics
Slide 6/35, Friday, March 4th, 2011
Any
λ
such that
A
T
λ
=
c
and
λ
≥
0 is said to be
dual feasible
.
Any
x
such that
Ax
≥
b
is said to be
primal feasible
.
UCSD Center for Computational Mathematics
Slide 7/35, Friday, March 4th, 2011
If
x
is
primalfeasible
(i.e.,
Ax
≥
b
)
and
λ
is
dualfeasible
(i.e.,
A
T
λ
=
c
and
λ
≥
0) then
λ
T
(
Ax

b
) =
λ
T
{z}
≥
0
(
Ax

b

{z
}
≥
0
)
≥
0
Some rearrangement gives
λ
T
(
Ax

b
) =
λ
T
Ax

λ
T
b
=
c
T
x

b
T
λ
≥
0
.
UCSD Center for Computational Mathematics
Slide 8/35, Friday, March 4th, 2011
It follows that for a given
x
such that
Ax
≥
b
,
c
T
x
≥
b
T
λ
for all
λ
such that
A
T
λ
=
c
, and
λ
≥
0.
UCSD Center for Computational Mathematics
Slide 9/35, Friday, March 4th, 2011
The
smallest value
of
c
T
x
is
c
T
x
*
. It follows that
c
T
x
*
≥
b
T
λ
for all
λ
such that
A
T
λ
=
c
, and
λ
≥
0
The
largest value
of
b
T
λ
is deﬁned by the linear program
maximize
λ
∈
R
m
b
T
λ
subject to
A
T
λ
=
c
,
λ
≥
0
.
This is called the
dual linear program
.
UCSD Center for Computational Mathematics
Slide 10/35, Friday, March 4th, 2011
The dual linear program is
maximize
λ
∈
R
m
b
T
λ
subject to
A
T
λ
=
c
,
λ
≥
0
.
The constraints of this dual problem are in
standard form
The dual constraint matrix is the transpose of the primal
constraint matrix.
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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