This preview shows pages 1–3. Sign up to view the full content.
Lecture 19
Solving LPs in Standard Form
UCSD Math 171A: Numerical Optimization
Philip E. Gill
http://ccom.ucsd.edu/~peg/math171
Wednesday, February 25th, 2009
Recap: Linear programs in standard form
minimize
x
∈
R
n
c
T
x
subject to
Ax
=
b
,
x
≥
0
The working set for the mixedconstraint simplex method has the
form:
A
k
=
A
I
k
!
←
all rows of
A
←
n

m
rows of
In
and
b
k
=
b
0
!
←
all elements of
b
←
n

m
components of 0
n
UCSD Center for Computational Mathematics
Slide 2/31, Wednesday, February 25th, 2009
Recap: Linear programs in standard form
We can deﬁne a column permutation matrix
P
such that
A
I
k
!
P
=
A
k
P
=
B
N
0
I
n

m
!
where:
B
is the matrix of
basic
columns (nonsingular)
N
is the matrix of
nonbasic
columns
The permutation moves the basic variables to the front.
UCSD Center for Computational Mathematics
Slide 3/31, Wednesday, February 25th, 2009
The column permutation applied to a row vector produces a row
vector with basic part ﬁrst.
c
T
P
=
(
c
T
B
c
T
N
)
⇒
P
T
c
=
±
c
B
c
N
²
x
T
P
=
(
x
T
B
x
T
N
)
⇒
P
T
x
=
±
x
B
x
N
²
p
T
P
=
(
p
T
B
p
T
N
)
⇒
P
T
p
=
±
p
B
p
N
²
UCSD Center for Computational Mathematics
Slide 4/31, Wednesday, February 25th, 2009
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
AP
=
(
B
N
)
and
I
k
P
=
(
0
I
n

m
)
4
=
(
0
I
N
)
so that
±
A
I
k
²
P
=
±
B
N
0
I
N
²
UCSD Center for Computational Mathematics
Slide 5/31, Wednesday, February 25th, 2009
Result
P
is an
orthogonal matrix
, i.e.,
P
T
P
=
I
=
PP
T
and det(
P
) =
±
1.
Applying the column permutation
P
T
to
(
c
T
B
c
T
N
)
returns
c
T
:
(
c
T
B
c
T
N
)
=
c
T
P
⇒
(
c
T
B
c
T
N
)
P
T
=
c
T
PP
T
=
c
T
Also
P
±
c
B
c
N
²
=
PP
T
c
=
c
UCSD Center for Computational Mathematics
Slide 6/31, Wednesday, February 25th, 2009
The vertex
x
k
deﬁned by the simplex working set satisﬁes
A
k
x
k
=
b
k
⇒
A
k
PP
T
x
k
=
b
k
⇒
(
A
k
P
)(
P
T
x
k
) =
b
k
⇒
±
B
N
0
I
N
²±
x
B
x
N
²
=
±
b
0
²
Bx
B
+
Nx
N
=
b
x
N
= 0
⇒
Bx
B
=
b
Note: to avoid clutter, we don’t put a suﬃx on
B
or
N
.
UCSD Center for Computational Mathematics
Slide 7/31, Wednesday, February 25th, 2009
Result
The matrix
A
k
is nonsingular
if and only if
B
is nonsingular.
Proof: From the deﬁnition of
A
k
:
det(
A
k
) =
±
det(
A
k
) det(
P
)
=
±
det(
A
k
P
)
=
±
det
±
B
N
0
I
N
²
=
±
det(
B
) det(
I
N
)
=
±
det(
B
)
.
UCSD Center for Computational Mathematics
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 10/23/2010 for the course MATH 171a taught by Professor Staff during the Winter '08 term at UCSD.
 Winter '08
 staff
 Math

Click to edit the document details