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Math 171A: Linear Programming
Lecture 22
The Simplex Method for Standard Form
Philip E. Gill
c
±
2011
http://ccom.ucsd.edu/~peg/math171a
Monday, February 28th, 2011
Recap: Simplex method for LPs in standard form
minimize
x
∈
R
n
c
T
x
subject to
Ax
=
b

{z
}
equality constraints
,
x
≥
0

{z
}
simple bounds
We apply “mixedconstraint” simplex with full matrix
±
A
I
²
Each vertex is a feasible basic solution of
Ax
=
b
Simplex method involves solving with an
m
×
m
nonsingular
B
UCSD Center for Computational Mathematics
Slide 2/38, Monday, February 28th, 2011
k
th iteration
STEP 1:
Check for optimality
(Implicitly solve
A
T
k
λ
k
=
c
)
Solve
B
T
π
=
c
B
and set
z
N
=
c
N

N
T
π
.
If [
z
N
]
i
≥
0 for
i
= 1, 2, .
. . ,
n

m
, then
STOP
.
Otherwise, deﬁne [
z
N
]
s
= min(
z
N
).
The
s
th nonbasic variable (i.e.,
x
ν
s
) will become basic.
N
=
{
ν
1
, ν
2
, . . . ,
ν
s
↑
s
th element of
N
, . . . , ν
n

m
}
UCSD Center for Computational Mathematics
Slide 3/38, Monday, February 28th, 2011
STEP 2:
Compute the search direction
(Implicitly solve
A
k
p
k
=
e
m
+
s
)
B
N
0
I
N
!
p
B
p
N
!
=
0
e
s
!
A step along
p
k
increases
x
ν
s
but keeps other nonbasics ﬁxed at 0
UCSD Center for Computational Mathematics
Slide 4/38, Monday, February 28th, 2011
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View Full Document STEP 2: (continued)
Compute the search direction
Solve
Bp
B
+
Np
N
= 0
p
N
=
e
s
Bp
B
=

Np
N
=

Ne
s
=

(
s
th column of
N
)
=

(column
ν
s
of
A
)
=

a
ν
s
⇒
we solve
Bp
B
=

a
ν
s
UCSD Center for Computational Mathematics
Slide 5/38, Monday, February 28th, 2011
STEP 3:
Step to an adjacent vertex
If we take a step
α
along the vector
p
B
p
N
!
=
p
B
e
s
!
Then the basic variables change to
x
B
+
α
p
B
⇒
all
the basic variables change
UCSD Center for Computational Mathematics
Slide 6/38, Monday, February 28th, 2011
STEP 3: (continued)
Step to an adjacent vertex
What happens to the nonbasic variables?
x
N
+
α
p
N
=
x
N
+
α
e
s
=
x
N
+
0
.
.
.
0
α
0
.
.
.
0
=
0
.
.
.
0
α
0
.
.
.
0
←
row
s
⇒
all nonbasics remain at 0 except
x
ν
s
, which wants to
increase
.
UCSD Center for Computational Mathematics
Slide 7/38, Monday, February 28th, 2011
STEP 3: (continued)
Step to an adjacent vertex
The changed variables are
x
B
and
x
ν
s
(which increases from 0 to
α
)
We must ensure that
x
B
+
α
p
B
≥
0
σ
i
=
[
x
B
]
i

[
p
B
]
i
if [
p
B
]
i
<
0
+
∞
if [
p
B
]
i
≥
0
This is known as the
min ratio test
.
Deﬁne
α
= min
{
σ
i
}
If
α
= +
∞
, the LP is unbounded,
STOP
.
UCSD Center for Computational Mathematics
Slide 8/38, Monday, February 28th, 2011
STEP 3: (continued)
Step to an adjacent vertex
If
α
=
σ
t
then
[
x
B
]
t
+
σ
t
[
p
B
]
t
= 0
⇒
the
t
th basic becomes nonbasic
t
points to index
β
t
of
B
s
points to index
ν
s
of
N
[
x
B
]
t
→
0
⇒
x
β
t
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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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