IE 534 Linear Programming
Lecture 15: The Simplex Algorithm (5)
Lizhi Wang
September 26, 2012
Lizhi Wang ([email protected])
IE 534 Linear Programming
September 26, 2012
1 / 20
Outline
1
Review
2
How to check the unboundedness of an LP?
3
How to find a better fbp?
Lizhi Wang ([email protected])
IE 534 Linear Programming
September 26, 2012
2 / 20
The nuts and bolts
1
What exactly is a “corner point”?
2
Is the optimal solution always basic?
3
How to find a basic solution?
4
How to find a fbp to start from? What if LP is infeasible?
5
How to check the optimality of a fbp?
6
How to check the unboundedness of an LP?
7
How to find a better fbp?
8
Is the Simplex algorithm guaranteed to terminate finitely?
9
Is the Simplex algorithm guaranteed to give the correct answer?
10
How efficient is the Simplex algorithm?
Lizhi Wang ([email protected])
IE 534 Linear Programming
September 26, 2012
3 / 20
The Simplex diagram
Lizhi Wang ([email protected])
IE 534 Linear Programming
September 26, 2012
4 / 20
The Simplex dictionary
For any fbp
(
B
,
N
)
, the Simplex standard form can be written as
max
ζ
=
c
>
x
s
.
t
.
A
x
=
b
x
≥
0
⇒
max
ζ
=
c
>
B
x
B
+
c
>
N
x
N
s
.
t
.
A
B
x
B
+
A
N
x
N
=
b
x
B
≥
0
,
x
N
≥
0
A Simplex dictionary is defined as
ζ
=
c
>
B
A

1
B
b
+ (
c
>
N

c
>
B
A

1
B
A
N
)
x
N
x
B
=
A

1
B
b

A

1
B
A
N
x
N
.
The term
(
c
>
N

c
>
B
A

1
B
A
N
)
is called the
reduced cost
.
Lizhi Wang ([email protected])
IE 534 Linear Programming
September 26, 2012
5 / 20
Optimality check
Proposition
For a fbp
(
B
,
N
)
, if
(
c
>
N

c
>
B
A

1
B
A
N
)
≤
0
, then the fbs
(
x
B
=
A

1
B
b,
x
N
= 0)
is optimal.
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 Spring '12
 lizhiwang
 Operations Research, Linear Programming, Optimization, Simplex algorithm, Lizhi Wang