IE 534 Linear Programming
Lecture 13: The Simplex Algorithm (3)
Lizhi Wang
October 2, 2015
Lizhi Wang ([email protected])
IE 534 Linear Programming
October 2, 2015
1 / 12
The nuts and bolts
1
What exactly is a “corner point”?
2
Does the optimal solution always occur at a corner point?
3
How to find a corner point to start from?
4
How to check the optimality of a corner point?
5
How to find a better corner point if the current one is not optimal?
6
How to identify an infeasible or unbounded LP?
7
Is the Simplex algorithm guaranteed to terminate finitely?
8
Is the Simplex algorithm guaranteed to give the correct answer?
9
How efficient is the Simplex algorithm?
Lizhi Wang ([email protected])
IE 534 Linear Programming
October 2, 2015
2 / 12
No, but...
No
Some LPs may not even have a basic solution.
An optimal solution may not be a basic solution.
But
A standard form LP always has a basic solution.
If the optimal solution is unique, then it must be a basic solution.
If a standard form LP has infinitely many optimal solutions, then
one of them must be a basic solution.
Lizhi Wang ([email protected])
IE 534 Linear Programming
October 2, 2015
3 / 12
The nuts and bolts
1
What exactly is a “corner point”?
2
Does the optimal solution always occur at a corner point?
3
How to find a corner point to start from?
4
How to check the optimality of a corner point?
5
How to find a better corner point if the current one is not optimal?
6
How to identify an infeasible or unbounded LP?
7
Is the Simplex algorithm guaranteed to terminate finitely?
8
Is the Simplex algorithm guaranteed to give the correct answer?
9
How efficient is the Simplex algorithm?
Lizhi Wang ([email protected])
IE 534 Linear Programming
October 2, 2015
5 / 12
Starting basic partition
Define
LP0
as
max
x,w
{
ζ
=
c
>
x
:
Ax
+
w
=
b, x
≥
0
, w
≥
0
}
.
LP0
is equivalent to
max
x
{
ζ
=
c
>
x
:
A
x
=
b,
x
≥
0
}
with
c
=
c
0
m
×
1
,
A
=
A
I
m
×
m
, and
x
=
x
w
∈
R
n
+
m
.
(
x
= 0
, w
=
b
)
is a basic solution to
LP0
.
If
b
≥
0
, then the following basic partition
(
B
0
,
N
0
)
is a fbp
B
0
=
{
n
+ 1
, ..., n
+
m
}
,
N
0
=
{
1
, ..., n
}
,
A
B
0
=
I
m
×
m
,
A
N
0
=
A,
x
B
0
=
A

1
B
0
b
=
b,
x
N
0
= 0
n
×
1
.
Otherwise we proceed using a technique called the twophase
method, in which we consider an auxiliary problem
LP1
.
Lizhi Wang ([email protected])
IE 534 Linear Programming
October 2, 2015
6 / 12
Example 1
Consider the following
LP0
instance
max
x
ζ
= 13
x
1
+ 7
x
2

12
x
3
s
.
t
.
2
x
1
+ 3
x
2

x
3
≤
5

4
x
1

7
x
2
+ 2
x
3
≤

11
3
x
1

4
x
2

2
x
3
≤

8
x
1
, x
2
, x
3
≥
0
.
⇒
max
x,w
ζ
= 13
x
1
+ 7
x
2

12
x
3
s
.
t
.
2
x
1
+ 3
x
2

x
3
+
w
4
=
5

4
x
1

7
x
2
+ 2
x
3
+
w
5
=

11
3
x
1

4
x
2

2
x
3
+
w
6
=

8
x
1
, x
2
, x
3
, w
4
, w
5
, w
6
≥
0
.
Consider the following auxiliary problem
LP1
min
x,w,t
t
7
+
t
8
s
.
t
.
2
x
1
+ 3
x
2

x
3
+
w
4
=
5

4
x
1

7
x
2
+ 2
x
3
+
w
5

t
7
=

11
3
x
1

4
x
2

2
x
3
+
w
6

t
8
=

8
x
1
, x
2
, x
3
, w
4
, w
5
, w
6
, t
7
, t
8
≥
0
.
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 Fall '10
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 Operations Research, Linear Programming, Optimization, Simplex algorithm, Lizhi Wang