Lecture 18
Linear Programs in Standard Form
UCSD Math 171A: Numerical Optimization
Philip E. Gill
http://ccom.ucsd.edu/~peg/math171
Monday, February 23rd, 2009
Linear programs in standard form
minimize
x
∈
R
n
c
T
x
subject to
Ax
=
b

{z
}
equality constraints
,
x
≥
0

{z
}
simple bounds
The matrix
A
is
m
×
n
with shape
A
=
Often,
n
±
m
.
We will show that the two
n
×
n
systems
A
T
k
λ
k
=
c
and
A
k
p
k
=
e
s
are equivalent to
two systems of order m
.
UCSD Center for Computational Mathematics
Slide 2/36, Monday, February 23rd, 2009
Every linear program can be written in standard form.
For example, suppose the constraints are in the format
Ax
≥
b
.
⇒
there are no simple bounds (i.e., the
x
j
are “free variables”)
There are two steps involved in reformulating the constraints in
standard form.
1
First we convert all the free variables into bounded variables
2
Then we convert the general inequalities into equalities
UCSD Center for Computational Mathematics
Slide 3/36, Monday, February 23rd, 2009
Deﬁne new nonnegative variables
u
i
and
v
i
such that
x
i
=
u
i

v
i
,
u
i
≥
0
,
v
i
≥
0
Then
Ax
=
A
(
u

v
) =
Au

Av
=
(
A

A
)
±
u
v
²
≥
b
This gives the linear program with
n
0
= 2
n
variables:
minimize
x
0
∈
R
n
0
c
0
T
x
0
subject to
A
0
x
0
≥
b
0
,
x
0
≥
0
with
A
0
=
(
A

A
)
,
x
0
=
±
u
v
²
,
c
0
=
±
c

c
²
and
b
0
=
b
UCSD Center for Computational Mathematics
Slide 4/36, Monday, February 23rd, 2009
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View Full DocumentNow we assume that every variable is simply bounded, i.e.,
x
i
≥
0.
minimize
x
1
,
x
2
,
x
3

6
x
1

9
x
2

5
x
3
subject to
2
x
1
+ 3
x
2
+
x
3
≤
5
x
1
+ 2
x
2
+
x
3
≥
3
x
1
,
x
2
,
x
3
≥
0
Consider a new variable
x
4
and the constraint
2
x
1
+ 3
x
2
+
x
3
+
x
4
= 5
For feasible
x
1
,
x
2
,
x
3
and for
x
4
≥
0
, then
2
x
1
+ 3
x
2
+
x
3
≤
5
x
4
is called a
slack variable
.
UCSD Center for Computational Mathematics
Slide 5/36, Monday, February 23rd, 2009
Consider a new variable
x
5
and the constraint
x
1
+ 2
x
2
+
x
3

x
5
= 3
For feasible
x
1
,
x
2
,
x
3
and
x
5
≥
0
, then
x
1
+ 2
x
2
+
x
3
≥
3
x
5
is called a
surplus variable
.
UCSD Center for Computational Mathematics
Slide 6/36, Monday, February 23rd, 2009
minimize
x
1
,
x
2
,
x
3
,
x
4
,
x
5

6
x
1

9
x
2

5
x
3
subject to
2
x
1
+ 3
x
2
+
x
3
+
x
4
= 5
x
1
+ 2
x
2
+
x
3

x
5
= 3
x
1
,
x
2
,
x
3
,
x
4
,
x
5
≥
0
“Slack variable” is a generic name for both slack and surplus
variables
An equality constraint
a
T
x
=
b
needs neither a slack nor
surplus variable.
UCSD Center for Computational Mathematics
Slide 7/36, Monday, February 23rd, 2009
Optimality conditions for an LP in standard form
minimize
x
∈
R
n
c
T
x
subject to
Ax
=
b
x
≥
0
⇒
minimize
x
∈
R
n
c
T
x
subject to
Ax
=
b
Dx
≥
f
with
±
D
=
I
f
= 0
The “full” vector of residuals at a feasible point is:
A
D
!
x
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 staff
 Math, Linear Programming, Optimization, UCSD Center for Computational Mathematics, UCSD Center, Computational Mathematics

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