Now 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