∇
f
(
x
) +
m
X
i
=1
p
i
∇
G
i
(
x
) = 0
n
×
1
(5)
0
≤
G
(
x
)
⊥
p
≥
0
.
(6)
Here ‘
⊥
’ indicates the complementarity of two vectors.
If vectors
a
∈
R
m
and
b
∈
R
m
have the same dimensions, then
0
≤
a
⊥
b
≥
0
means that: (i)
a
≥
0
, (ii)
b
≥
0
, and (iii)
a
>
b
= 0
.
Two vectors that satisfy condition (iii) are said to be
complementary or perpendicular to each other.
Lizhi Wang ([email protected])
IE 534 Linear Programming
September 7, 2012
12 / 18
KKT conditions for linear programming
For linear programming, KKT conditions are both sufficient and
necessary for optimality
For any given
x
0
, it is an optimal solution to the LP if and only if
there exists
p
0
that satisfies the KKT conditions
Lizhi Wang ([email protected])
IE 534 Linear Programming
September 7, 2012
13 / 18
KKT conditions for nonlinear programming
For general nonlinear programming, KKT conditions are neither
sufficient nor necessary for optimality
Solution
(
x
1
= 1
, x
2
= 0)
does not satisfy KKT conditions, but it is
optimal to
max
x
1
s
.
t
.
x
2
+ (
x
1

1)
3
≤
0
x
2
≥
0
.
Solution
(
x
1
= 0
.
5
, x
2
= 0
.
25)
satisfies KKT conditions, but it is not
optimal to
min
x
2
s
.
t
.
x
2
1

x
1
+
x
2
≥
0
0
≤
x
1
≤
2
.
Lizhi Wang ([email protected])
IE 534 Linear Programming
September 7, 2012
14 / 18
KKT conditions for the standard form LP
max
{
c
>
x
:
Ax
≤
b, x
≥
0
}
max
f
(
x
) =
c
>
x
:
G
(
x
) =

A
m
×
n
I
n
×
n
x
+
b
m
×
1
0
n
×
1
≥
0
(
m
+
n
)
×
1
∇
f
(
x
) +
m
X
i
=1
p
i
∇
G
i
(
x
) = 0
n
×
1
(7)
0
≤
G
(
x
)
⊥
p
≥
0
.
(8)
c
+

A
>
I
y
μ
= 0
(9)
0
≤
b

Ax
x
⊥
y
μ
≥
0
(10)
0
≤
b

Ax
x
⊥
y
A
>
y

c
≥
0
.
(11)
Lizhi Wang ([email protected])
IE 534 Linear Programming
September 7, 2012
15 / 18
LP optimality condition
0
≤
b

Ax
x
⊥
y
A
>
y

c
≥
0
.
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 Spring '12
 lizhiwang
 Operations Research, Linear Programming, Optimization, Lizhi Wang