F x m x i 1 p i g i x 0 n 1 5 g x p 6 here indicates

∇ 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 .