This document discusses linear programming (LP) duality. It begins by introducing LP duality and establishing that the optimal values of a primal and dual LP are equal. It then presents the dual LP formulation and complementary slackness conditions. The document explores relationships between primal and dual feasible regions and optimal solutions. It introduces the Farkas lemma and uses it to prove LP duality. Applications of duality theory including LP infeasibility and unboundedness are also discussed.

1. 2016/10/11
3.4
D3

2. 2016/10/11

3. 2016/10/11
•
•
•
•

4. 2016/10/11
Abstract
• LP
• LP = LP
• ( : ( 3.13) LP ≤ LP )
• LP
• Farkas
• LP LP
• c A

5. 2016/10/11
3.18 LP LP
LP
LP LP (dual LP)
LP
LP (primal LP)
max c>
x : Ax  b
max c>
x : Ax  b
min{y>
b : A>
y = c, y 0}

6. 2016/10/11
3.19
LP LP ( )
:

7. 2016/10/11
3.19
LP LP ( )
: LP
LP
max
8
<
:
b>
y :
0
@
A>
A>
I
1
A y 
0
@
c
c
0
1
A
9
=
;
max c>
x : Ax  b
min{y>
b : A>
y = c, y 0}
min
8
<
:
z>
c z0>
c : (A A I)
0
@
z
z0
w
1
A = b, z, z0
, w 0
9
=
;

8. 2016/10/11
3.19
LP LP ( )
( ):
x0
min c>
x0
: Ax0
w = b, w 0
x0>
= z0>
z>
max c>
x0
: Ax0
+ w = b, w 0
w
LP
max{c>
x0
: Ax0
 b}
min
8
<
:
z>
c z0>
c : (A A I)
0
@
z
z0
w
1
A = b, z, z0
, w 0
9
=
;

9. 2016/10/11
3.20
: LP
LP LP
3.19
LP
,
LP LP
P D P := {x : Ax  b}
max{c>
x : x 2 P} = min{y>
b : y 2 D}
DP
D := {y : A>
y = c, y 0}
D

10. 2016/10/11
3.20
:
3.10
D
B>
:= (A A I), d>
:= (c>
c>
0)
rank(B) = m
D
,
LP LP
P D P := {x : Ax  b}
max{c>
x : x 2 P} = min{y>
b : y 2 D}
DP
D := {y : A>
y = c, y 0}
D := {y : B>
y  d}

11. 2016/10/11
3.20
,
LP LP
( ):
( 3.13)
→
P D P := {x : Ax  b}
max{c>
x : x 2 P} = min{y>
b : y 2 D}
DP
D y
min{y>
b : y 2 D}
P
min{y>
b : y 2 D} = max{ y>
b : y 2 D}
max{ y>
b : y 2 D}
y
D := {y : A>
y = c, y 0}

12. 2016/10/11
3.20
,
LP LP
( ): LP LP
,
LP 3.19
( LP)
P D P := {x : Ax  b}
max{c>
x : x 2 P} = min{y>
b : y 2 D}
DP
min{y>
b : y 2 D}
zy
max{c>
x : x 2 P}
y>
b = c>
z
D := {y : A>
y = c, y 0}

13. 2016/10/11
3.21
LP
(a) - (c)
(a)
(b)
(c)
: 3.20 (a) (b)
3.13 (b) (a)
(b) (c)
max c>
x : Ax  b
y>
(b Ax) = 0
y>
(b Ax) = y>
b y>
Ax = y>
b c>
x = 0
y>
b = c>
x
x
x
y
y
min{y>
b : A>
y = c, y 0}

14. 2016/10/11
3.21
LP
(a) - (c)
(a)
(b)
(c)
: 3.20 (a) (b)
3.13 (b) (a)
(b) (c)
max c>
x : Ax  b
y>
(b Ax) = 0
y>
(b Ax) = y>
b y>
Ax = y>
b c>
x = 0
y>
b = c>
x
x
x
y
y
min{y>
b : A>
y = c, y 0}

15. 2016/10/11
• (complementary
slackness condition)
• :
•
• 0 →
y>
(b Ax) = 0
x⇤
2 P = {x : Ax  b} max{c>
x : x 2 P}
c A0
x⇤
= b0
A0
A
y>
0 A0
y>
A = y>
A0

16. 2016/10/11
3.22
,
:
P := {x : Ax  b} Z ✓ P
z 2 Z z max{c>
x : x 2 P}
c C
A0
x  b0
z 2 Z A0
z = b0
Ax  b
A0C
A0
A0
z

17. 2016/10/11
3.22
,
:
( )
P := {x : Ax  b} Z ✓ P
z 2 Z z max{c>
x : x 2 P}
c C
A0
x  b0
z 2 Z A0
z = b0
Ax  b
A0C
A0
z 2 conv(Z)
conv(·)
z =
P|Z|
i=1 zi/|Z| OK

18. 2016/10/11
3.22
,
( ): ,
P := {x : Ax  b} Z ✓ P
z 2 Z z max{c>
x : x 2 P}
c C
A0
x  b0
z 2 Z A0
z = b0
Ax  b
A0C
z 2 conv(Z)
c>
z =
Pk
i=1 ic>
zi = max{c>
x : x 2 P}
Pk
i=1 i = max{c>
x : x 2 P}
z =
Pk
i=1 izi (zi 2 Z)
Pk
i=1 i = 1 z 2 P
(∵ )

19. 2016/10/11
3.22
,
( ): 3.21
P := {x : Ax  b} Z ✓ P
z 2 Z z max{c>
x : x 2 P}
c C
A0
x  b0
z 2 Z A0
z = b0
Ax  b
A0C
Az  b
z c
A0

20. 2016/10/11
3.22
,
( ):
P := {x : Ax  b} Z ✓ P
z 2 Z z max{c>
x : x 2 P}
c C
A0
x  b0
z 2 Z A0
z = b0
Ax  b
A0C
c A0
A0
x = b0 a0
x = 0
z 2 Z
a0
z = 0
= max{a0>
x : x 2 P} a0
2 C

21. 2016/10/11
3.22
,
( ):
P := {x : Ax  b} Z ✓ P
z 2 Z z max{c>
x : x 2 P}
c C
A0
x  b0
z 2 Z A0
z = b0
Ax  b
A0C
c A0
a0
2 C
A0C

22. 2016/10/11
3.23
LP
(a) - (c)
(a)
(b)
(c)
: (a) (b):
(b) (c):
y>
(b Ax) = 0
y>
b = c>
x
x
x
y
y
min{c>
x : Ax b, x 0} max{y>
b : y>
A  c, y 0}
(c>
A>
y)x = 0
max
⇢
( c>
)x :
✓
A
I
◆
x 
✓
b
0
◆
y>
b = c>
x
≤0≥0 ≥0 ≥0
y>
(b Ax)  0  (c>
y>
A)x
y>
(b Ax) = (c>
y>
A)x

23. 2016/10/11
•
(primal complementary slackness condition)
•
(dual complementary slackness condition)
(c>
A>
y)x = 0
y>
(b Ax) = 0

24. 2016/10/11
• LP LP
• LP
(unbounded)
(infeasible)

25. 2016/10/11
Abstract
• LP
• LP = LP
• ( : ( 3.13) LP ≤ LP )
• LP
• Farkas
• LP → LP
• c A

26. 2016/10/11
3.24
(=
)
:
0
0
Ax  b
A>
y = 0 y 0
y>
b 0
x
Ax  b
min{y>
b : A>
y = 0, 0  y  1}
A>
y = 0 y 0 y>
b 0
x
1
min{1>
w : Ax w  b, w 0}

27. 2016/10/11
3.24
(=
)
(1):
0
LP 0 0
Ax  b
A>
y = 0 y 0
y>
b 0
x
Ax  b x
min{1>
w : Ax w  b, w 0}
w

28. 2016/10/11
3.24
(=
)
(2):
0
0
Ax  b
A>
y = 0 y 0
y>
b 0
x
min{y>
b : A>
y = 0, 0  y  1}
min{1>
w : Ax w  b, w 0}

29. 2016/10/11
3.24
(=
)
(2):
0
0
Ax  b
A>
y = 0 y 0
y>
b 0
x
min{y>
b : A>
y = 0, 0  y  1}
min{1>
w : Ax w  b, w 0}
max
⇢
0 1
✓
x
w
◆
:
✓
A I
0 I
◆ ✓
x
w
◆

✓
b
0
◆
min
⇢
(y>
z>
)
✓
b
0
◆
:
✓
A>
0
I I
◆ ✓
y
z
◆
=
✓
0
1
◆
, y, z 0
= min{y>
b : A>
y = 0, y + z  1, y, z 0}
=
=

30. 2016/10/11
3.24
(=
)
(2)-2:
0
0
,
→
Ax  b
A>
y = 0 y 0
y>
b 0
x
min{y>
b : A>
y = 0, 0  y  1}
min{1>
w : Ax w  b, w 0}
x = 0, w = |b| y = 0
|b|i = |bi|

31. 2016/10/11
3.24
(=
)
(3):
0
( )
( ) 0
Ax  b
A>
y = 0 y 0
y>
b 0
x
min{y>
b : A>
y = 0, 0  y  1}
A>
y = 0 y 0 y>
b 0
0  y  1 y = cy0
 1

32. 2016/10/11
3.25
: , 3.24
Ax  b
y 0 y>
b 0
x 0
A>
y 0
✓
A
I
◆
x 
✓
b
0
◆
x 0
A>
I
✓
y
w
◆
= A>
y w = 0
✓
y
w
◆
0
y>
w
✓
b
0
◆
= y>
b 0
w

33. 2016/10/11
: , 3.25
3.26 Farkas
y>
b 0
x 0
A>
y 0
Ax = b
y
✓
A
A
◆
x 
✓
b
b
◆
x 0
A>
A>
✓
y1
y2
◆
= A>
(y1 y2) 0
✓
y1
y2
◆
0
y>
1 y>
2
✓
b
b
◆
= (y>
1 y>
2 )b 0
y = y1 y2

34. 2016/10/11
3.27
LP LP
LP LP
: ( LP LP )
: LP (= )
LP 1
3.13 ( ) LP
LP
x
y
c>
x  y>
b

35. 2016/10/11
3.27
LP LP
LP LP
: (LP LP )
1 LP
: LP LP
Farkas
A>
y = c
y 0
Az 0 z>
c < 0 z 2 P
x⇤

36. 2016/10/11
3.27
LP LP
LP LP
( ): (LP LP )
Az 0 z>
c < 0 z 2 P
x⇤
A(x⇤
z) = Ax⇤
Az  bx⇤
x⇤
z
c>
(x⇤
z) > cx⇤

37. 2016/10/11
LP LP
• LP LP 4
• LP LP
• LP LP
• LP LP
• LP LP

38. 2016/10/11
3.28
LP
:
LP LP
( ) 3.27 2
( ) 3.27 1
max c>
x : Ax  b
c A
A>
y = c y 0
c A

39. 2016/10/11
3.29
: ( ) 3.12
( ) 3.12
C = {x 2 Rn
: Ax  0} C
✓
A
I
◆
n
M
ej b0
= ±ej b0
My = b0 yb0

40. 2016/10/11
3.29
: ( )
3.12
(= )
a1, . . . , atC
A
b1, . . . , bs
a1, . . . , at
b1, . . . , bs B
C = {x : Bx  0}
D := {x : Ax  0}

41. 2016/10/11
3.29
: ( )
b1, . . . , bsD := {x : Ax  0}
bi 2 D b>
j ai = a>
i bj  0i, j
ai 2 {x : Bx  0} C ✓ {x : Bx  0}
w 2 {x : Bx  0} C
C = {x : Bx  0}
w

42. 2016/10/11
3.29
: ( ) :
Farkas (3.26)
( )
w 2 {x : Bx  0} C
w /2 C
A>
v = w
w a1, . . . , at
Ay 0 y>
w < 0
v 0
yy 2 D
b1, . . . , bsD y = Bz
z 0 0 < y>
w = z>
B>
w  0
≤0≥0

43. 2016/10/11
• LP := LP = LP
•
• LP
• Farkas
• LP LP
• LP
•
c 2 A