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2016/10/11 3.4 D3

LP Duality

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Page 1: LP Duality

2016/10/11

3.4

D3

Page 2: LP Duality

2016/10/11

Page 3: LP Duality

2016/10/11

Page 4: LP Duality

2016/10/11

Abstract• LP

• LP = LP

• ( : ( 3.13) LP ≤ LP )

• LP

• Farkas

• LP LP

• c A

Page 5: LP Duality

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}

Page 6: LP Duality

2016/10/11

3.19 LP LP ( )

:

Page 7: LP Duality

2016/10/11

3.19 LP LP ( )

: LP

LP

max

8<

:�b>y :

0

@A>

�A>

�I

1

A y

0

@c�c0

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

@zz0

w

1

A = �b, z, z0, w � 0

9=

;

Page 8: LP Duality

2016/10/11

3.19 LP LP ( )

( ):

x

0

min��c

>x

0 : �Ax

0 � w = �b, w � 0

x

0> = z

0> � z

>

max

�c

>x

0: Ax

0+ w = b, w � 0

w

LPmax{c>x0

: Ax

0 b}

min

8<

:z>c� z0>c : (A �A � I)

0

@zz0

w

1

A = �b, z, z0, w � 0

9=

;

Page 9: LP Duality

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

Page 10: LP Duality

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}

Page 11: LP Duality

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}

Page 12: LP Duality

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}

Page 13: LP Duality

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}

Page 14: LP Duality

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}

Page 15: LP Duality

2016/10/11

• (complementary

slackness condition)

• :

• 0 →

y

>(b�Ax) = 0

x

⇤ 2 P = {x : Ax b} max{c>x : x 2 P}

c A

0x

⇤ = b

0 A0A

y> � 0 A0 y>A = y>A0

Page 16: LP Duality

2016/10/11

3.22 ,

:

P := {x : Ax b} Z ✓ P

z 2 Z zmax{c>x : x 2 P}

c C

A

0x b

0 z 2 Z A0z = b0 Ax b

A0C

A0

A0 z

Page 17: LP Duality

2016/10/11

3.22 ,

:

( )

P := {x : Ax b} Z ✓ P

z 2 Z zmax{c>x : x 2 P}

c C

A

0x b

0 z 2 Z A0z = b0 Ax b

A0C

A0z 2 conv(Z)

conv(·)

z =P|Z|

i=1 zi/|Z| OK

Page 18: LP Duality

2016/10/11

3.22 ,

( ): ,

P := {x : Ax b} Z ✓ P

z 2 Z zmax{c>x : x 2 P}

c C

A

0x b

0 z 2 Z A0z = b0 Ax b

A0C

z 2 conv(Z)

c

>z =

Pki=1 �ic

>zi = max{c>x : x 2 P}

Pki=1 �i = max{c>x : x 2 P}

z =Pk

i=1 �izi (zi 2 Z)

Pki=1 �i = 1 z 2 P

(∵ )

Page 19: LP Duality

2016/10/11

3.22 ,

( ): 3.21

P := {x : Ax b} Z ✓ P

z 2 Z zmax{c>x : x 2 P}

c C

A

0x b

0 z 2 Z A0z = b0 Ax b

A0C

Az b

z c

A0

Page 20: LP Duality

2016/10/11

3.22 ,

( ):

P := {x : Ax b} Z ✓ P

z 2 Z zmax{c>x : x 2 P}

c C

A

0x b

0 z 2 Z A0z = b0 Ax b

A0C

c A0

A

0x = b

0a

0x = �

0 z 2 Z

a

0z = �

0= max{a0>x : x 2 P} a0 2 C

Page 21: LP Duality

2016/10/11

3.22 ,

( ):

P := {x : Ax b} Z ✓ P

z 2 Z zmax{c>x : x 2 P}

c C

A

0x b

0 z 2 Z A0z = b0 Ax b

A0C

c A0 a0 2 C

A0C

Page 22: LP Duality

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 ≥0y

>(b�Ax) 0 (c> � y

>A)x

y

>(b�Ax) = (c> � y

>A)x

Page 23: LP Duality

2016/10/11

(primal complementary slackness condition) •

(dual complementary slackness condition)

(c> �A

>y)x = 0

y

>(b�Ax) = 0

Page 24: LP Duality

2016/10/11

• LP LP

• LP

(unbounded)

(infeasible)

Page 25: LP Duality

2016/10/11

Abstract• LP

• LP = LP

• ( : ( 3.13) LP ≤ LP )

• LP

• Farkas

• LP → LP

• c A

Page 26: LP Duality

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}

Page 27: LP Duality

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

Page 28: LP Duality

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}

Page 29: LP Duality

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>)

✓b0

◆:

✓A> 0

�I � I

◆✓yz

◆=

✓0�1

◆, y, z � 0

= min{y>b : A>y = 0, y + z 1, y, z � 0}

=

=

Page 30: LP Duality

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|

Page 31: LP Duality

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

Page 32: LP Duality

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

�✓yw

◆= A>y � w = 0

✓yw

◆� 0

�y> w

�✓b0

◆= y>b � 0

w

Page 33: LP Duality

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>�

✓y1y2

◆= A>(y1 � y2) � 0

✓y1y2

◆� 0

�y>1 y>2

�✓ b�b

◆= (y>1 � y>2 )b � 0

y = y1 � y2

Page 34: LP Duality

2016/10/11

3.27 LP LP

LP LP

: ( LP LP )

: LP (= )

LP 1

3.13 ( ) LP

LP

x

y

c

>x y

>b

Page 35: LP Duality

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

Page 36: LP Duality

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 b

x

x

⇤ � z

c

>(x⇤ � z) > cx

Page 37: LP Duality

2016/10/11

LP LP

• LP LP 4

• LP LP

• LP LP

• LP LP

• LP LP

Page 38: LP Duality

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

Page 39: LP Duality

2016/10/11

3.29

: ( ) 3.12

( ) 3.12C = {x 2 Rn : Ax 0} C

✓AI

◆n

M

ej b0 = ±ej b0

My = b0 yb0

Page 40: LP Duality

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}

Page 41: LP Duality

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

Page 42: LP Duality

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

y�y 2 D

b1, . . . , bsD �y = Bz

z � 0 0 < �y>w = z>B>w 0≤0≥0

Page 43: LP Duality

2016/10/11

• LP := LP = LP

• LP

• Farkas

• LP LP

• LP

c 2 A