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2016/10/11
3.4
D3
2016/10/11
2016/10/11
•
•
•
•
2016/10/11
Abstract• LP
• LP = LP
• ( : ( 3.13) LP ≤ LP )
• LP
• Farkas
• LP LP
• c A
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}
2016/10/11
3.19 LP LP ( )
:
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=
;
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=
;
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
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}
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}
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}
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}
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}
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
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
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
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
(∵ )
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
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
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
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
2016/10/11
•
(primal complementary slackness condition) •
(dual complementary slackness condition)
(c> �A
>y)x = 0
y
>(b�Ax) = 0
2016/10/11
• LP LP
• LP
(unbounded)
(infeasible)
2016/10/11
Abstract• LP
• LP = LP
• ( : ( 3.13) LP ≤ LP )
• LP
• Farkas
• LP → LP
• c A
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}
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
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}
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}
=
=
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|
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
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
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
2016/10/11
3.27 LP LP
LP LP
: ( LP LP )
: LP (= )
LP 1
3.13 ( ) LP
LP
x
y
c
>x y
>b
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
⇤
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
⇤
2016/10/11
LP LP
• LP LP 4
• LP LP
• LP LP
• LP LP
• LP LP
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
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
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}
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
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
2016/10/11
• LP := LP = LP
•
• LP
• Farkas
• LP LP
• LP
•
c 2 A