Substitutes and Complements in Network Flows Viewed as Discrete - - PowerPoint PPT Presentation

SLIDE 1

Substitutes and Complements in Network Flows Viewed as Discrete Convexity

K a z u

• M

u r

• t

a ( U n i v . T

• k

y

• ,

J a p a n ) A k i y

• s

h i S h i

• u

r a ( Z I B , G e r m a n y ; T

• h
• k

u U n i v . , J a p a n )

SLIDE 2

Overview of Our Result

M a x w e i g h t c i r c u l a t i

• n

p r

• b

l e m

N:

v e r t e x

• a

r c i n c i d e n c e m a t r i x

• f

d i g r a p h G=(V,A)

w(

a) : w e i g h t

• f

a∈A, c( a) : c a p a c i t y

• f

a∈A O u r I n t e r e s t : c

• m

b i n a t

• r

i a l p r

• p

e r t i e s

• f
• p

t i m a l v a l u e f u n c t i

• n

f u n c t i

• n

i n p r

• b

l e m p a r a m e t e r s : w e i g h t w & c a p a c i t y c

SLIDE 3

Overview of Our Result

G

a l e

• P
• l

i t

• f

( 1 9 8 1 )

F* i

s s u b m

• d

u l a r / s u p e r m

• d

u l a r w . r . t . s

• m

e p a r a m e t e r s

w

e l l

• k

n

• w

n r e s u l t i n p a r a m e t r i c L P

F* i

s c

• n

v e x / c

• n

c a v e w . r . t . s

• m

e p a r a m e t e r s

O

u r R e s u l t :

p

r

• v

i d e a b e t t e r u n d e r s t a n d i n g f r

• m

t h e v i e w p

• i

n t

• f

“d i s c r e t e c

• n

v e x a n a l y s i s ” ( M u r

• t

a 1 9 9 6 )

F* i

s M

• c
• n

v e x / L

• c
• n

v e x &M

• c
• n

c a v e / L

• c
• n

c a v e

n

i c e p r

• p

e r t i e s i n M a t h e m a t i c a l E c

• n
• m

i c s

SLIDE 4

Sub-/Supermodularity of F*

P: “p a r a l l e l ” a r c s e t , S: “s e r i e s ” a r c s e t F* i s ( 1 ) s u b m

• d

u l a r w . r . t . { w( a) | a∈P} a n d { c( a) | a∈P} ( 2 ) s u p e r m

• d

u l a r w . r . t . { w( a) | a∈S} a n d { c( a) | a∈S} T h e

• r

e m ( G a l e

• P
• l

i t

• f

1 9 8 1 )

SLIDE 5

“Parallel” & “Series” Arcs

P⊆A:

a “p a r a l l e l ” a r c s e t ∀C: u n d i r e c t e d s i m p l e c y c l e i n G, a r c s i n P∩C h a v e d i f f e r e n t d i r e c t i

• n

s i n C “p a r a l l e l ” a r c s e t

SLIDE 6

“Parallel” & “Series” Arcs

P⊆A:

a “p a r a l l e l ” a r c s e t ∀C: u n d i r e c t e d s i m p l e c y c l e i n G, a r c s i n P∩C h a v e d i f f e r e n t d i r e c t i

• n

s i n C

SLIDE 7

“Parallel” & “Series” Arcs

P⊆A:

a “p a r a l l e l ” a r c s e t ∀C: u n d i r e c t e d s i m p l e c y c l e i n G, a r c s i n P∩C h a v e d i f f e r e n t d i r e c t i

• n

s i n C

SLIDE 8

“Parallel” & “Series” Arcs

P⊆A:

a “p a r a l l e l ” a r c s e t ∀C: u n d i r e c t e d s i m p l e c y c l e i n G, a r c s i n P∩C h a v e d i f f e r e n t d i r e c t i

• n

s i n C

SLIDE 9

“Parallel” & “Series” Arcs

S⊆A:

a “s e r i e s ” a r c s e t ∀C: u n d i r e c t e d s i m p l e c y c l e i n G, a r c s i n S∩C h a v e t h e s a m e d i r e c t i

• n

i n C “s e r i e s ” a r c s e t

SLIDE 10

“Parallel” & “Series” Arcs

S⊆A:

a “s e r i e s ” a r c s e t ∀C: u n d i r e c t e d s i m p l e c y c l e i n G, a r c s i n S∩C h a v e t h e s a m e d i r e c t i

• n

i n C

SLIDE 11

“Parallel” & “Series” Arcs

S⊆A:

a “s e r i e s ” a r c s e t ∀C: u n d i r e c t e d s i m p l e c y c l e i n G, a r c s i n S∩C h a v e t h e s a m e d i r e c t i

• n

i n C

SLIDE 12

Sub-/Supermodularity of F*

P: “p a r a l l e l ” a r c s e t , S: “s e r i e s ” a r c s e t F* i s ( 1 ) s u b m

• d

u l a r w . r . t . { w( a) | a∈P} a n d { c( a) | a∈P} ( 2 ) s u p e r m

• d

u l a r w . r . t . { w( a) | a∈S} a n d { c( a) | a∈S} T h e

• r

e m ( G a l e

• P
• l

i t

• f

1 9 8 1 )

SLIDE 13

Convexity/Concavity of F*

w

e l l

• k

n

• w

n r e s u l t i n p a r a m e t r i c L P F* i s ( 1 ) c

• n

v e x w . r . t . { w( a) | a∈A} ( 2 ) c

• n

c a v e w . r . t . { c( a) | a∈A} ( A: s e t

• f

a l l a r c s i n G) P r

• p
• s

i t i

• n

SLIDE 14

Summary of Previous Results

P:

“p a r a l l e l ” a r c s e t , S: “s e r i e s ” a r c s e t

F* i

s s u b m

• d

u l a r & c

• n

v e x w . r . t . { w( a) | a∈P}

F* i

s s u b m

• d

u l a r & c

• n

c a v e w . r . t . { c( a) | a∈P}

F* i

s s u p e r m

• d

u l a r & c

• n

v e x w . r . t . { w( a) | a∈S}

F* i

s s u p e r m

• d

u l a r & c

• n

c a v e w . r . t . { c( a) | a∈S} a l l c

• m

b i n a t i

• n
• f

s u b m

• d

u l a r i t y / s u p e r m

• d

u l a r i t y a n d c

• n

v e x i t y / c

• n

c a v i t y

SLIDE 15

Our Result

P:

“p a r a l l e l ” a r c s e t , S: “s e r i e s ” a r c s e t

F* i

s s u b m

• d

u l a r & c

• n

v e x w . r . t . { w( a) | a∈P}

F* i

s s u b m

• d

u l a r & c

• n

c a v e w . r . t . { c( a) | a∈P}

F* i

s s u p e r m

• d

u l a r & c

• n

v e x w . r . t . { w( a) | a∈S}

F* i

s s u p e r m

• d

u l a r & c

• n

c a v e w . r . t . { c( a) | a∈S} c

• n

s e q u e n c e

• f

d i s c r e t e c

• n

v e x i t y / c

• n

c a v i t y : M

• c
• n

v e x i t y / M

• c
• n

c a v i t y a n d L

• c
• n

v e x i t y / L

• c
• n

c a v i t y

SLIDE 16

M-convexity & L-convexity

M

• c
• n

v e x a n d L

• c
• n

v e x f u n c t i

• n

s

i

n t r

• d

u c e d b y M u r

• t

a ( 1 9 9 6 )

p

l a y c e n t r a l r

• l

e i n t h e

• r

y

• f

“d i s c r e t e c

• n

v e x a n a l y s i s ”

t

h e

• r

e t i c a l f r a m e w

• r

k f

• r

w e l l

• s
• l

v e d c

• m

b i n a t

• r

i a l

• p

t i m i z a t i

• n

p r

• b

l e m s

d

e e p r e l a t i

• n

s h i p w i t h ( p

• l

y ) m a t r

• i

d t h e

• r

y

c

• n

c e p t s f

• r

f u n c t i

• n
• v

e r i n t e g e r l a t t i c e Zn l a t e r e x t e n d e d t

• f

u n c t i

• n

s

• v

e r r e a l s p a c e Rn

e

n j

• y

n i c e p r

• p

e r t i e s a s “d i s c r e t e c

• n

v e x i t y ”

SLIDE 17

Fundamental Properties of M-convexity & L-convexity

M

• c
• n

v e x , L

• c
• n

v e x ( c a n b e e x t e n d e d t

• )
• r

d i n a r y c

• n

v e x f u n c t i

• n

s

M

• c
• n

v e x s u p e r m

• d

u l a r

L

• c
• n

v e x s u b m

• d

u l a r

n

i c e p r

• p

e r t i e s i n M a t h e m a t i c a l E c

• n
• m

i c s

SLIDE 18

Our Main Result

P: “p a r a l l e l ” a r c s e t , S: “s e r i e s ” a r c s e t F* i s ( 1 ) L

• c
• n

v e x w . r . t . { w( a) | a∈P} s u b m

• d

u l a r & c

• n

v e x ( 2 ) M

• c
• n

c a v e w . r . t . { c( a) | a∈P} s u b m

• d

u l a r & c

• n

c a v e ( 3 ) M

• c
• n

v e x w . r . t . { w( a) | a∈S} s u p e r m

• d

u l a r & c

• n

v e x ( 4 ) L

• c
• n

c a v e w . r . t . { c( a) | a∈S} s u p e r m

• d

u l a r & c

• n

c a v e T h e

• r

e m

SLIDE 19

Implication in Math Economics

s

u p p

• s

e F* r e p r e s e n t s u t i l i t y f u n c t i

• n

w . r . t . s

• m

e g

• d

s

F* :

s u b m

• d

u l a r g

• d

s a r e s u b s t i t u t e s e . g . , c

• f

f e e a n d t e a

F* :

s u p e r m

• d

u l a r g

• d

s a r e c

• m

p l e m e n t s e . g . , l

• c

k a n d k e y

M

• c
• n

c a v i t y

• g

r

• s

s s u b s t i t u t e s p r

• p

e r t y s i n g l e i m p r

• v

e m e n t c

• n

d i t i

• n

n

• c
• m

p l e m e n t a r i t y c

• n

d i t i

• n

G a l e P

• l

i t

• f

( 1 9 8 1 ) F* h a s n i c e r p r

• p

e r t y t h a n s u b

• /

s u p e r m

• d

u l a r i t y

SLIDE 20

Definition of M-convex Function

f is M-convex i x y

( , 1 , , , ) ( , 1 , ,

• 1

, ) ( ,

• 1

, , 1 , ) ( ,

• 1

, , , )

ｊ

SLIDE 21

Definition of L-convex Function

g is L-convex p q

SLIDE 22

Extension to Linear Program

L P

• v

e r U n i m

• d

u l a r L i n e a r S p a c e t

• t

a l l y u n i m

• d

u l a r m a t r i x s a m e r e s u l t h

• l

d s a s F*

SLIDE 23

Extension to Separable Concave Program

S e p a r a b l e C

• n

c a v e P r

• g

r a m

• v

e r U n i m

• d

u l a r L i n e a r S p a c e t

• t

a l l y u n i m

• d

u l a r m a t r i x s e p a r a b l e c

• n

c a v e f n P : “p a r a l l e l ” a r c s e t , S: “s e r i e s ” a r c s e t FSC i s ( 1 ) M

• c
• n

c a v e w . r . t . { c( a) | a∈P} ( 2 ) L

• c
• n

c a v e w . r . t . { c( a) | a∈S}

SLIDE 24

Thanks for your attention!