Inte

grat

ing

Sol

utio

n M

etho

ds

thro

ugh

Dua

lity

John

Hoo

ker

US

-Mex

ico

Wor

ksho

p on

Opt

imiz

atio

nO

axac

a, J

anua

ry 2

011

Slid

e 2

Zap

otec

Dua

lity

mas

kLa

te c

lass

ical

pe

riod

Rep

rese

nts

life/

deat

h,

day/

nigh

t, he

at/c

old

See

a s

imila

r mas

k in

Cen

tro

Cul

tura

l S

anto

Dom

ingo

Slid

e 3

Lalit

aM

ahat

ripur

asun

dari

repr

esen

ting

Bra

hman

-Atm

an

cons

ciou

snes

s (m

ind/

body

)

Dua

lity

Two

pers

pect

ives

on

the

sam

e ph

enom

enon

Com

plem

enta

rity

Min

glin

g of

opp

osite

s

Yin

/yan

gD

aois

m

Alh

aram

azda

Zor

oast

rian

god

Goo

d/ev

il

Slid

e 4

Goa

l

•D

esig

n a

gene

ral-p

urpo

se

solv

er th

at e

xplo

its

prob

lem

-spe

cific

str

uctu

re.

Slid

e 5

Goa

l

•D

esig

n a

gene

ral-p

urpo

se

solv

er th

at e

xplo

its

prob

lem

-spe

cific

str

uctu

re.

•A

ppro

ach:

•F

ind

com

mon

alg

orith

mic

str

uctu

re

acro

ss s

olut

ion

met

hods

•A

djus

t spe

cific

ele

men

ts o

f thi

s st

ruct

ure

to s

uit t

he p

robl

em.

Slid

e 6

Goa

l

•D

esig

n a

gene

ral-p

urpo

se

solv

er th

at e

xplo

its

prob

lem

-spe

cific

str

uctu

re.

•A

ppro

ach:

•F

ind

com

mon

alg

orith

mic

str

uctu

re

acro

ss s

olut

ion

met

hods

•A

djus

t spe

cific

ele

men

ts o

f thi

s st

ruct

ure

to s

uit t

he p

robl

em.

•T

hesi

s:•

Suc

cess

ful c

ombi

nato

rial o

ptim

izat

ion

met

hods

use

a

prim

al-d

ual-d

ual

solu

tion

stra

tegy

.•

The

dua

ls c

an b

e de

sign

ed to

exp

loit

prob

lem

str

uctu

re.

•T

his

pers

pect

ive

can

unify

met

hods

an

d le

ad to

new

one

s.

Slid

e 7

Out

line

•T

wo

dual

ities

•E

xam

ple:

Inte

grat

ed a

ppro

ach

to IP

•F

ocus

on

infe

renc

e du

ality

•E

xam

ple:

SA

T

•E

xam

ples

of i

nteg

rate

d so

lvin

g•

Pie

cew

ise

linea

r op

timiz

atio

n•

Pla

nnin

g an

d di

sjun

ctiv

e sc

hedu

ling

•P

lann

ing

and

cum

ulat

ive

sche

dulin

g•

Sin

gle-

faci

lity

sche

dulin

g•

Tru

ss s

truc

ture

des

ign

(non

linea

r)

•C

oncl

usio

n

Slid

e 8

Sur

vivi

ng th

is ta

lk

•W

e m

ust c

over

a w

ide

varie

ty o

f pro

blem

s to

mak

e th

e po

int.

•F

ocus

on

the

stru

ctur

al p

aral

lels

•R

athe

r th

an th

e de

tails

of e

ach

prob

lem

Slid

e 9

Tw

o du

aliti

es

•In

fere

nce

and

rela

xatio

n ca

n ex

ploi

t pro

blem

str

uctu

re.

•T

hey

wor

k to

geth

er w

ith s

earc

h in

mos

t com

bina

toria

l op

timiz

atio

n m

etho

ds.

•S

earc

h, in

fere

nce,

and

rel

axat

ion

are

linke

d by

tw

o un

derly

ing

dual

ities

:•

Dua

lity

of s

earc

h an

d in

fere

nce

•D

ualit

y of

sea

rch

and

rela

xatio

n

Slid

e 10

Tw

o w

ays

to e

xplo

it st

ruct

ure

•In

fere

nce

•U

se k

now

ledg

e of

pro

blem

str

uctu

re to

rev

eal h

idde

n in

form

atio

n.

•T

his

can

excl

ude

unpr

omis

ing

area

s of

the

sear

ch s

pace

..

Sea

rch

spac

e

Sol

utio

n

Red

uced

sea

rch

spac

e Infe

rred

con

stra

int

Slid

e 11

Tw

o w

ays

to e

xplo

it st

ruct

ure

•R

elax

atio

n•

Use

kno

wle

dge

of p

robl

em s

truc

ture

to d

esig

n a

larg

er b

ut

sim

pler

se

arch

spa

ce.

•S

olut

ion

of r

elax

atio

n m

ay b

e ne

ar s

olut

ion

of o

rigin

al p

robl

em

and

prov

ides

a b

ound

.

Sea

rch

spac

eR

elax

ed s

earc

h sp

ace

Sol

utio

nS

olut

ion

of r

elax

atio

n

Slid

e 12

Tw

o du

aliti

es

•D

ualit

y of

sea

rch

and

infe

renc

e.•

Sea

rch

look

s fo

r a

cert

ifica

te o

f fea

sibi

lity

.•

The

infe

renc

e du

al

look

s fo

r a

cert

ifica

te (p

roof

) of

infe

asib

ility

(or

optim

ality

).

•D

ualit

y of

sea

rch

and

rela

xatio

n.•

Sea

rch

enum

erat

es re

stric

tions

of

the

prob

lem

.•

The

rel

axat

ion

dual

enu

mer

ates

(par

amet

eriz

ed) r

elax

atio

ns

of

the

prob

lem

.

Slid

e 13

Prim

al-d

ual-d

ual m

etho

ds

•P

rimal

met

hods

.•

Enu

mer

ate

poss

ible

sol

utio

ns

(in g

ener

al, p

robl

em r

estr

ictio

ns).

•D

ual m

etho

ds.

•E

num

erat

e pr

oofs

or

rel

axat

ions

.

•P

rimal

-dua

l met

hods

.•

Sol

ve th

e pr

imal

and

a d

ual

sim

ulta

neou

sly.

•P

rimal

-dua

l-dua

l met

hods

.•

Sol

ve th

e pr

imal

and

bot

h du

als

sim

ulta

neou

sly.

•S

trat

egy

of m

ost s

ucce

ssfu

l com

bina

toria

l opt

imiz

atio

n m

etho

ds.

Slid

e 14

Cla

ssic

al s

olut

ion

met

hods

•C

P s

olve

r•

Sea

rch:

B

ranc

hing

•In

fere

nce:

F

llter

ing

•R

elax

atio

n:

Dom

ain

stor

e

•M

ILP

sol

ver

•S

earc

h:

Bra

nchi

ng•

Infe

renc

e:

Cut

ting

plan

es, p

reso

lve,

red

uced

cos

t var

iabl

e fix

ing

•R

elax

atio

n:

LP, L

agra

ngea

n

•B

ende

rs•

Sea

rch:

E

num

erat

e su

bpro

blem

s.•

Infe

renc

e:

Ben

ders

cut

s•

Rel

axat

ion:

M

aste

r pr

oble

m

Slid

e 15

Cla

ssic

al s

olut

ion

met

hods

•G

loba

l opt

imiz

atio

n•

Sea

rch:

E

num

erat

e bo

xes

•In

fere

nce:

D

omai

n re

duct

ion,

dua

l-bas

ed v

aria

ble

boun

ding

•R

elax

atio

n:

Con

vexi

ficat

ion

•S

AT

•S

earc

h:

Bra

nchi

ng, d

ynam

ic b

ackt

rack

ing,

etc

.•

Infe

renc

e:

Con

flict

cla

uses

•

Rel

axat

ion:

S

ame

as r

estr

ictio

n

•Lo

cal s

earc

h•

Sea

rch:

E

num

erat

e ne

ighb

orho

ods.

•In

fere

nce:

T

abu

list,

etc.

•R

elax

atio

n:

Sam

e as

res

tric

tion

++

++

++

≥+

++

≤∈

12

34

12

34

12

34

min

90

6050

40

75

43

42

8

{0,1

,2,3

}i

xx

xx

xx

xx

xx

xx

x

Exa

mpl

e: In

tegr

ated

app

roac

h to

IP

Slid

e 16

++

++

++

≥+

++

≤∈

12

34

12

34

12

34

min

90

6050

40

75

43

42

8

{0,1

,2,3

}i

xx

xx

xx

xx

xx

xx

x

Infe

renc

e: B

ound

s pr

opag

atio

n

Sim

ple

cons

trai

ntpr

ogra

mm

ing

tech

niqu

e

−⋅

−⋅

−⋅

≥=

1

425

34

33

31

7x

Slid

e 17

++

++

++

≥+

++

≤∈

∈

12

34

12

34

12

34

12

34

min

90

6050

40

75

43

42

8

{1,2

,3},

,,

{0,1

,2,3

}

xx

xx

xx

xx

xx

xx

xx

xx

Bou

nds

prop

agat

ion

−⋅

−⋅

−⋅

≥=

1

425

34

33

31

7x

Red

uced

do

mai

n

Slid

e 18

Sim

ple

cons

trai

ntpr

ogra

mm

ing

tech

niqu

e

++

++

++

≥+

++

≤∈

∈

12

34

12

34

12

34

12

34

min

90

6050

40

75

43

42

8

{1,2

,3},

,,

{0,1

,2,3

}

xx

xx

xx

xx

xx

xx

xx

xx

Bou

nds

prop

agat

ion

−⋅

−⋅

−⋅

≥=

1

425

34

33

31

7x

Red

uced

do

mai

n

In g

ener

al:

Bou

nds

prop

agat

ion

aim

s fo

r bo

unds

con

sist

ency

Dom

ain

filte

ring

aim

s fo

r hy

pera

rcco

nsis

tenc

y (G

AC

)S

lide

19

Sim

ple

cons

trai

ntpr

ogra

mm

ing

tech

niqu

e

Rel

axat

ion:

Lin

ear p

rogr

amm

ing

Rep

lace

dom

ains

w

ith b

ound

s

Slid

e 20

++

++

++

≥+

++

≤≤

≤≥

12

34

12

34

12

34

1

min

90

6050

40

75

43

42

8

03,

1i

xx

xx

xx

xx

xx

xx

xx

++

++

++

≥+

++

≤≤

≤≥

12

34

12

34

12

34

1

min

90

6050

40

75

43

42

8

03,

1i

xx

xx

xx

xx

xx

xx

xx

Infe

renc

e: c

uttin

g pl

anes

(val

id in

equa

litie

s)

We

can

crea

te a

tigh

ter

rela

xatio

n w

ith t

he a

dditi

on o

f cu

tting

pla

nes

.

Slid

e 21

++

++

++

≥+

++

≤≤

≤≥

12

34

12

34

12

34

1

min

90

6050

40

75

43

42

8

03,

1i

xx

xx

xx

xx

xx

xx

xx

{1,2

} is

a p

acki

ng

…be

caus

e 7x

1+

5x 2

alon

e ca

nnot

sat

isfy

the

ineq

ualit

y,

even

with

x1

= x

2=

3.

Infe

renc

e: c

uttin

g pl

anes

(val

id in

equa

litie

s)

Slid

e 22

++

++

++

≥+

++

≤≤

≤≥

12

34

12

34

12

34

1

min

90

6050

40

75

43

42

8

03,

1i

xx

xx

xx

xx

xx

xx

xx

{1,2

} is

a p

acki

ng

{}

−⋅

+⋅

+≥

=

3

4

42(7

35

3)

2m

ax4,

3x

x

So,

+≥

−⋅

+⋅

34

43

42(7

35

3)

xx

whi

ch im

plie

s

Gen

eral

inte

ger

knap

sack

cut

Infe

renc

e: c

uttin

g pl

anes

(val

id in

equa

litie

s)

Slid

e 23

++

++

++

≥+

++

≤≤

≤≥

12

34

12

34

12

34

1

min

90

6050

40

75

43

42

8

03,

1i

xx

xx

xx

xx

xx

xx

xx

Cut

ting

plan

es (v

alid

ineq

ualit

ies)

Max

imal

Pac

king

sK

naps

ack

cuts

{1,2

}x 3

+ x

4≥

2

{1,3

}x 2

+ x

4≥

2

{1,4

}x 2

+ x

3≥

3

Kna

psac

k cu

ts c

orre

spon

ding

to n

onm

axim

al

pack

ings

can

be

nonr

edun

dant

.S

lide

24

++

++

++

≥+ +

≥+

≥+

≥++

≤≤

≤≥

12

34

12

34

1 342

3

2

1

23

4

4

min

90

6050

40

75

43

42

8

03,

1

2 2 3

i

xx

xx

xx

xx

xx

x

xx

xx

xx

x

xx

Con

tinuo

us r

elax

atio

n w

ith c

uts

Opt

imal

val

ue o

f 523

.3 is

a lo

wer

bou

nd o

n op

timal

val

ue

of o

rigin

al p

robl

em.

Kna

psac

k cu

ts

Slid

e 25

Prim

al-d

ual-d

ual m

etho

d

Sea

rch:

br

anch

ing

Infe

renc

e:

Bou

nds

prop

agat

ion,

cut

ting

plan

es

Rel

axat

ion:

LP

, dom

ain

stor

e

Slid

e 26

Prim

al-d

ual-d

ual m

etho

dx 1

∈{

123

}x 2

∈{0

123}

x 3∈

{012

3}x 4

∈{0

123}

x =

(2⅓

,3,2⅔

,0)

valu

e =

523⅓

Pro

paga

te b

ound

s an

d so

lve

rela

xatio

n.

Sin

ce s

olut

ion

of

rela

xatio

n is

in

feas

ible

, bra

nch.

Slid

e 27

Bra

nch

on a

va

riabl

e w

ith

noni

nteg

ral v

alue

in

the

rela

xatio

n.

x 1∈

{ 1

23}

x 2∈

{012

3}x 3

∈{0

123}

x 4∈

{012

3}x

= (

2⅓,3

,2⅔

,0)

valu

e =

523⅓

x 1∈

{1,2

}x 1

= 3

Prim

al-d

ual-d

ual m

etho

d

Slid

e 28

Pro

paga

te b

ound

s an

d so

lve

rela

xatio

n.

Sin

ce r

elax

atio

n is

infe

asib

le,

back

trac

k.

x 1∈

{ 1

23}

x 2∈

{012

3}x 3

∈{0

123}

x 4∈

{012

3}x

= (

2⅓,3

,2⅔

,0)

valu

e =

523⅓

x 1∈

{ 1

2 }

x 2∈

{

23}

x 3∈

{ 1

23}

x 4∈

{ 1

23}

infe

asib

lere

laxa

tion

x 1∈

{1,2

}x 1

= 3

Prim

al-d

ual-d

ual m

etho

d

Slid

e 29

Pro

paga

te b

ound

s an

d so

lve

rela

xatio

n.

Bra

nch

on

noni

nteg

ral

varia

ble.

x 1∈

{ 1

23}

x 2∈

{012

3}x 3

∈{0

123}

x 4∈

{012

3}x

= (

2⅓,3

,2⅔

,0)

valu

e =

523⅓

x 1∈

{ 1

2 }

x 2∈

{

23}

x 3∈

{ 1

23}

x 4∈

{ 1

23}

infe

asib

lere

laxa

tion

x 1∈

{

3}

x 2∈

{012

3}x 3

∈{0

123}

x 4∈

{012

3}x

= (

3,2.

6,2,

0)va

lue

= 5

26

x 1∈

{1,2

}x 1

= 3

x 2∈

{0,1

,2}x 2

= 3

Prim

al-d

ual-d

ual m

etho

d

Slid

e 30

Bra

nch

agai

n.

x 1∈

{ 1

23}

x 2∈

{012

3}x 3

∈{0

123}

x 4∈

{012

3}x

= (

2⅓,3

,2⅔

,0)

valu

e =

523⅓

x 1∈

{ 1

2 }

x 2∈

{

23}

x 3∈

{ 1

23}

x 4∈

{ 1

23}

infe

asib

lere

laxa

tion

x 1∈

{

3}

x 2∈

{012

3}x 3

∈{0

123}

x 4∈

{012

3}x

= (

3,2.

6,2,

0)va

lue

= 5

26

x 1∈

{

3}

x 2∈

{012

}x 3

∈{

123

}x 4

∈{0

123}

x =

(3,

2,2¾

,0)

valu

e =

527

½

x 1∈

{1,2

}x 1

= 3

x 2∈

{0,1

,2}x 2

= 3

x 3∈

{1,2

}x 3

= 3

Prim

al-d

ual-d

ual m

etho

d

Slid

e 31

Sol

utio

n of

re

laxa

tion

is in

tegr

al.

Thi

s be

com

es th

e in

cum

bent

so

lutio

n.

x 1∈

{ 1

23}

x 2∈

{012

3}x 3

∈{0

123}

x 4∈

{012

3}x

= (

2⅓,3

,2⅔

,0)

valu

e =

523⅓

x 1∈

{ 1

2 }

x 2∈

{

23}

x 3∈

{ 1

23}

x 4∈

{ 1

23}

infe

asib

lere

laxa

tion

x 1∈

{

3}

x 2∈

{012

3}x 3

∈{0

123}

x 4∈

{012

3}x

= (

3,2.

6,2,

0)va

lue

= 5

26

x 1∈

{

3}

x 2∈

{012

}x 3

∈{

123

}x 4

∈{0

123}

x =

(3,

2,2¾

,0)

valu

e =

527

½

x 1∈

{

3}

x 2∈

{ 1

2 }

x 3∈

{ 1

2 }

x 4∈

{ 1

23}

x =

(3,

2,2,

1)va

lue

= 5

30fe

asib

le s

olut

ionx 1

∈{1

,2}

x 1=

3

x 2∈

{0,1

,2}x 2

= 3

x 3∈

{1,2

}x 3

= 3

Prim

al-d

ual-d

ual m

etho

d

Slid

e 32

Sol

utio

n is

no

nint

egra

l, bu

t us

e bo

und

to

back

trac

k,

x 1∈

{ 1

23}

x 2∈

{012

3}x 3

∈{0

123}

x 4∈

{012

3}x

= (

2⅓,3

,2⅔

,0)

valu

e =

523⅓

x 1∈

{ 1

2 }

x 2∈

{

23}

x 3∈

{ 1

23}

x 4∈

{ 1

23}

infe

asib

lere

laxa

tion

x 1∈

{

3}

x 2∈

{012

3}x 3

∈{0

123}

x 4∈

{012

3}x

= (

3,2.

6,2,

0)va

lue

= 5

26

x 1∈

{

3}

x 2∈

{012

}x 3

∈{

123

}x 4

∈{0

123}

x =

(3,

2,2¾

,0)

valu

e =

527

½

x 1∈

{

3}

x 2∈

{ 1

2 }

x 3∈

{ 1

2 }

x 4∈

{ 1

23}

x =

(3,

2,2,

1)va

lue

= 5

30fe

asib

le s

olut

ion

x 1∈

{

3}

x 2∈

{012

}x 3

∈{

3

}x 4

∈{0

12 }

x =

(3,

1½,3

,½)

valu

e =

530

back

trac

kdu

e to

bou

nd

x 1∈

{1,2

}x 1

= 3

x 2∈

{0,1

,2}x 2

= 3

x 3∈

{1,2

}x 3

= 3

Prim

al-d

ual-d

ual m

etho

d

Slid

e 33

Ano

ther

fea

sibl

e so

lutio

n fo

und.

Sea

rch

term

inat

es

with

incu

mbe

nt a

s op

timum

.

x 1∈

{ 1

23}

x 2∈

{012

3}x 3

∈{0

123}

x 4∈

{012

3}x

= (

2⅓,3

,2⅔

,0)

valu

e =

523⅓

x 1∈

{ 1

2 }

x 2∈

{

23}

x 3∈

{ 1

23}

x 4∈

{ 1

23}

infe

asib

lere

laxa

tion

x 1∈

{

3}

x 2∈

{012

3}x 3

∈{0

123}

x 4∈

{012

3}x

= (

3,2.

6,2,

0)va

lue

= 5

26

x 1∈

{

3}

x 2∈

{012

}x 3

∈{

123

}x 4

∈{0

123}

x =

(3,

2,2¾

,0)

valu

e =

527

½

x 1∈

{

3}

x 2∈

{

3}

x 3∈

{012

}x 4

∈{0

12 }

x =

(3,

3,0,

2)va

lue

= 5

30fe

asib

le s

olut

ion

x 1∈

{

3}

x 2∈

{ 1

2 }

x 3∈

{ 1

2 }

x 4∈

{ 1

23}

x =

(3,

2,2,

1)va

lue

= 5

30fe

asib

le s

olut

ion

x 1∈

{

3}

x 2∈

{012

}x 3

∈{

3

}x 4

∈{0

12 }

x =

(3,

1½,3

,½)

valu

e =

530

back

trac

kdu

e to

bou

nd

x 1∈

{1,2

}x 1

= 3

x 2∈

{0,1

,2}x 2

= 3

x 3∈

{1,2

}x 3

= 3

Prim

al-d

ual-d

ual m

etho

d

Slid

e 34

Slid

e 35

Exa

mpl

e of

sea

rch/

infe

renc

e du

ality

•W

e so

lved

the

infe

renc

e du

al

by s

olvi

ng t

he p

rimal

(se

arch

) pr

oble

m.

•F

rom

top

dow

n, th

e tr

ee is

a s

earc

h.

•F

rom

bot

tom

up,

the

tree

is a

pro

ofof

opt

imal

ity.

Slid

e 36

Exa

mpl

e of

sea

rch/

infe

renc

e du

ality

•W

e so

lved

the

infe

renc

e du

al

by s

olvi

ng t

he p

rimal

(se

arch

) pr

oble

m.

•F

rom

top

dow

n, th

e tr

ee is

a s

earc

h.

•F

rom

bot

tom

up,

the

tree

is a

pro

ofof

opt

imal

ity.

•C

uttin

g pl

anes

, dom

ain

filte

ring

are

step

s in

the

proo

f.

Slid

e 37

Exa

mpl

e of

sea

rch/

infe

renc

e du

ality

•W

e so

lved

the

infe

renc

e du

al

by s

olvi

ng t

he p

rimal

(se

arch

) pr

oble

m.

•F

rom

top

dow

n, th

e tr

ee is

a s

earc

h.

•F

rom

bot

tom

up,

the

tree

is a

pro

ofof

opt

imal

ity.

•C

uttin

g pl

anes

, dom

ain

filte

ring

are

step

s in

the

proo

f.•

The

tree

als

o en

code

s a

solu

tion

of a

rel

axat

ion

dual

(s

ubad

ditiv

edu

al).

Slid

e 38

Exa

mpl

e of

sea

rch/

infe

renc

e du

ality

•W

e so

lved

the

infe

renc

e du

al

by s

olvi

ng t

he p

rimal

(se

arch

) pr

oble

m.

•F

rom

top

dow

n, th

e tr

ee is

a s

earc

h.

•F

rom

bot

tom

up,

the

tree

is a

pro

ofof

opt

imal

ity.

•C

uttin

g pl

anes

, dom

ain

filte

ring

are

step

s in

the

proo

f.•

The

tree

als

o en

code

s a

solu

tion

of a

rel

axat

ion

dual

(s

ubad

ditiv

edu

al).

•W

e ca

n al

so s

olve

the

prim

al (

sear

ch)

prob

lem

by

solv

ing

the

infe

renc

e du

al•

Infe

renc

e du

al g

ener

ates

nog

oods

for

nogo

od-b

ased

sea

rch

.

Slid

e 39

Rel

axat

ion

and

infe

renc

e du

ality

•A

ll(?)

opt

imiz

atio

n du

als

are

inst

ance

s of

bot

h.

Out

line:

•R

elax

atio

n du

al•

Exa

mpl

e: S

urro

gate

dua

l•

Exa

mpl

e: L

agra

ngea

ndu

al

•In

fere

nce

dual

•E

xam

ple:

LP

dua

l•

Exa

mpl

e: L

agra

ngea

ndu

al•

Exa

mpl

e: S

urro

gate

dua

l

Slid

e 40

Rel

axat

ion

dual

∈m

in(

)f

x

xS

max

()

u

uUθ

∈

Prim

alD

ual

Fea

sibl

e se

t

Pro

blem

with

re

laxe

d ep

igra

ph,

para

met

eriz

ed

by u

{}

()

min

(,

)|(

)u

fx

ux

Su

θ=

∈w

here

()

,(

,)

(),

all

x

Su

Sf

xu

fx

S⊃

≤∈

Slid

e 41

Exa

mpl

e: S

urro

gate

dua

l

max

()

0

u

u

θ≥

Prim

alD

ual

{}

()

min

()|

()

0,u

fx

ugx

xS

θ=

≥∈

whe

re

min

()

()

0

fx

gx

xS≥

∈

•R

epla

ce c

onst

rain

ts g

(x)

≥0

with

a s

urro

gate

ug(

x) ≥

0,

obje

ctiv

e fu

nctio

n un

chan

ged.

•LP

dua

l is

a sp

ecia

l cas

e.

Slid

e 42

Exa

mpl

e: L

agra

ngea

ndu

al

max

()

0

u

u

θ≥

Prim

alD

ual

{}

()

min

()

()|

uf

xug

xx

Sθ

=−

∈w

here

min

()

()

0

fx

gx

xS≥

∈

•R

epla

ce o

bjec

tive

func

tion

f(x)

with

a lo

wer

bou

nd f(

x) −

ug(x

),

ineq

ualit

y co

nstr

aint

s dr

oppe

d.•

LP d

ual i

s a

spec

ial c

ase.

Slid

e 43

Infe

renc

e du

al

∈m

in(

)f

x

xS

∈⇒

≥∈

max

()

P

v

xS

vf

x

PP

Prim

alD

ual

Fea

sibl

e se

tF

amily

of

proo

fs

(infe

renc

e m

etho

d)

Fol

low

s us

ing

proo

f P

•D

ual i

s de

fined

rel

ativ

e to

an

infe

renc

e m

etho

d.

•S

tron

g du

ality

app

lies

if in

fere

nce

met

hod

is c

ompl

ete

.

Slid

e 44

Exa

mpl

e: L

P d

ual

≥≥

min

0cx

Ax

b

x

≥

⇒

≥

≥

∈

max

0

P

v

Ax

bcx

vx

PP

Prim

alD

ual

Non

nega

tive

linea

r co

mbi

natio

n +

dom

inat

ion

0 d

omin

ates

iff

fo

r so

me

0

xuA

xub

cxv

Ax

bcx

vu

≥≥

≥≥

⇒≥

≥

Slid

e 45

Exa

mpl

e: L

P d

ual

≥≥

min

0cx

Ax

b

x

≥

⇒

≥

≥

∈

max

0

P

v

Ax

bcx

vx

PP

Prim

alD

ual

Non

nega

tive

linea

r co

mbi

natio

n +

dom

inat

ion

uA≤

c a

nd u

b≥

v

0 d

omin

ates

iff

fo

r so

me

0

xuA

xub

cxv

Ax

bcx

vu

≥≥

≥≥

⇒≥

≥

Slid

e 46

Exa

mpl

e: L

P d

ual

≥≥

min

0cx

Ax

b

x

≥

⇒

≥

≥

∈

max

0

P

v

Ax

bcx

vx

PP

Prim

alD

ual

Non

nega

tive

linea

r co

mbi

natio

n +

dom

inat

ion

•In

fere

nce

met

hod

is c

ompl

ete

(ass

umin

g fe

asib

ility

) du

e to

Far

kas

Lem

ma.

•S

o w

e ha

ve s

tron

g du

ality

(a

ssum

ing

feas

ibili

ty).m

ax

0ub

uAc

u

=≤ ≥

uA≤

c a

nd u

b≥

v

Slid

e 47

Exa

mpl

e: L

agra

ngea

n du

al

≥∈

min

()

()

0

fx

gx

xS

≥⇒

≥∈

max (

)(

)P

v

gx

bf

xv

PP

Prim

alD

ual

Non

nega

tive

linea

r co

mbi

natio

n +

dom

inat

ion

()

0 d

omin

ates

(

)0

()

0(

)iff

fo

r so

me

0

xS

ugx

fx

vg

xf

xv

u

∈≥

−≥

≥⇒

≥≥

Slid

e 48

Exa

mpl

e: L

agra

ngea

n du

al

≥∈

min

()

()

0

fx

gx

xS

≥⇒

≥∈

max (

)(

)P

v

gx

bf

xv

PP

Prim

alD

ual

Non

nega

tive

linea

r co

mbi

natio

n +

dom

inat

ion

ug(x

)≤

f(x)

−v

for

all x

∈S

Tha

t is,

v ≤

f(x)

−ug

(x)

for

all x

∈S

Or

{

}m

in(

)(

)x

Sv

fx

ugx

∈≤

−

()

0 d

omin

ates

(

)0

()

0(

)iff

fo

r so

me

0

xS

ugx

fx

vg

xf

xv

u

∈≥

−≥

≥⇒

≥≥

Slid

e 49

Exa

mpl

e: L

agra

ngea

n du

al

≥∈

min

()

()

0

fx

gx

xS

≥⇒

≥∈

max (

)(

)P

v

gx

bf

xv

PP

Prim

alD

ual

Non

nega

tive

linea

r co

mbi

natio

n +

dom

inat

ion

•In

fere

nce

met

hod

is in

com

plet

e

{}

0m

axm

in(

)(

)x

Su

fx

ugx

∈≥

=−

ug(x

)≤

f(x)

−v

for

all x

∈S

Tha

t is,

v ≤

f(x)

−ug

(x)

for

all x

∈S

Or

{

}m

in(

)(

)x

Sv

fx

ugx

∈≤

−

Slid

e 50

Exa

mpl

e: S

urro

gate

dua

l

≥∈

min

()

()

0

fx

gx

xS

≥⇒

≥∈

max (

)(

)P

v

gx

bf

xv

PP

Prim

alD

ual

Non

nega

tive

linea

r co

mbi

natio

n +

impl

icat

ion

()

0 i

mpl

ies

()

()

0(

)iff

fo

r s

ome

0

xS

ugx

fx

vg

xf

xv

u

∈≥

≥≥

⇒≥

≥

Slid

e 51

Exa

mpl

e: S

urro

gate

dua

l

≥∈

min

()

()

0

fx

gx

xS

≥⇒

≥∈

max (

)(

)P

v

gx

bf

xv

PP

Prim

alD

ual

Non

nega

tive

linea

r co

mbi

natio

n +

impl

icat

ion

Any

x

∈S

with

ug(

x)≥ 0

sat

isfie

s f(

x) ≥

v

So,

min

{f(

x) │

ug(x

)≤

0, x

∈S

} ≥

v

()

0 i

mpl

ies

()

()

0(

)iff

fo

r s

ome

0

xS

ugx

fx

vg

xf

xv

u

∈≥

≥≥

⇒≥

≥

Slid

e 52

Exa

mpl

e: S

urro

gate

dua

l

≥∈

min

()

()

0

fx

gx

xS

≥⇒

≥∈

max (

)(

)P

v

gx

bf

xv

PP

Prim

alD

ual

Non

nega

tive

linea

r co

mbi

natio

n +

impl

icat

ion

•In

fere

nce

met

hod

is in

com

plet

e

{}

0m

axm

in(

)|(

)0

xS

uf

xug

x∈

≥=

≥

Any

x

∈S

with

ug(

x)≥ 0

sat

isfie

s f(

x) ≥

v

So,

min

{f(

x) │

ug(x

)≤

0, x

∈S

} ≥

v

Slid

e 53

Foc

us o

n in

fere

nce

dual

ity

Out

line:

•N

ogoo

d-ba

sed

sear

ch•

Sol

utio

n of

infe

renc

e du

al p

rovi

des

nogo

ods

to g

uide

sea

rch.

•B

ende

rs c

uts

and

conf

lict c

laus

es

in S

AT

are

exa

mpl

es.

Slid

e 54

Foc

us o

n in

fere

nce

dual

ity

Out

line:

•N

ogoo

d-ba

sed

sear

ch•

Sol

utio

n of

infe

renc

e du

al p

rovi

des

nogo

ods

to g

uide

sea

rch.

•B

ende

rs c

uts

and

conf

lict c

laus

es

in S

AT

are

exa

mpl

es.

•E

xam

ple:

SA

T•

DP

L +

con

flict

cla

uses

•P

artia

l ord

er d

ynam

ic b

ackt

rack

ing

Slid

e 55

Foc

us o

n in

fere

nce

dual

ity

Out

line:

•N

ogoo

d-ba

sed

sear

ch•

Sol

utio

n of

infe

renc

e du

al p

rovi

des

nogo

ods

to g

uide

sea

rch.

•B

ende

rs c

uts

and

conf

lict c

laus

es

in S

AT

are

exa

mpl

es.

•E

xam

ple:

SA

T•

DP

L +

con

flict

cla

uses

•P

artia

l ord

er d

ynam

ic b

ackt

rack

ing

•E

xam

ples

: In

tegr

ated

pro

blem

sol

ving

•In

clud

ing

mor

e ex

ampl

es o

f log

ic-b

ased

Ben

ders

Slid

e 56

Nog

ood-

base

d se

arch

•N

ogoo

dsre

cord

sea

rch

expe

rienc

e•

Eac

h so

lutio

n ex

amin

ed g

ener

ates

a n

ogoo

d.•

Nex

t sol

utio

n m

ust s

atis

fy c

urre

nt n

ogoo

dse

t.•

Bra

nchi

ng s

earc

h is

a s

peci

al c

ase.

•N

ogoo

dsca

n be

der

ived

by

solv

ing

infe

renc

e du

al o

f a

subp

robl

em.

•S

ubpr

oble

mca

n be

def

ined

by

fixin

g so

me

varia

bles

to c

urre

nt

valu

es in

the

sear

ch.

Slid

e 57

Exa

mpl

e: L

ogic

-bas

ed B

ende

rs

Par

titio

n va

riabl

es x

,yan

d se

arch

ove

r va

lues

of x

Sub

prob

lem

re

sults

from

fix

ing

x

∈m

in(

,)

(,

)fx

y

xy

S∈

min

(,

)

(,

)fx

y

xy

S

x

Hoo

ker

1994

Hoo

ker

& O

ttoss

on20

03

Slid

e 58

Exa

mpl

e: L

ogic

-bas

ed B

ende

rs

Par

titio

n va

riabl

es x

,yan

d se

arch

ove

r va

lues

of x

Sub

prob

lem

re

sults

from

fix

ing

x

Let p

roof

P b

e so

lutio

n of

sub

prob

lem

infe

renc

e d

ual f

or

∈m

in(

,)

(,

)fx

y

xy

S∈

min

(,

)

(,

)fx

y

xy

S

=x

x

Hoo

ker

1994

Hoo

ker

& O

ttoss

on20

03

Slid

e 59

Exa

mpl

e: L

ogic

-bas

ed B

ende

rs

Par

titio

n va

riabl

es x

,yan

d se

arch

ove

r va

lues

of x

Sub

prob

lem

re

sults

from

fix

ing

x

Let p

roof

P b

e so

lutio

n of

sub

prob

lem

infe

renc

e d

ual f

or

The

n P

deriv

es a

low

er b

ound

∈m

in(

,)

(,

)fx

y

xy

S∈

min

(,

)

(,

)fx

y

xy

S

=x

x

Hoo

ker

1994

Hoo

ker

& O

ttoss

on20

03

(,

)B

Px

x

Slid

e 60

Exa

mpl

e: L

ogic

-bas

ed B

ende

rs

Par

titio

n va

riabl

es x

,yan

d se

arch

ove

r va

lues

of x

Sub

prob

lem

re

sults

from

fix

ing

x

Let p

roof

P b

e so

lutio

n of

sub

prob

lem

infe

renc

e d

ual f

or

The

n P

deriv

es a

low

er b

ound

The

sam

e pr

oof P

prov

ides

a lo

wer

bou

nd B

(P,x

) fo

r an

y x

∈m

in(

,)

(,

)fx

y

xy

S∈

min

(,

)

(,

)fx

y

xy

S

=x

x

Hoo

ker

1994

Hoo

ker

& O

ttoss

on20

03

(,

)B

Px

x

Slid

e 61

Exa

mpl

e: L

ogic

-bas

ed B

ende

rs

Par

titio

n va

riabl

es x

,yan

d se

arch

ove

r va

lues

of x

Sub

prob

lem

re

sults

from

fix

ing

x

Let p

roof

P b

e so

lutio

n of

sub

prob

lem

infe

renc

e d

ual f

or

The

n P

deriv

es a

low

er b

ound

The

sam

e pr

oof P

prov

ides

a lo

wer

bou

nd B

(P,x

) fo

r an

y x

Add

the

Ben

ders

cut

v≥

B(P

,x)

to th

e m

aste

r pr

oble

m

∈m

in(

,)

(,

)fx

y

xy

S∈

min

(,

)

(,

)fx

y

xy

S

=x

x

Hoo

ker

1994

Hoo

ker

& O

ttoss

on20

03

(,

)B

Px

min

Ben

ders

cut

s

v

x

Slid

e 62

Exa

mpl

e: L

ogic

-bas

ed B

ende

rs

Par

titio

n va

riabl

es x

,yan

d se

arch

ove

r va

lues

of x

Sub

prob

lem

re

sults

from

fix

ing

x

Let p

roof

P b

e so

lutio

n of

sub

prob

lem

infe

renc

e d

ual f

or

The

n P

deriv

es a

low

er b

ound

The

sam

e pr

oof P

prov

ides

a lo

wer

bou

nd B

(P,x

) fo

r an

y x

Add

the

Ben

ders

cut

v≥

B(P

,x)

to th

e m

aste

r pr

oble

m

Sol

ve m

aste

r pr

oble

m fo

r ne

xt

∈m

in(

,)

(,

)fx

y

xy

S∈

min

(,

)

(,

)fx

y

xy

S

=x

x

Hoo

ker

1994

Hoo

ker

& O

ttoss

on20

03

(,

)B

Px

min

Ben

ders

cut

s

v

x

Slid

e 63

Cla

ssic

al B

ende

rs

Par

titio

n va

riabl

es x

,yan

d se

arch

ove

r va

lues

of x

Sub

prob

lem

re

sults

from

fix

ing

x

min

()

()

,0

xfx

cy

gx

Ay

b

xD

y++

≥∈

≥(

)

min

()

()

0fx

cy

Ay

bg

x

y

u

+≥

−≥

Slid

e 64

Cla

ssic

al B

ende

rs

Par

titio

n va

riabl

es x

,yan

d se

arch

ove

r va

lues

of x

Sub

prob

lem

re

sults

from

fix

ing

x

Let u

be

a du

al s

olut

ion

of s

ubpr

oble

mfo

r

=

xx

min

()

()

,0

xfx

cy

gx

Ay

b

xD

y++

≥∈

≥(

)

min

()

()

0fx

cy

Ay

bg

x

y

u

+≥

−≥

Slid

e 65

Cla

ssic

al B

ende

rs

Par

titio

n va

riabl

es x

,yan

d se

arch

ove

r va

lues

of x

Sub

prob

lem

re

sults

from

fix

ing

x

Let u

be

a du

al s

olut

ion

of s

ubpr

oble

mfo

r

The

n u

is p

roof

of b

ound

=x

x

x

min

()

()

,0

xfx

cy

gx

Ay

b

xD

y++

≥∈

≥(

)

min

()

()

0fx

cy

Ay

bg

x

y

u

+≥

−≥

(,

)(

)(

())

Bu

xf

xu

bg

x=

+−

Slid

e 66

Cla

ssic

al B

ende

rs

Par

titio

n va

riabl

es x

,yan

d se

arch

ove

r va

lues

of x

Sub

prob

lem

re

sults

from

fix

ing

x

Let u

be

a du

al s

olut

ion

of s

ubpr

oble

mfo

r

The

n u

is p

roof

of b

ound

Als

o u

is p

roof

of b

ound

B(u

,x)

= f(

x) +

u(b

−g(

x))

for

any

x

=x

x

min

()

()

,0

xfx

cy

gx

Ay

b

xD

y++

≥∈

≥(

)

min

()

()

0fx

cy

Ay

bg

x

y

u

+≥

−≥

(,

)(

)(

())

Bu

xf

xu

bg

x=

+−

Slid

e 67

Cla

ssic

al B

ende

rs

Par

titio

n va

riabl

es x

,yan

d se

arch

ove

r va

lues

of x

Sub

prob

lem

re

sults

from

fix

ing

x

Let u

be

a du

al s

olut

ion

of s

ubpr

oble

mfo

r

The

n u

is p

roof

of b

ound

Als

o u

is p

roof

of b

ound

B(u

,x)

= f(

x) +

u(b

−g(

x))

for

any

x

Add

Ben

ders

cut

v≥

= f(

x) +

u(b

−g(

x))

to m

aste

r pr

oble

m:

=x

x

min

()

()

,0

xfx

cy

gx

Ay

b

xD

y++

≥∈

≥(

)

min

()

()

0fx

cy

Ay

bg

x

y

u

+≥

−≥

(,

)(

)(

())

Bu

xf

xu

bg

x=

+−

min

Ben

ders

cut

s

v

Slid

e 68

Cla

ssic

al B

ende

rs

Par

titio

n va

riabl

es x

,yan

d se

arch

ove

r va

lues

of x

Sub

prob

lem

re

sults

from

fix

ing

x

Let u

be

a du

al s

olut

ion

of s

ubpr

oble

mfo

r

The

n u

is p

roof

of b

ound

Als

o u

is p

roof

of b

ound

B(u

,x)

= f(

x) +

u(b

−g(

x))

for

any

x

Add

Ben

ders

cut

v≥

= f(

x) +

u(b

−g(

x))

to m

aste

r pr

oble

m:

Sol

ve m

aste

r pr

oble

m fo

r ne

xt

=x

x

x

min

()

()

,0

xfx

cy

gx

Ay

b

xD

y++

≥∈

≥(

)

min

()

()

0fx

cy

Ay

bg

x

y

u

+≥

−≥

(,

)(

)(

())

Bu

xf

xu

bg

x=

+−

min

Ben

ders

cut

s

v

Slid

e 69

Exa

mpl

e: S

AT

•S

olve

SA

T b

y ch

rono

logi

cal b

ackt

rack

ing

+ u

nit c

laus

e ru

le =

DP

L

•T

o ge

t nog

ood,

sol

ve in

fere

nce

dual

at

cur

rent

nod

e.•

Sol

ve d

ual w

ith u

nit c

laus

e ru

le

•P

roce

ss n

ogoo

dse

t (m

aste

r pr

oble

m)

with

par

alle

l res

olut

ion

•N

ogoo

dse

t is

a re

laxa

tion

of th

e pr

oble

m.

Slid

e 70

42

32

41

31

43

2

43

1

65

2

65

2

65

1

65

1

xx

xx

xx

xx

xx

x

xx

x

xx

x

xx

x

xx

x

xx

x

∨∨∨∨∨

∨∨

∨∨

∨∨

∨∨

∨∨

∨

Fin

d a

satis

fyin

g so

lutio

n.

Exa

mpl

e: S

AT

Slid

e 71

=1

0x =

20

x =3

0x =

40

x =5

0x

DP

L w

ith c

hron

olog

ical

bac

ktra

ckin

g

Bra

nch

to h

ere.

Sol

ve s

ubpr

oble

m w

ith u

nit c

laus

e ru

le, w

hich

pro

ves

infe

asib

ility

.

(x1,

… x

5) =

(0,

…, 0

) cr

eate

s th

e in

feas

ibili

ty.

∨∨

∨∨

12

34

5x

xx

xx

Slid

e 72

=1

0x =

20

x =3

0x =

40

x =5

0x

Gen

erat

e no

good

.

Bra

nch

to h

ere.

Sol

ve s

ubpr

oble

m w

ith u

nit c

laus

e ru

le, w

hich

pro

ves

infe

asib

ility

.

(x1,

… x

5) =

(0,

…, 0

) cr

eate

s th

e in

feas

ibili

ty.

DP

L w

ith c

hron

olog

ical

bac

ktra

ckin

g

Slid

e 73

=1

0x =

20

x =3

0x =

40

x =5

0x

⋅∨

∨∨

∨∨

∨∨

∨1

23

45

12

34

5

Rel

axat

ion

Sol

utio

n of

N

ogoo

ds

0(0

,0,0

,0,0

,)

1

kk

kR

R

xx

xx

x

xx

xx

x

Con

flict

cla

use

appe

ars

as n

ogoo

d in

duce

d by

sol

utio

n of

Rk.

∨∨

∨∨

12

34

5x

xx

xx

DP

L w

ith c

hron

olog

ical

bac

ktra

ckin

g

Mas

ter

prob

lem

Slid

e 74

=1

0x =

20

x =3

0x =

40

x =5

0x

⋅∨

∨∨

∨∨

∨∨

∨⋅

∨∨

∨∨

12

34

5

12

34

51

23

45

Rel

axat

ion

Sol

utio

n of

N

ogoo

ds

0(0

,0,0

,0,0

,)

1(0

,0,0

,0,1

,)

kk

kR

R

xx

xx

x

xx

xx

xx

xx

xx

Go

to s

olut

ion

that

sol

ves

rela

xatio

n, w

ith p

riorit

y to

0=

51

x

DP

L w

ith c

hron

olog

ical

bac

ktra

ckin

g

Mas

ter

prob

lem

Slid

e 75

=1

0x =

20

x =3

0x =

40

x =5

0x

⋅∨

∨∨

∨∨

∨∨

∨⋅

∨∨

∨∨

12

34

5

12

34

51

23

45

Rel

axat

ion

Sol

utio

n of

N

ogoo

ds

0(0

,0,0

,0,0

,)

1(0

,0,0

,0,1

,)

kk

kR

R

xx

xx

x

xx

xx

xx

xx

xx

Pro

cess

nog

ood

set w

ith

para

llel r

esol

utio

n

∨∨

∨∨

∨∨

∨∨

12

34

5

12

34

5

xx

xx

x

xx

xx

x

∨∨

∨1

23

4x

xx

xpa

ralle

l-abs

orbs

∨∨

∨∨

∨∨

∨∨

12

34

5

12

34

5

xx

xx

x

xx

xx

x

=5

1x

DP

L w

ith c

hron

olog

ical

bac

ktra

ckin

g

Mas

ter

prob

lem

Slid

e 76

=1

0x =

20

x =3

0x =

40

x =5

0x

⋅∨

∨∨

∨∨

∨∨

∨⋅

∨∨

∨∨

∨∨

∨

12

34

5

12

34

51

23

45

12

34

Rel

axat

ion

Sol

utio

n of

N

ogoo

ds

0(0

,0,0

,0,0

,)

1(0

,0,0

,0,1

,)

2

kk

kR

R

xx

xx

x

xx

xx

xx

xx

xx

xx

xx

Pro

cess

nog

ood

set w

ith

para

llel r

esol

utio

n

∨∨

∨∨

∨∨

∨∨

12

34

5

12

34

5

xx

xx

x

xx

xx

x

∨∨

∨1

23

4x

xx

xpa

ralle

l-abs

orbs

∨∨

∨∨

∨∨

∨∨

12

34

5

12

34

5

xx

xx

x

xx

xx

x

=5

1x

DP

L w

ith c

hron

olog

ical

bac

ktra

ckin

g

Mas

ter

prob

lem

Slid

e 77

=1

0x =

20

x =3

0x =

40

x =5

0x

⋅∨

∨∨

∨∨

∨∨

∨⋅

∨∨

∨∨

∨∨

∨⋅

12

34

5

12

34

51

23

45

12

34

Rel

axat

ion

Sol

utio

n of

N

ogoo

ds

0(0

,0,0

,0,0

,)

1(0

,0,0

,0,1

,)

2(0

,0,0

,1,0

,)

kk

kR

R

xx

xx

x

xx

xx

xx

xx

xx

xx

xx

Sol

ve r

elax

atio

n ag

ain,

con

tinue

.

So

back

trac

king

is n

ogoo

d-ba

sed

sear

ch

with

par

alle

l res

olut

ion

DP

L w

ith c

hron

olog

ical

bac

ktra

ckin

g

Mas

ter

prob

lem

Slid

e 78

Exa

mpl

e: S

AT

+ c

onfli

ct c

laus

es

•U

se s

tron

ger

nogo

ods

= c

onfli

ct c

laus

es.

•N

ogoo

dsru

le o

ut o

nly

bran

ches

that

pla

y a

role

in u

nit c

laus

e re

futa

tion.

Slid

e 79

01

=x 0

2=

x 03

=x 0

4=

x 05

=x

Bra

nch

to h

ere.

Uni

t cla

use

rule

pr

oves

infe

asib

ility

.

(x1,

x 5)

= (

0,0)

is o

nly

prem

ise

of u

nit

clau

se p

roof

.

DP

L w

ith c

onfli

ct c

laus

es

Slid

e 80

01

=x 0

2=

x 03

=x 0

4=

x 05

=x

∨1

5x

x

⋅∨

∨1

5

15

Rel

axat

ion

Sol

utio

n of

N

ogoo

ds

0(0

,0,0

,0,0

,)

1

kk

kR

R

xx

xx

Con

flict

cla

use

appe

ars

as n

ogoo

d in

duce

d by

sol

utio

n of

Rk.

DP

L w

ith c

onfli

ct c

laus

es

Slid

e 81

=1

0x =

20

x =3

0x =

40

x =5

0x

=5

1x

∨1

5x

x∨

25

xx

⋅∨

∨⋅

∨1

5

15

25

Rel

axat

ion

Sol

utio

n of

N

ogoo

ds

0(0

,0,0

,0,0

,)

1(0

,0,0

,0,1

,)

kk

kR

R

xx

xx

xx

Mas

ter

prob

lem

DP

L w

ith c

onfli

ct c

laus

es

Slid

e 82

=1

0x =

20

x =3

0x =

40

x =5

0x

=5

1x

∨1

5x

x∨

25

xx

∨1

2x

x

⋅∨

∨⋅

∨∨

15

15

25

12

Rel

axat

ion

Sol

utio

n of

N

ogoo

ds

0(0

,0,0

,0,0

,)

1(0

,0,0

,0,1

,)

2

kk

kR

R

xx

xx

xx

xx

∨ ∨1

5

25

xx

xx

para

llel-r

esol

ve to

yie

ld∨

12

xx

∨1

2x

xpa

ralle

l-abs

orbs

∨ ∨1

5

25

xx

xx

DP

L w

ith c

onfli

ct c

laus

es

Mas

ter

prob

lem

Slid

e 83

⋅∨

∨⋅

∨∨

⋅⋅⋅⋅

∨

15

15

25

12

12

1

Rel

axat

ion

Sol

utio

n of

N

ogoo

ds

0(0

,0,0

,0,0

,)

1(0

,0,0

,0,1

,)

2(0

,1,

,,,)

3

kk

kR

R

xx

xx

xx

xx

xx

x

=1

0x =

20

x=

21

x

=3

0x =

40

x =5

0x

15

=x

∨1

5x

x∨

25

xx

∨1

2x

x

∨1

2x

x 1x

∨ ∨1

2

12

xx

xx

para

llel-r

esol

ve to

yie

ld1x

DP

L w

ith c

onfli

ct c

laus

es

Slid

e 84

=1

0x

=1

1x

=2

0x =

30

x =4

0x =

50

x=

51

x

∨1

5x

x∨

25

xx

∨1

2x

x

21

xx

∨ 1x

1x

⋅∨

∨⋅

∨∨

⋅⋅⋅⋅

∨⋅⋅

⋅⋅⋅

∅

15

15

25

12

12

11

Rel

axat

ion

Sol

utio

n of

N

ogoo

ds

0(0

,0,0

,0,0

,)

1(0

,0,0

,0,1

,)

2(0

,1,

,,,)

3(1

,,,

,,)

4

kk

kR

R

xx

xx

xx

xx

xx

xx

Sea

rch

term

inat

es

DP

L w

ith c

onfli

ct c

laus

es

Slid

e 85

SA

T +

par

tial o

rder

dyn

amic

bac

ktra

ckin

g

•F

orge

t abo

ut th

e br

anch

ing

tree

•Le

t no

good

sdi

rect

the

sear

ch in

a m

ore

gene

ral w

ay.

•S

imila

r m

etho

d us

ed in

rec

ent S

AT

sol

vers

•S

olve

rel

axat

ion

by s

elec

ting

a so

lutio

n th

at c

onfo

rms

to

nogo

ods.

•C

onfo

rm =

take

s op

posi

te s

ign

than

in n

ogoo

ds.

•M

ore

free

dom

than

in b

ranc

hing

.

Gin

sber

g &

McA

llist

er 1

993,

199

4

Slid

e 86

⋅∨

∨5

1

15

Rel

axat

ion

Sol

utio

n of

N

ogoo

ds

0(0

,0,0

,0,0

,)

1

kk

kR

R

xx

xx

Arb

itrar

ily c

hoos

e on

e va

riabl

e to

be

last

Par

tial O

rder

Dyn

amic

Bac

ktra

ckin

g

Slid

e 87

⋅∨

∨5

1

15

Rel

axat

ion

Sol

utio

n of

N

ogoo

ds

0(0

,0,0

,0,0

,)

1

kk

kR

R

xx

xx

Arb

itrar

ily c

hoos

e on

e va

riabl

e to

be

last

Oth

er v

aria

bles

are

pe

nulti

mat

e

Par

tial O

rder

Dyn

amic

Bac

ktra

ckin

g

Slid

e 88

⋅∨

∨⋅⋅

⋅⋅

51

15

Rel

axat

ion

Sol

utio

n of

N

ogoo

ds

0(0

,0,0

,0,0

,)

1(1

,,,,

0,)

2

kk

kR

R

xx

xx

Sin

ce x

5is

pen

ultim

ate

in a

t lea

st o

ne

nogo

od, i

t mus

t con

form

to n

ogoo

ds.

It m

ust t

ake

valu

e op

posi

te it

s si

gn in

the

nogo

ods.

x 5w

ill h

ave

the

sam

e si

gn in

all

nogo

ods

whe

re it

is p

enul

timat

e.

Thi

s al

low

s m

ore

free

dom

than

chr

onol

ogic

al

back

trac

king

.

Par

tial O

rder

Dyn

amic

Bac

ktra

ckin

g

Slid

e 89

⋅∨

∨⋅⋅

⋅⋅

∨5

1

15

51

Rel

axat

ion

Sol

utio

n of

N

ogoo

ds

0(0

,0,0

,0,0

,)

1(1

,,,,

0,)

2

kk

kR

R

xx

xx

xx

Sin

ce x

5is

pen

ultim

ate

in a

t lea

st o

ne

nogo

od, i

t mus

t con

form

to n

ogoo

ds.

It m

ust t

ake

valu

e op

posi

te it

s si

gn in

the

nogo

ods.

x 5w

ill h

ave

the

sam

e si

gn in

all

nogo

ods

whe

re it

is p

enul

timat

e.

Thi

s al

low

s m

ore

free

dom

than

chr

onol

ogic

al

back

trac

king

.

Cho

ice

of la

st

varia

ble

is a

rbitr

ary

but m

ust b

e co

nsis

tent

with

pa

rtia

l ord

er im

plie

d by

pre

viou

s ch

oice

s.

Par

tial O

rder

Dyn

amic

Bac

ktra

ckin

g

Slid

e 90

⋅∨

∨⋅⋅

⋅⋅

∨5

1

15

51

5

Rel

axat

ion

Sol

utio

n of

N

ogoo

ds

0(0

,0,0

,0,0

,)

1(1

,,,,

0,)

2

kk

kR

R

xx

xx

xx

x

Sin

ce x

5is

pen

ultim

ate

in a

t lea

st o

ne

nogo

od, i

t mus

t con

form

to n

ogoo

ds.

It m

ust t

ake

valu

e op

posi

te it

s si

gn in

the

nogo

ods.

x 5w

ill h

ave

the

sam

e si

gn in

all

nogo

ods

whe

re it

is p

enul

timat

e.

Thi

s al

low

s m

ore

free

dom

than

chr

onol

ogic

al

back

trac

king

.

∨ ∨5

1

51

xx

xx

Par

alle

l-res

olve

to

yie

ld5x

Cho

ice

of la

st

varia

ble

is a

rbitr

ary

but m

ust b

e co

nsis

tent

with

pa

rtia

l ord

er im

plie

d by

pre

viou

s ch

oice

s.

Par

tial O

rder

Dyn

amic

Bac

ktra

ckin

g

Slid

e 91

⋅∨

∨⋅⋅

⋅⋅

∨⋅

⋅⋅⋅

∨

∨

51

15

51

55

2

5

52

Rel

axat

ion

Sol

utio

n of

N

ogoo

ds

0(0

,0,0

,0,0

,)

1(1

,,,,

0,)

2(

,0,,

,1,

)

3

kk

kR

R

xx

xx

xx

xx

x

x

xx

does

not

par

alle

l-res

olve

with

5x

∨5

2x

xbe

caus

e x 5

is n

ot la

st in

bot

h cl

ause

s

Par

tial O

rder

Dyn

amic

Bac

ktra

ckin

g

Slid

e 92

⋅∨

∨⋅⋅

⋅⋅

∨⋅

⋅⋅⋅

∨

⋅⋅⋅

⋅

∨

∅

51

15

51

55

2

52

52

Rel

axat

ion

Sol

utio

n of

N

ogoo

ds

0(0

,0,0

,0,0

,)

1(1

,,,,

0,)

2(

,0,,

,1,

)

3(

,1,,

,1,)

4

kk

kR

R

xx

xx

xx

xx

x

xx

xx

Mus

t con

form

Sea

rch

term

inat

esPar

tial O

rder

Dyn

amic

Bac

ktra

ckin

g

Slid

e 93

Exa

mpl

es o

f int

egra

ted

solv

ing

•P

iece

wis

e lin

ear

optim

izat

ion.

•Ill

ustr

ates

mod

elin

g w

ith m

etac

onst

rain

t

•P

lann

ing

& s

ched

ulin

g -

Logi

c-ba

sed

Ben

ders

•P

lann

ing

and

disj

unct

ive

sche

dulin

g•

Pla

nnin

g an

d cu

mul

ativ

e sc

hedu

ling

•S

ingl

e-fa

cilit

y sc

hedu

ling

•T

russ

str

uctu

re d

esig

n -

Glo

bal o

ptim

izat

ion.

•S

olve

d by

SIM

PL,

a p

roto

type

inte

grat

ed s

olve

r.

Aro

n, H

ooke

r, &

Y

unes

2004

, 201

0

Slid

e 94

Pie

cew

ise

linea

r opt

imiz

atio

n

max

()

ii

i

ii

fx

xC

≤∑

∑

Eac

h f i

is a

pie

cew

ise

linea

r se

mic

ontin

uous

func

tion

Max

imiz

e a

piec

ewis

e lin

ear

sepa

rabl

e fu

nctio

n su

bjec

t to

budg

et

cons

trai

nt.

Slid

e 95

Sem

icon

tinuo

us p

iece

wis

e lin

ear

func

tion

f(x)

Pie

cew

ise

linea

r opt

imiz

atio

n

Slid

e 96

Inte

grat

ed a

ppro

ach

•S

earc

h: B

ranc

hon

var

iabl

es

•R

elax

atio

n:

Use

a s

peci

aliz

ed c

onve

x hu

ll re

laxa

tion

for

piec

ewis

e lin

ear

func

tions

(no

nee

d fo

r 0-

1 m

odel

or

SO

S2)

.

•In

fere

nce:

Bou

nds

prop

agat

ion

.O

ttoss

on,

Tho

rste

inss

on&

H

ooke

r 19

99

Pie

cew

ise

linea

r opt

imiz

atio

n

Slid

e 97In

tegr

ated

m

odel

()

max

piec

ewis

e,

,,

,,

,al

l

ii

ii

ii

ii

ii

u

xC

xu

LU

cd

i

≤∑

∑

Met

acon

stra

int

(glo

bal c

onst

rain

t in

CP

)

Pie

cew

ise

linea

r opt

imiz

atio

n

Slid

e 98

Sem

icon

tinuo

us p

iece

wis

e lin

ear

func

tion

f(x)

Tig

ht li

near

re

laxa

tion

afte

r pr

opag

atio

n

Pie

cew

ise

linea

r opt

imiz

atio

n

Slid

e 99

Sem

icon

tinuo

us p

iece

wis

e lin

ear

func

tion

f(x)

Tig

hter

re

laxa

tion

afte

r br

anch

ing

Val

ue o

f x in

sol

utio

n of

cur

rent

line

ar r

elax

atio

n

Pie

cew

ise

linea

r opt

imiz

atio

n

Slid

e 10

0

SIM

PL

mod

el

Pie

cew

ise

linea

r opt

imiz

atio

n

Slid

e 10

1

SIM

PL

mod

el

Pie

cew

ise

linea

r opt

imiz

atio

n

Pie

cew

ise

linea

r m

etac

onst

rain

t.

Slid

e 10

2

Com

puta

tion

time

(sec

) vs

. num

ber

of jo

bs

Pie

cew

ise

linea

r opt

imiz

atio

n

Slid

e 10

3

•Ass

ign

jobs

to m

achi

nes,

and

sch

edul

e th

e m

achi

nes

assi

gned

to

eac

h m

achi

ne w

ithin

tim

e w

indo

ws.

•T

he o

bjec

tive

is to

min

imiz

e pr

oces

sing

cos

t, m

akes

pan

, or

tota

l tar

dine

ss.

Pla

nnin

g an

d di

sjun

ctiv

e sc

hedu

ling

Slid

e 10

4

Job

Dat

aE

xam

ple

Ass

ign

5 jo

bs to

2 m

achi

nes.

Sch

edul

e jo

bs a

ssig

ned

to e

ach

mac

hine

with

out

over

lap

(dis

junc

tive

sche

dulin

g)

Mac

hine

A

Mac

hine

B

Pla

nnin

g an

d di

sjun

ctiv

e sc

hedu

ling

Slid

e 10

5

()

min

, al

l

noO

verla

p(

),(

),

all

j

j

xj

j

jj

jx

j

jj

ijj

c

rs

dp

j

sx

ip

xi

i

≤≤

−

==

∑S

tart

tim

e of

job

j Tim

e w

indo

ws

Jobs

can

not o

verla

p

Pla

nnin

g an

d di

sjun

ctiv

e sc

hedu

ling

Inte

grat

ed

mod

el

Mac

hine

ass

igne

d to

job

j

Her

e w

e m

inim

ize

proc

essi

ng c

ost.

Slid

e 10

6

Pla

nnin

g an

d di

sjun

ctiv

e sc

hedu

ling

Inte

grat

ed a

ppro

ach

•S

earc

h: E

num

erat

e su

bpro

blem

s (d

efin

ed b

y as

sign

ing

jobs

to

mac

hine

s)

•R

elax

atio

n:

Enu

mer

ate

mas

ter

prob

lem

s (w

hich

ass

ign

jobs

to

mac

hine

s)

•In

fere

nce:

Gen

erat

e no

good

s (lo

gic-

base

d B

ende

rs c

uts)

, w

hich

are

add

ed to

mas

ter

prob

lem

.

Slid

e 10

7

•Ass

ign

the

jobs

in th

e m

aste

r pr

oble

m, t

o be

sol

ved

by M

ILP

.

•S

ched

ule

the

jobs

in th

e su

bpro

blem

, to

be s

olve

d by

CP

.

Pla

nnin

g an

d di

sjun

ctiv

e sc

hedu

ling

Inte

grat

ed a

ppro

ach

Jain

& G

ross

man

n 20

01

Slid

e 10

8

Pla

nnin

g an

d di

sjun

ctiv

e sc

hedu

ling

The

sub

prob

lem

dec

oupl

es in

to a

se

para

te s

ched

ulin

g pr

oble

m o

n ea

ch m

achi

ne.

In th

is p

robl

em, t

he s

ubpr

oble

m

is a

feas

ibili

ty p

robl

em.

•Ass

ign

the

jobs

in th

e m

aste

r pr

oble

m, t

o be

sol

ved

by M

ILP

.

•S

ched

ule

the

jobs

in th

e su

bpro

blem

, to

be s

olve

d by

CP

.

Inte

grat

ed a

ppro

ach

Jain

& G

ross

man

n 20

01

Slid

e 10

9

Pla

nnin

g an

d di

sjun

ctiv

e sc

hedu

ling

The

sub

prob

lem

dec

oupl

es in

to a

se

para

te s

ched

ulin

g pr

oble

m o

n ea

ch m

achi

ne.

In th

is p

robl

em, t

he s

ubpr

oble

m

is a

feas

ibili

ty p

robl

em.

•Ass

ign

the

jobs

in th

e m

aste

r pr

oble

m, t

o be

sol

ved

by M

ILP

.

•S

ched

ule

the

jobs

in th

e su

bpro

blem

, to

be s

olve

d by

CP

.

Inte

grat

ed a

ppro

ach

•S

olve

infe

renc

e du

al

of s

ubpr

oble

m to

gen

erat

e no

good

s (lo

gic-

base

d B

ende

rs c

uts)

, whi

ch a

re a

dded

to m

aste

r pr

oble

m.

Slid

e 11

0

For

a fi

xed

assi

gnm

ent

th

e su

bpro

blem

on

each

mac

hine

iis

()

, al

l w

ith

noO

verla

p(

),(

)

jj

jj

xj

j

jj

ijj

rs

dp

jx

i

sx

ip

xi

≤≤

−=

==

x

Inte

grat

ed

mod

el

Pla

nnin

g an

d di

sjun

ctiv

e sc

hedu

ling

()

min

, al

l

noO

verla

p(

),(

),

all

j

j

xj

j

jj

jx

j

jj

ijj

c

rs

dp

j

sx

ip

xi

i

≤≤

−

==

∑S

tart

tim

e of

job

j

Tim

e w

indo

ws

Jobs

can

not o

verla

p

Slid

e 11

1

Sup

pose

we

assi

gn jo

bs 1

,2,3

,5 to

mac

hine

A in

iter

atio

n k.

We

can

prov

e th

at th

ere

is n

o fe

asib

le s

ched

ule.

Edg

e fin

ding

de

rives

infe

asib

ility

by

reas

onin

g on

ly w

ith jo

bs 2

,3,5

. S

o th

ese

jobs

alo

ne c

reat

e in

feas

ibili

ty.

So

we

have

a B

ende

rs c

ut(

)2

35

xx

xA

¬=

==

Pla

nnin

g an

d di

sjun

ctiv

e sc

hedu

ling

Logi

c-ba

sed

Ben

ders

app

roac

h

Slid

e 11

2

We

wan

t the

mas

ter

prob

lem

to b

e an

MIL

P, w

hich

is g

ood

for

assi

gnm

ent p

robl

ems.

So

we

writ

e th

e B

ende

rs c

ut

Usi

ng 0

-1 v

aria

bles

:2

35

2A

AA

xx

x+

+≤

= 1

if jo

b 5

is

assi

gned

to

mac

hine

A

Pla

nnin

g an

d di

sjun

ctiv

e sc

hedu

ling

Logi

c-ba

sed

Ben

ders

app

roac

h

()

23

5x

xx

A¬

==

=

Slid

e 11

3

The

mas

ter

prob

lem

is a

rel

axat

ion

, for

mul

ated

as

an M

ILP

:

{}

23

5

min

2

rela

xatio

n of

sub

prob

lem

0,1ij

ijij

AA

A

ij

cx

xx

x

x

++

≤

∈∑B

ende

rs c

ut fr

om m

achi

ne A

Pla

nnin

g an

d di

sjun

ctiv

e sc

hedu

ling

Slid

e 11

4

Pla

nnin

g an

d di

sjun

ctiv

e sc

hedu

ling

•K

eys

to s

ucce

ss:

•S

tren

gthe

n B

ende

rs c

uts

by s

olvi

ng s

ubpr

oble

ms

for

a ne

ighb

orho

od o

f cur

rent

mas

ter

solu

tion.

•U

se r

elax

atio

nof

sub

prob

lem

in th

e m

aste

r.

Slid

e 11

5

Pla

nnin

g an

d di

sjun

ctiv

e sc

hedu

ling

SIM

PL

mod

el

Slid

e 11

6

Pla

nnin

g an

d di

sjun

ctiv

e sc

hedu

ling

Com

puta

tiona

l res

ults

–Lo

ng p

roce

ssin

g tim

es

SIM

PL

resu

lts a

re s

imila

r to

orig

inal

han

d-co

ded

resu

lts.

Jobs

Mac

hine

sM

ILP

(CP

LEX

11)

Nod

es

Sec

.S

IMP

LB

ende

rsIte

r.

Cut

s

Sec

.

32

10.

002

10.

00

73

10.

00

1316

0.12

123

3,35

16.

626

350.

73

155

2,77

98.

820

290.

83

205

33,3

2188

213

825.

4

225

352,

309

10,5

6369

989.

6

Yun

es, A

ron

&

Hoo

ker

2010

Slid

e 11

7

Pla

nnin

g an

d di

sjun

ctiv

e sc

hedu

ling

Com

puta

tiona

l res

ults

–S

hort

pro

cess

ing

times

*out

of m

emor

y

Jobs

Mac

hine

sM

ILP

(C

PLE

X 1

1)N

odes

Sec

.S

IMP

LB

ende

rsIte

r.

Cut

s

S

ec.

32

10.

011

00.

00

73

10.

02

10

0.00

123

499

0.98

10

0.01

155

529

2.6

21

0.06

205

250,

047

369

65

0.28

225

> 2

7.5

mil.

>48

hr

912

0.42

255

> 5

.4 m

il.>

19

hr*

1721

1.09

Yun

es, A

ron

&

Hoo

ker

2010

Slid

e 11

8

Pla

nnin

g an

d cu

mul

ativ

e sc

hedu

ling

Con

side

r a

cum

ulat

ive

sche

dulin

g co

nstr

aint

:

()

12

31

23

12

3cu

mul

ativ

e(

,,

),(

,,

),(

,,

),s

ss

pp

pc

cc

C

A fe

asib

le s

olut

ion:

Slid

e 11

9

Edg

e fin

ding

for

cum

ulat

ive

sche

dulin

g

We

can

dedu

ce th

at jo

b 3

mus

t fin

ish

afte

r th

e ot

hers

fini

sh:

{}

31,

2>

Bec

ause

the

tota

l ene

rgy

requ

ired

exce

eds

the

area

bet

wee

n th

e ea

rlies

t rel

ease

tim

e an

d th

e la

ter

dead

line

of jo

bs 1

,2:

()

3{1

,2}

{1,2

}{1

,2,3

}e

eC

LE

+>

⋅−

Slid

e 12

0

Edg

e fin

ding

for

cum

ulat

ive

sche

dulin

g

We

can

dedu

ce th

at jo

b 3

mus

t fin

ish

afte

r th

e ot

hers

fini

sh:

{}

31,

2>

Bec

ause

the

tota

l ene

rgy

requ

ired

exce

eds

the

area

bet

wee

n th

e ea

rlies

t rel

ease

tim

e an

d th

e la

ter

dead

line

of jo

bs 1

,2:

()

3{1

,2}

{1,2

}{1

,2,3

}e

eC

LE

+>

⋅−

Tota

l ene

rgy

requ

ired

= 2

29

5

8

Slid

e 12

1

Edg

e fin

ding

for

cum

ulat

ive

sche

dulin

g

We

can

dedu

ce th

at jo

b 3

mus

t fin

ish

afte

r th

e ot

hers

fini

sh:

{}

31,

2>

Bec

ause

the

tota

l ene

rgy

requ

ired

exce

eds

the

area

bet

wee

n th

e ea

rlies

t rel

ease

tim

e an

d th

e la

ter

dead

line

of jo

bs 1

,2:

()

3{1

,2}

{1,2

}{1

,2,3

}e

eC

LE

+>

⋅−

Tota

l ene

rgy

requ

ired

= 2

29

5

8A

rea

avai

labl

e =

20

Slid

e 12

2

Edg

e fin

ding

for

cum

ulat

ive

sche

dulin

g

We

can

dedu

ce th

at jo

b 3

mus

t fin

ish

afte

r th

e ot

hers

fini

sh:

{}

31,

2>

We

can

upda

te th

e re

leas

e tim

e of

job

3 to

3{1

,2}

{1,2

}{1

,2}

3

()(

)Je

Cc

LE

Ec

−−

−+

Ene

rgy

avai

labl

e fo

r jo

bs 1

,2 if

sp

ace

is le

ft fo

r jo

b 3

to s

tart

any

time

= 1

0

10

Slid

e 12

3

Edg

e fin

ding

for

cum

ulat

ive

sche

dulin

g

We

can

dedu

ce th

at jo

b 3

mus

t fin

ish

afte

r th

e ot

hers

fini

sh:

{}

31,

2>

We

can

upda

te th

e re

leas

e tim

e of

job

3 to

3{1

,2}

{1,2

}{1

,2}

3

()(

)Je

Cc

LE

Ec

−−

−+

Ene

rgy

avai

labl

e fo

r jo

bs 1

,2 if

sp

ace

is le

ft fo

r jo

b 3

to s

tart

any

time

= 1

0

10E

xces

s en

ergy

re

quire

d by

jobs

1,

2 =

4

4

Slid

e 12

4

Edg

e fin

ding

for

cum

ulat

ive

sche

dulin

g

We

can

dedu

ce th

at jo

b 3

mus

t fin

ish

afte

r th

e ot

hers

fini

sh:

{}

31,

2>

We

can

upda

te th

e re

leas

e tim

e of

job

3 to

3{1

,2}

{1,2

}{1

,2}

3

()(

)Je

Cc

LE

Ec

−−

−+

Ene

rgy

avai

labl

e fo

r jo

bs 1

,2 if

sp

ace

is le

ft fo

r jo

b 3

to s

tart

any

time

= 1

0

10E

xces

s en

ergy

re

quire

d by

jobs

1,

2 =

4

4M

ove

up jo

b 3

rele

ase

time

4/2

= 2

uni

ts

beyo

nd E

{1,2

}

E3

Slid

e 12

5

Edg

e fin

ding

for

cum

ulat

ive

sche

dulin

g

In g

ener

al, i

f (

){

}{

}J

kJ

Jk

eC

LE

∪∪

>⋅

−

then

k>

J, a

nd u

pdat

e E

kto

()(

)0

()(

)m

ax

Jk

JJ

Jk

JJ

JJ

Jk

eC

cL

E

eC

cL

EE

c′

′′

′′

′′

′ ⊂−

−−

>

−−

−+

In g

ener

al, i

f (

){

}{

}J

kJ

kJ

eC

LE

∪∪

>⋅

−

then

k<

J, a

nd u

pdat

e L k

to

()(

)0

()(

)m

in

Jk

JJ

Jk

JJ

JJ

Jk

eC

cL

E

eC

cL

EL

c′

′′

′′

′′

′ ⊂−

−−

>

−−

−−

Slid

e 12

6

Edg

e fin

ding

for

cum

ulat

ive

sche

dulin

g

Key

res

ult:

The

re is

an

O(n

2 ) a

lgor

ithm

that

find

s al

l app

licat

ions

of t

he

edge

find

ing

rule

s.

Nui

tjen

& A

arts

1994

, 199

6B

aptis

te, L

e P

ape

& N

uitje

n20

01

Slid

e 12

7

Oth

er p

ropa

gatio

n ru

les

for

cum

ulat

ive

sche

dulin

g

•E

xten

ded

edge

find

ing.

•T

imet

ablin

g.

•N

ot-f

irst/n

ot-la

st r

ules

.

•E

nerg

etic

rea

soni

ng.

Slid

e 12

8

•A

ssig

n jo

bs to

faci

litie

s (e

.g. f

acto

ries)

, with

cum

ulat

ive

sche

dulin

g in

ea

ch fa

cilit

y.

Pla

nnin

g an

d cu

mul

ativ

e sc

hedu

ling

Slid

e 12

9

{}

{}

*(1

)m

axm

in ii

i

iij

ijj

jj

Jj

Jj

J

MM

px

dd

∈∈

∈

≥−

−+

−

∑

•A

ssig

n jo

bs to

faci

litie

s (e

.g. f

acto

ries)

, with

cum

ulat

ive

sche

dulin

g in

ea

ch fa

cilit

y.

•B

ende

rs c

ut fo

r m

inim

um m

akes

pan

, with

a c

umul

ativ

e sc

hedu

ling

subp

robl

em:

Pla

nnin

g an

d cu

mul

ativ

e sc

hedu

ling

Min

imum

mak

espa

n on

mac

hine

ifo

r jo

bs

curr

ently

ass

igne

d

Jobs

cur

rent

ly

assi

gned

to m

achi

ne i

Hoo

ker

2004

Slid

e 13

0

Pla

nnin

g an

d cu

mul

ativ

e sc

hedu

ling

Min

imum

Cos

tC

ompu

tatio

n tim

e (s

ec)

vs. n

umbe

r of

jobs

4 fa

cilit

ies

(eac

h po

int =

5 in

stan

ces)

Ran

out

of

mem

ory

Com

puta

tion

term

inat

ed

afte

r 72

00 s

ec

Slid

e 13

1

Pla

nnin

g an

d cu

mul

ativ

e sc

hedu

ling

Min

imum

Mak

espa

nC

ompu

tatio

n tim

e (s

ec)

vs. n

umbe

r of

jobs

4 fa

cilit

ies

(eac

h po

int =

5 in

stan

ces)

Com

puta

tion

term

inat

ed

afte

r 72

00 s

ec

Slid

e 13

2

•S

teel

pro

duct

ion

sche

dulin

g

•B

atch

sch

edul

ing

in a

che

mic

al p

lant

(B

AS

F)

•A

ssem

bly

line

sche

dulin

g (C

itroë

n/P

ugeo

t)

•Lo

catio

n-al

loca

tion

•D

ispa

tchi

ng o

f aut

omat

ed g

uide

d ve

hicl

es

•Q

ueui

ng d

esig

n &

con

trol

•R

eal-t

ime

sche

dulin

g of

com

pute

r pr

oces

ses

•T

raffi

c di

vers

ion

•H

ome

heal

th c

are

deliv

ery*

•S

port

s sc

hedu

ling

Oth

er a

pplic

atio

ns o

f log

ic-b

ased

Ben

ders

*Cire

& H

ooke

r 20

10

Slid

e 13

3

•C

an w

e ap

ply

deco

mpo

sitio

n to

pur

e sc

hedu

ling

prob

lem

s?

•C

onsi

der

sing

le-m

achi

ne s

ched

ulin

g w

ith a

long

tim

e ho

rizon

.

Sin

gle-

faci

lity

Sch

edul

ing

Slid

e 13

4

•C

an w

e ap

ply

deco

mpo

sitio

n to

pur

e sc

hedu

ling

prob

lem

s?

•C

onsi

der

sing

le-m

achi

ne s

ched

ulin

g w

ith a

long

tim

e ho

rizon

.

•S

egm

ente

d pr

oble

m: j

obs

mus

t fin

ish

with

in a

seg

men

t.

•S

egm

ent b

ound

arie

s ar

e w

eeke

nds

or s

hutd

own

times

.

•M

aste

r pr

oble

m a

ssig

ns jo

bs to

seg

men

ts.

Sin

gle-

faci

lity

Sch

edul

ing

z 0z 1

z 2z 3

z 4

Cob

an&

Hoo

ker

2010

Slid

e 13

5

•B

ende

rs c

ut fr

om s

egm

ent i

(min

mak

espa

npr

oble

m):

•

Sin

gle-

faci

lity

sche

dulin

g ()

()

{}

{}

()

{}

{}

′′′

∈∈

′′

∈∈

′∈

′′

∈∈

≥−

−−

−−

≤−

∈

≤−

−

≤−

∑∑

∑

**

11

1,

..m

axm

in1

max

min

ii

ii

i

ii

ij

iji

iij

jJ

jJ

iij

i

ij

jij

jJ

jJ

jJ

ij

jj

Jj

J

MM

py

wM

y

qy

jJ

wd

dy

wd

d

J i′

= {

jobs

with

rel

ease

tim

esbe

fore

sta

rt o

f seg

men

t i}

J i′′

= {

jobs

with

rel

ease

tim

esaf

ter

star

t of s

egm

ent i

}= 1

whe

n jo

b j

is a

ssig

ned

to

segm

ent i

Slid

e 13

6

Sin

gle-

faci

lity

sche

dulin

g

Seg

men

ted

Pro

blem

Com

puta

tion

time

(sec

) vs

. num

ber

of jo

bsT

ime

horiz

on p

ropo

rtio

nal t

o no

. of j

obs

Com

puta

tion

term

inat

ed

afte

r 60

0 se

c

Slid

e 13

7

•U

nseg

men

ted

prob

lem

: job

s ca

n be

sch

edul

ed a

nytim

e w

ithin

th

eir

time

win

dow

s.

•S

egm

ent b

ound

arie

s ar

e on

ly f

or d

ecom

posi

tion

purp

oses

.

Sin

gle-

faci

lity

sche

dulin

g

z 0z 1

z 2z 3

z 4

Cob

an&

Hoo

ker

2011

Slid

e 13

8

•M

aste

r pr

oble

m m

ust e

ncod

e w

heth

er a

job

is s

plit

betw

een

segm

ents

.

•

Sin

gle-

faci

lity

sche

dulin

g

−−

−−

−−

≥

=+

++

≤≤

≤

≤+

>≤

+<

≤+

>

∑ ∑∑

∑0

12

3

12

3

11,

,21,

,3

21,

,11,

,3

31,

,31,

,2

min

1, a

ll

, a

ll ,

1,

1,

1,

, a

ll 1,

all

, a

ll ,

all

, a

ll 1,

all

iji ij

ijij

ijij

ijij

ijj

jj

iji

ji

j

iji

ji

j

iji

ji

j

M

yj

yy

yy

yi

j

yy

y

yy

yi

j

yy

yi

nj

yy

yi

{}

++

+

≤+

<≤

≤≤

==

==

≤

−

∈

∑∑

∑

∑

31,

,31,

,1

01

2

13

23

31

, a

ll ,

all

1,

1,

1,

0, a

ll

, a

ll

Ben

ders

cut

s, r

elax

atio

n

,0,

1

iji

ji

j

ijij

iji

ii

ijij

njnj

jij

ii

i

ijijk

j

yy

yi

nj

yy

y

yy

yy

j

py

jz

z

yy

Slid

e 13

9

•B

ende

rs c

uts

are

gene

rate

d fo

r se

vera

l cas

es.

•B

ende

rs c

ut w

hen

a jo

b j 1

com

plet

es a

t sta

rt o

fse

gmen

t and

som

e jo

b st

arts

at i

ts r

elea

se ti

me:

•

Sin

gle-

faci

lity

sche

dulin

g

()

()

()

()(

){

}

11

1

11

11

**

min

min

min

min

11

1

0,1

i

ii

iij

jJ

ijij

iij

ii

ijij

ij

iji

i

iMM

My

xx

px

p

xx

pp

xp

η

ηε

η

η

∈

≥−

−−

−+

∆≤

++

∆−

−

−+

∆≥

−−

+−

∆−

∈

∑

{}

0

**

**

min

|i

jij

jk

jJ

sr

ss

δ∈

=−

<

Sta

rt ti

me

of jo

b j

in

solu

tion

of

subp

robl

em

Sta

rt ti

me

of

first

job

k fo

r w

hich

sk*

= r

k

Rel

ease

tim

e

{}

0

**

*m

inm

in|

,,

ij

jk

jj

kj

ij

Jp

ps

sr

pr

pδ

∈=

>+

≤≤

∆+

= 0

whe

n re

optim

izin

gsc

hedu

le o

n se

gmen

t do

esn’

t cha

nge

job

orde

r

Slid

e 14

0

Sin

gle-

faci

lity

sche

dulin

g

Uns

egm

ente

dP

robl

emC

ompu

tatio

n tim

e (s

ec)

vs. n

umbe

r of

jobs

Tim

e ho

rizon

pro

port

iona

l to

no. o

f job

s

Com

puta

tion

term

inat

ed

afte

r 60

0 se

c

Slid

e 14

1

2 de

g. fr

eedo

m0

deg.

free

dom

Tota

l 8 d

egre

es o

f fre

edom

Load

Trus

s S

truc

ture

Des

ign

Sel

ect s

ize

of e

ach

bar

(pos

sibl

y ze

ro)

to s

uppo

rt th

e lo

ad w

hile

m

inim

izin

g w

eigh

t.

10-b

ar c

antil

ever

trus

s

Slid

e 14

2

ih=

leng

th o

f bar

i

jp

= lo

ad a

long

d.f.

j

s i=

forc

e al

ong

bar

v i=

elo

ngat

ion

of b

ar

d j=

nod

e di

spla

cem

ent

Ai=

cro

ss-s

ectio

nal

ar

ea o

f bar

Trus

s S

truc

ture

Des

ign

Not

atio

n

Slid

e 14

3

} M

inim

ize

tota

l wei

ght

} E

quili

briu

m

} C

ompa

tibili

ty

} H

ooke

’s la

w

} E

long

atio

n bo

unds

} D

ispl

acem

ent b

ound

s

} Lo

gica

l dis

junc

tion

Are

a m

ust b

e on

e of

sev

eral

dis

cret

e va

lues

Aik

Con

stra

ints

can

be

impo

sed

for

mul

tiple

load

ing

cond

ition

s

nonl

inea

r

min ..

cos

,all

cos

,all

,all

,all

,all

()

\/

ii

i

iji

ji

ijj

ij i

ii

ii L

Ui

ii

LU

jj

j

ki

ik

hA

st

sp

j

dv

i

EA

vs

ih v

vv

i

dd

dj

AA

θ θ

= =

=

≤≤

≤≤ =

∑ ∑ ∑

Trus

s S

truc

ture

Des

ign

Slid

e 14

4

Intr

oduc

ing

new

var

iabl

es li

near

izes

the

pro

blem

but

mak

es it

m

uch

larg

er.

0-1

varia

bles

indi

catin

g si

ze

of b

ar i

Elo

ngat

ion

varia

ble

disa

ggre

gate

d by

bar

si

ze

Hoo

ke’s

law

be

com

es li

near

min ..

cos

,all

cos

,all

,all

,all

,all

1,al

l

iik

iki

k

iji

ji

ijj

ikj

k

iik

iki

ki L

Ui

ii

LU

jj

j

ikk

hA

y

st

sp

j

dv

i

EA

vs

ih v

vv

i

dd

dj

yi

θ θ

= = =

≤≤

≤≤

=

∑∑

∑ ∑∑

∑

∑

Trus

s S

truc

ture

Des

ign

MIL

P

mod

el

Slid

e 14

5

Inte

grat

ed a

ppro

ach

•S

earc

h: B

ranc

h by

spl

ittin

g th

e ra

nge

of a

reas

Ai (n

o ne

ed fo

r 0-

1 va

riabl

es).

•R

elax

atio

n:

Gen

erat

e a

quas

i-rel

axat

ion

, whi

ch is

line

ar a

nd

muc

h sm

alle

r th

an M

ILP

mod

el.

•In

fere

nce:

Use

logi

c cu

ts.

Trus

s S

truc

ture

Des

ign

Bol

lapr

agad

a, G

hatta

s&

Hoo

ker

2001

Slid

e 14

6

The

orem

Sup

pose

we

min

imiz

e cx

subj

ect t

o g(

x,y)≤

0.

If g(

x,y)

is s

emih

omog

eneo

usin

x∈

Rn

and

conc

ave

in s

cala

r y,

th

en th

e fo

llow

ing

is a

qua

si-r

elax

atio

nof

g(x

,y)

≤0:

Trus

s S

truc

ture

Des

ign

12

1

2

12

(,

)(

,)

0

(1)

(1)

LU

LU

LU

gx

yg

xy

xx

x

xx

x

xx

x

αα

αα

+≤

≤≤

−≤

≤−

=+

Hoo

ker

2005

12

1

2

12

(,

)(

,)

0

(1)

(1)

LU

LU

LU

gx

yg

xy

xx

x

xx

x

xx

x

αα

αα

+≤

≤≤

−≤

≤−

=+

Not

a v

alid

rel

axat

ion,

and

it m

ay c

onta

in d

iffer

ent

varia

bles

, B

ut it

s op

timal

val

ue is

a v

alid

low

er

boun

d on

the

optim

al v

alue

of t

he o

rigin

al p

robl

em.

The

orem

Sup

pose

we

min

imiz

e cx

subj

ect t

o g(

x,y)≤

0.

If g(

x,y)

is s

emih

omog

eneo

usin

x∈

Rn

and

conc

ave

in s

cala

r y,

th

en th

e fo

llow

ing

is a

qua

si-r

elax

atio

nof

g(x

,y)

≤0:

Slid

e 14

7

Trus

s S

truc

ture

Des

ign

Hoo

ker

2005

12

1

2

12

(,

)(

,)

0

(1)

(1)

LU

LU

LU

gx

yg

xy

xx

x

xx

x

xx

x

αα

αα

+≤

≤≤

−≤

≤−

=+

(,

)(

,)

gx

yg

xy

αα

≤fo

r al

l x, y

and

α∈

[0,1

]

(0,

)0

gy

=fo

r al

l y

The

orem

Sup

pose

we

min

imiz

e cx

subj

ect t

o g(

x,y)≤

0.

If g(

x,y)

is s

emih

omog

eneo

usin

x∈

Rn

and

conc

ave

in s

cala

r y,

th

en th

e fo

llow

ing

is a

qua

si-r

elax

atio

nof

g(x

,y)

≤0:

Slid

e 14

8

Trus

s S

truc

ture

Des

ign

Hoo

ker

2005

The

orem

Sup

pose

we

min

imiz

e cx

subj

ect t

o g(

x,y)≤

0.

If g(

x,y)

is s

emih

omog

eneo

usin

x∈

Rn

and

conc

ave

in s

cala

r y,

th

en th

e fo

llow

ing

is a

qua

si-r

elax

atio

nof

g(x

,y)

≤0:

Slid

e 14

9

Trus

s S

truc

ture

Des

ign

12

1

2

12

(,

)(

,)

0

(1)

(1)

LU

LU

LU

gx

yg

xy

xx

x

xx

x

xx

x

αα

αα

+≤

≤≤

−≤

≤−

=+

Bou

nds

on y

Hoo

ker

2005

The

orem

(JN

H 2

005)

Sup

pose

we

min

imiz

e cx

sub

ject

to g

(x,y

)≤

0.

If g(

x,y)

is s

emih

omog

eneo

us in

x∈

Rn

and

conc

ave

in s

cala

r y,

th

en th

e fo

llow

ing

is a

qua

si-r

elax

atio

nof

g(x

,y)

≤0:

Slid

e 15

0

Trus

s S

truc

ture

Des

ign

12

1

2

12

(,

)(

,)

0

(1)

(1)

LU

LU

LU

gx

yg

xy

xx

x

xx

x

xx

x

αα

αα

+≤

≤≤

−≤

≤−

=+

Bou

nds

on x

Hoo

ker

2005

The

orem

(JN

H 2

005)

Sup

pose

we

min

imiz

e cx

sub

ject

to g

(x,y

)≤

0.

If g(

x,y)

is s

emih

omog

eneo

us in

x∈

Rn

and

conc

ave

in s

cala

r y,

th

en th

e fo

llow

ing

is a

qua

si-r

elax

atio

nof

g(x

,y)

≤0:

Slid

e 15

1

Trus

s S

truc

ture

Des

ign

12

1

2

12

(,

)(

,)

0

(1)

(1)

LU

LU

LU

gx

yg

xy

xx

x

xx

x

xx

x

αα

αα

+≤

≤≤

−≤

≤−

=+

has

the

form

g(x

,y)

= 0

with

gse

mih

omog

enou

sin

xbe

caus

e w

e ca

n w

rite

it

ii

ii

i

EA

vs

h=

0i

ii

ii

EA

vs

h−

=

with

x =

(A

i,si),

y

= v

i.

Slid

e 15

2

Hoo

ke’s

law

is

linea

rized

Elo

ngat

ion

boun

ds

split

into

2 s

ets

of

boun

ds

01

01

0

1

min

[(1

)]

..co

s,a

ll

cos

,all

()

,all

,all

(1)

(1),

all

,all

01,

all

LU

ii

ii

ii

iji

ji

ijj

ii

j

LU

ii

ii

ii

i LU

ii

ii

iL

Ui

ii

ii

LU

jj

j

i

hA

yA

y

st

sp

j

dv

vi

EA

vA

vs

ih v

yv

vy

i

vy

vv

yi

dd

dj

yi

θ θ

+−

= =+

+=

≤≤

−≤

≤−

≤≤

≤≤

∑ ∑ ∑

Trus

s S

truc

ture

Des

ign

So

we

have

a q

uasi

-rel

axat

ion

of th

e tr

uss

prob

lem

:

Slid

e 15

3

Trus

s S

truc

ture

Des

ign

Logi

c cu

ts

v i0

and

v i1

mus

t hav

e sa

me

sign

in a

feas

ible

sol

utio

n.

If no

t, w

e br

anch

by

addi

ng lo

gic

cuts

01

01

,0,

,0

ii

ii

vv

vv

≤≥

Slid

e 15

4

Trus

s S

truc

ture

Des

ign

SIM

PL

mod

el

Slid

e 15

5

Load

Trus

s S

truc

ture

Des

ign

10-b

ar c

antil

ever

trus

s

Slid

e 15

6

Trus

s S

truc

ture

Des

ign

Com

puta

tiona

l res

ults

(se

cond

s)

No.

bar

sLo

ads

BA

RO

NC

PLE

X 1

1H

and

code

dS

IMP

L

101

5.3

0.40

0.03

0.08

101

3.8

0.26

0.02

0.07

101

8.1

0.83

0.16

0.49

101

8.8

1.2

0.22

0.63

102

244.

90.

641.

84

102*

327

146

145

65

102*

2067

1087

600

651

*plu

s di

spla

cem

ent b

ound

s

Yun

es, A

ron

&

Hoo

ker

2010

Slid

e 15

7

Trus

s S

truc

ture

Des

ign

25-b

ar p

robl

em

Slid

e 15

8

Trus

s S

truc

ture

Des

ign

72-b

ar p

robl

em

Slid

e 15

9

Trus

s S

truc

ture

Des

ign

Com

puta

tiona

l res

ults

(se

cond

s)

No.

bar

sLo

ads

BA

RO

NC

PLE

X 1

1H

and

code

dS

IMP

L

252

3,30

244

4420

722

3,37

620

833

28

902

21,0

1157

013

192

108

2>

24

hr*

3208

1907

1720

200

2>

24

hr*

> 2

4 hr

*>

24

hr**

> 2

4 hr

***

* no

feas

ible

sol

utio

n fo

und

** b

est f

easi

ble

solu

tion

has

cost

32,

748

***

best

feas

ible

sol

utio

n ha

s co

st 3

2,70

0

Slid

e 16

0

Sum

mar

y

•C

ombi

nato

rial o

ptim

izat

ions

met

hods

tend

to u

se a

com

mon

pr

imal

/dua

l/dua

l ap

proa

ch.

•T

his

can

be a

bas

is fo

r a

gene

ral-p

urpo

se s

olve

r•

…if

the

prob

lem

is m

odel

ed w

ith m

etac

onst

rain

ts.

Slid

e 16

1

Sum

mar

y

•C

ombi

nato

rial o

ptim

izat

ions

met

hods

tend

to u

se a

com

mon

pr

imal

/dua

l/dua

l ap

proa

ch.

•T

his

can

be a

bas

is fo

r a

gene

ral-p

urpo

se s

olve

r.

•…

if th

e pr

oble

m is

mod

eled

with

met

acon

stra

ints

.

•T

his

mor

e ge

nera

l poi

nt o

f vie

w c

an s

ugge

st n

ew m

etho

ds.

•M

etho

ds b

ased

on

infe

renc

e du

ality

ill

ustr

ated

her

e.•

So

far,

new

met

hods

bas

ed o

n re

laxa

tion

dual

ity

are

prim

arily

di

scre

te•

…an

d/or

tree

s, m

ini-b

ucke

ts &

non

seria

lDP

, mul

tival

ued

deci

sion

dia

gram

s, e

tc.

Slid

e 16

2

Gra

cias

.

¿H

ay p

regu

ntas

?