- Risk inclusion ProgrammingMultistage...
57
Slide Number 1 Multistage Stochastic Programming John R. Birge University of Michigan Models - Long and short term - Risk inclusion Approximations - stages and scenarios Computation
Transcript of - Risk inclusion ProgrammingMultistage...
Slid
e Nu
mb
er 1
Mu
ltistage S
toch
astic P
rog
ramm
ing
Joh
n R
. Birg
eU
niversity o
f Mich
igan
Mo
dels - L
on
g an
d sh
ort term
- Risk in
clusio
nA
pp
roxim
ation
s - stages an
d scen
arios
Co
mp
utatio
n
Slid
e Nu
mb
er 2
OU
TL
INE
•Mo
tivation
- Sh
ort an
d L
on
g T
erm F
ramew
ork
•Lo
ng
-Term
: Fin
ance/cap
acity decisio
ns
–Pro
blem
s of u
ncertain
ty–G
eneral ap
pro
ach to
ward
risk - op
tion
s
•Sh
ort-T
erm: P
rod
uctio
n sch
edu
ling
–Typ
es of u
ncertain
ty–R
esults o
n cycles an
d m
atchin
g u
p–D
ifferent ro
le of risk
·Gen
eral Mo
del A
pp
roxim
ation
s•C
om
pu
tation
•Su
mm
ary
Slid
e Nu
mb
er 3
Len
gth
of H
orizo
n an
d
Decisio
ns
•L
ON
G T
ER
M H
OR
IZO
N D
EC
ISIO
NS
(YE
AR
S)
–S
TR
AT
EG
IES
–O
VE
RA
LL
CA
PA
CIT
Y–
PR
OD
UC
T M
IX–
SO
UR
CE
S O
F U
NC
ER
TA
INT
Y»
MA
RK
ET
»C
OM
PE
TIT
OR
S
•S
HO
RT
TO
ME
DIU
M T
ER
M D
EC
ISIO
NS
(< Y
EA
R)
–A
CT
UA
L P
RO
DU
CT
ION
–D
AIL
Y T
O M
ON
TH
LY
MIX
–V
AR
IAB
LE
PR
OD
UC
TIV
E C
AP
AC
ITY
Slid
e Nu
mb
er 4
Fin
ancial P
lann
ing
•G
OA
L: A
ccum
ulate $G
for tu
ition
Y years
from
no
w (L
on
g T
erm)
•A
ssum
e: –
$ W(0) - in
itial wealth
–K
- investm
ents
–co
ncave u
tility (piecew
ise linear)
GW
(Y)
Utility
RA
ND
OM
NE
SS
: return
s r(k,t) - for k in
perio
d t
wh
ere Y T
decisio
n p
eriod
s
Slid
e Nu
mb
er 5
FO
RM
UL
AT
ION
•S
CE
NA
RIO
S: σ ∈ Σ
–P
rob
ability, p
(σ)
–G
rou
ps, S
t1 , ..., StS
t at t
•M
UL
TIS
TA
GE
ST
OC
HA
ST
IC N
LP
FO
RM
:
max Σ
σ p(σ) ( U
(W( σ
, T) )
s.t. (for all σ
): Σk x(k,1, σ
) = W(o
) (initial)
Σk r(k,t-1, σ
) x(k,t-1, σ) - Σ
k x(k,t, σ) = 0 , all t >1;
Σk r(k,T
-1, σ) x(k,T
-1, σ) - W
( σ , T
) = 0, (final);
x(k,t, σ) ≥ 0, all k,t;
No
nan
ticipativity:
x(k,t, σ’) - x(k,t, σ) = 0 if σ
’, σ ∈ S
ti for all t, i, σ
’, σ T
his says d
ecision
cann
ot d
epen
d o
n fu
ture.
Slid
e Nu
mb
er 6
DA
TA
and
SO
LU
TIO
NS
•A
SS
UM
E:
–Y
=15 years–
G=$80,000
–T
=3 (5 year intervals)
–k=2 (sto
ck/bo
nd
s)
•R
eturn
s (5 year):–
Scen
ario A
: r(stock) = 1.25 r(b
on
ds)= 1.14
–S
cenario
B: r(sto
ck) = 1.06 r(bo
nd
s)= 1.12
•S
olu
tion
:P
ER
IOD
SC
EN
AR
IOS
TO
CK
BO
ND
S 1
1-8 41.5
13.5 2
1-4 65.1
2.17 2
5-8 36.7
22.4 3
1-2 83.8
0 3
3-4 0
71.4 3
5-6 0
71.4 3
7-8 64.0
0
Slid
e Nu
mb
er 7
MO
DE
L V
AL
UE
S
•C
OM
PA
RIS
ON
TO
ME
AN
VA
LU
ES
:–
RP
= -7 EM
S=-19 (all sto
ck investm
ents)
»V
SS
= RP
- EM
S = 12
•H
OR
IZO
N/P
ER
IOD
EF
FE
CT
S–
TR
UN
CA
TIO
N A
T 10 Y
EA
RS
»M
OR
E C
ON
SE
RV
AT
IVE
»H
EA
VY
BO
ND
INV
ES
TM
EN
T–
LO
NG
PE
RIO
DS
»M
OR
E M
EA
N E
FF
EC
T - L
ES
S D
IST
RIB
UT
ION
»H
EA
VY
ST
OC
K IN
VE
ST
ME
NT
•R
ES
UL
T–
NE
ED
TH
RE
E P
ER
IOD
S F
OR
HE
DG
ING
SO
LU
TIO
N
Slid
e Nu
mb
er 8
CA
PA
CIT
Y D
EC
ISIO
NS
•W
hat to
pro
du
ce?•
Wh
ere to p
rod
uce?
(Wh
en?
)•
Ho
w m
uch
to p
rod
uce?
A12 3
B
EX
AM
PL
E: M
odels 1,2, 3 ; Plants A
,B
Should B also build 2?
Slid
e Nu
mb
er 9
GO
AL
S
•A
DD
AS
MU
CH
VA
LU
E A
S P
OS
SIB
LE
• B
ut: h
ow
do
you
measu
re value?
- Net P
resent Values?
- Discounted C
ash Flow
s?
- Net P
rofit?
- Payback? IR
R?
Slid
e Nu
mb
er 10
Trad
ition
al Ap
pro
ach
•In
cremen
tal Decisio
n–
Ad
d C
apacity at B
for M
od
el 2?
•A
nalysis
–F
ind
expected
dem
and
for 2?
–U
se expected
dem
and
for 1,3
–=> D
iscou
nted
cash flo
ws
•R
esult: N
o m
od
el 2 at B–
Wh
y?
Slid
e Nu
mb
er 11
RO
LE
OF
UN
CE
RT
AIN
TY
•P
rob
lem: w
e do
no
t kno
w:
–w
hat th
e dem
and
will b
e –
ho
w m
uch
we really can
pro
du
ce in:
»1 d
ay, 1 week, 1 m
on
th, 1 year
–co
sts of in
pu
ts–
com
petito
r reaction
•R
esult: C
apacity fo
r 2 at B m
ay be u
seful if:
–d
eman
d fo
r 2 hig
her th
an exp
ected–
dem
and
for 3 lo
wer th
an exp
ected, d
eman
d fo
r 1 hig
her
–co
sts of 1 o
r 3 hig
her th
an exp
ected, co
sts of 2 lo
wer
–sh
ort ru
n cap
acity limit o
n 3
•E
ffect: New
capacity m
ay add
value
Slid
e Nu
mb
er 12
ME
AS
UR
ING
VA
LU
E
•S
UP
PO
SE
RIS
K N
EU
TR
AL
: (expected
cost)
ob
jective –
RE
SU
LT
: Do
es no
t corresp
on
d to
decisio
n m
aker p
reference
–D
ifficult to
assess real value th
is way
–
•R
ES
OL
UT
ION
: use eco
no
mic/fin
ancial
theo
ry:–
Cap
ital Asset P
ricing
Mo
del
–E
fficient M
arket Th
eory
–
•C
ON
SE
QU
EN
CE
: Fo
r finan
cial ob
jectives–
Kn
ow
ho
w to
assess based
on
risk
Slid
e Nu
mb
er 13
BA
SIC
S O
F C
AP
M
•R
ISK
/RE
TU
RN
TR
AD
EO
FF
:–
Investo
rs can d
iversify–
Firm
s need
no
t diversity
–A
ll investm
ents o
n secu
rity market lin
e
Risk
Return
NE
ED
: Symm
etric Risk
Slid
e Nu
mb
er 14
IMP
LIC
AT
ION
S F
OR
CA
PA
CIT
Y
DE
CIS
ION
S
•V
AL
IDIT
Y O
F S
YM
ME
TR
Y:
–U
nlikely:
»C
on
strained
resou
rces»
Co
rrelation
s amo
ng
dem
and
s
•A
LT
ER
NA
TIV
ES
?–
Op
tion
Th
eory
»A
llow
s for n
on
-symm
etric risk»
Exp
licitly con
siders co
nstrain
ts -»
Sell at a g
iven p
rice
•
Slid
e Nu
mb
er 15
US
E O
F O
PT
ION
S
•C
AP
AC
ITY
LIM
ITS
CU
T O
FF
PO
TE
NT
IAL
R
EV
EN
UE
LIK
E S
EL
LIN
G O
PT
ION
TO
C
OM
PE
TIT
OR
•V
AL
UE
S A
SY
MM
ET
RIC
RIS
K
•Assu
mp
tion
: risk free hed
ge
–Can
evaluate as if risk n
eutral
–As in
Black-S
cho
les mo
del
•Step
s with
capacity evalu
ation
:–A
dju
st revenu
e to risk-free eq
uivalen
t–D
iscou
nt at riskless rate
RE
SUL
TS F
RO
M F
INA
NC
E:
Slid
e Nu
mb
er 16
EV
AL
UA
TIN
G T
HE
OP
TIO
N
•C
AN
NO
T U
SE
EX
PE
CT
AT
ION
S (S
ING
LE
F
OR
EC
AS
TS
) AL
ON
E B
EC
AU
SE
OF
:•
Co
rrelated D
eman
d–
Mo
dels 1,2,3 sim
ilar
•C
apacity L
imit - cu
ts off reven
ue g
row
th–
=> Asym
metric p
ayoff
SalesC
apacity
Revenue
Slid
e Nu
mb
er 17
US
E W
ITH
A M
OD
EL
-S
toch
astic Pro
gram
min
g
•K
ey: Maxim
ize the A
dd
ed V
alue w
ith In
stalled
Cap
acity–
Mu
st cho
ose b
est mix o
f mo
dels assig
ned
to p
lants
–M
aximize E
xpected
Valu
e[ Σi Pro
fit (i) Pro
du
ction
(i)]–
sub
ject to: M
axSales(i) >= Σ j P
rod
uctio
n(i at j)
– Σ i P
rod
uctio
n(i at j) <= C
apacity (i)
– P
rod
uctio
n(i at j) <= C
apacity (i at j)
–P
rod
uctio
n(i at j) >= 0
•N
eed M
axSales(i) - u
ncertain
–C
apacity(i at j) - D
ecision
in F
irst Stag
e (no
w)
•F
IRS
T: C
on
struct sales scen
arios
Slid
e Nu
mb
er 18
Sales S
cenario
s
•D
ifficulty:
–M
any m
od
els–
Co
rrelation
s–
Hig
h V
ariance
•S
imp
lification
–G
raves, Jord
an–
Meth
od
for calcu
lation
with
kno
wn
distrib
utio
n
•S
imu
lation
–S
till need
distrib
utio
n
•B
ut u
nkn
ow
n d
istribu
tion
•=> U
se bo
un
din
g ap
pro
ximatio
ns
Slid
e Nu
mb
er 19
RE
SU
LT
S O
F O
PT
ION
-S
TO
CH
AS
TIC
PR
OG
RA
MM
ING
M
OD
EL
•G
IVE
S V
AL
UE
ME
AS
UR
E
•IN
CO
RP
OR
AT
ES
UN
CE
RT
AIN
TY
AN
D A
NY
A
VA
ILA
BL
E IN
FO
RM
AT
ION
•C
AN
BE
US
ED
FO
R V
AR
YIN
G M
OD
EL
L
IFE
TIM
ES
/PR
OD
UC
TIO
N P
ER
IOD
S•
INT
EG
RA
TE
S C
AP
AC
ITY
DE
CIS
ION
S
AC
RO
SS
FIR
M (N
OT
JUS
T W
ITH
IN 1 P
LA
NT
)•
CA
N U
SE
FO
R U
TIL
IZA
TIO
N/L
OS
T S
AL
ES
/O
TH
ER
WH
AT
-IF A
NA
LY
SE
S
Slid
e Nu
mb
er 20
GE
NE
RA
LIZ
AT
ION
S F
OR
O
TH
ER
LO
NG
-TE
RM
DE
CIS
ION
•S
TA
RT
: Elim
inate co
nstrain
ts on
pro
du
ction
–D
eman
d u
ncertain
ty remain
s - assum
e that is sym
metric
–C
an valu
e un
con
strained
revenu
e with
market rate, r:
1/(1+r) t ct x
t
IMP
LIC
AT
ION
S OF
RISK
NE
UT
RA
L H
ED
GE
: C
an model as if investors are risk neutral
=> value grows at riskfree rate, r
f
Future value: [1/(1+r) t c
t (1+r
f ) t xt ]
BU
T: T
his new quantity is constrained
Slid
e Nu
mb
er 21
CO
NS
TR
AIN
T M
OD
IFIC
AT
ION
•F
OR
ME
R C
ON
ST
RA
INT
S: A
t xt ≤ b
t
•N
OW
: At x
t (1+rf ) t/(1+r) t ≤ b
t
•
•xt •b
t
•xt (1+
rf ) t/(1+r) t
•bt
Slid
e Nu
mb
er 22
NE
W P
ER
IOD
t PR
OB
LE
M
•W
AN
T T
O F
IND
(presen
t value):
MA
X [ c
t xt (1+r
f ) t/(1+r) t | At x
t (1+rf ) t/(1+r) t ≤ b
]1/ (1+r
f ) t
EQ
UIV
AL
EN
T T
O:
MA
X [ c
t x | A
t x ≤ b (1+r) t/(1+r
f ) t]1/ (1+r) t
ME
AN
ING
: To com
pensate for lower risk w
ith constraints, constraints expand and risky discount is used
Slid
e Nu
mb
er 23
EX
TR
EM
E C
AS
ES
•A
LL
SL
AC
K C
ON
ST
RA
INT
S:
1/ (1+r) t M
AX
[ ct x
| At x ≤ b
(1+r) t/(1+rf ) t]
becomes equivalent to:
1/ (1+r) t M
AX
[ ct x
| At x ≤ b
]
i.e. same as if unconstrained - risky rate
NO
SLA
CK
:becom
es equivalent to:
1/ (1+r) t[c
t x= B-1b
(1+r) t/(1+rf ) t]=c
t B-1b
/(1+rf ) t
i.e. same as if determ
inistic- riskfree rate
Slid
e Nu
mb
er 24
OV
ER
AL
L R
ES
UL
TS
- LO
NG
-T
ER
M
•C
AN
AD
AP
T O
BJE
CT
IVE
TO
RIS
K•
US
E R
AT
E F
RO
M F
IRM
AS
WH
OL
E
–S
YM
ME
TR
IC R
ISK
–A
SS
UM
ES
INV
ES
T L
IKE
WH
OL
E F
IRM
•A
DJU
ST
AL
L C
ON
ST
RA
INT
S O
N R
EV
EN
UE
G
EN
ER
AT
OR
S B
Y R
AT
E R
AT
IOS
•E
ND
RE
SU
LT
SH
OU
LD
RE
FL
EC
T IN
VE
ST
OR
A
TT
ITU
DE
TO
WA
RD
INV
ES
TM
EN
T
Slid
e Nu
mb
er 25
OU
TL
INE
•Mo
tivation
- Sh
ort an
d L
on
g T
erm F
ramew
ork
•Lo
ng
-Term
: Fin
ance/cap
acity decisio
ns
–Pro
blem
s of u
ncertain
ty–G
eneral ap
pro
ach to
ward
risk - op
tion
s
•Sh
ort-T
erm: P
rod
uctio
n sch
edu
ling
–Typ
es of u
ncertain
ty–R
esults o
n cycles an
d m
atchin
g u
p–D
ifferent ro
le of risk
·Gen
eral Mo
del A
pp
roxim
ation
s•C
om
pu
tation
•Su
mm
ary
Slid
e Nu
mb
er 26
SH
OR
T-T
ER
M U
NC
ER
TA
INT
IES
•E
FF
EC
TIV
E C
AP
AC
ITY
LIM
ITE
D B
Y–
UN
CE
RT
AIN
YIE
LD
S - Q
UA
LIT
Y L
OS
S–
MA
CH
INE
BR
EA
KD
OW
NS
–V
AR
IAB
LE
PR
OD
UC
TIO
N R
AT
ES
–U
NF
OR
ES
EE
N O
RD
ER
S–
LA
CK
OF
MA
TE
RIA
L/S
UP
PL
IES
–L
OG
IST
ICA
L P
RO
BL
EM
S
•G
EN
ER
AL
FR
AM
EW
OR
K–
BA
SIC
OP
TIM
IZA
TIO
N P
RO
BL
EM
–
MU
ST
DE
FIN
E O
BJE
CT
IVE
S–
LO
OK
AT
ST
RU
CT
UR
E
Slid
e Nu
mb
er 27
Sh
ort T
erm M
od
el
•R
isk–
Un
iqu
e to situ
ation
(no
t market)
–S
olved
man
y times
–F
ocu
s on
expectatio
n (all u
niq
ue risk - d
iversifiable)
•S
olu
tion
time
–M
ust im
plem
ent d
ecision
s–
Real-tim
e franew
ork
–N
eed fo
r efficiency
•C
oo
rdin
ation
–M
aintain
con
sistency w
ith lo
ng
-term g
oals
Slid
e Nu
mb
er 28
GE
NE
RA
L M
UL
TIS
TA
GE
M
OD
EL
•F
OR
MU
LA
TIO
N:
MIN
E [ Σ
t=1 T ft (xt ,x
t+1 ) ]s.t. x
t ∈ X
t x
t no
nan
ticipative
P[ h
t (xt ,x
t+1 ) ≤ 0 ] ≥ a (chan
ce con
straint)
DE
FIN
ITIO
NS
:
xt - ag
greg
ate pro
du
ction
ft - defin
es transitio
n - o
nly if reso
urces availab
le an
d in
clud
es sub
traction
of d
eman
d
Slid
e Nu
mb
er 29
DY
NA
MIC
PR
OG
RA
MM
ING
V
IEW
•S
TA
GE
S: t=1,...,T
•S
TA
TE
S: x
t -> Bt x
t (or o
ther tran
sform
ation
)•
VA
LU
E F
UN
CT
ION
:∠Ψ
t (xt ) = E
[ψt (x
t ,ξt )] w
here
∠ξt is th
e rand
om
elemen
t and
∠ψt (x
t ,ξt ) = m
in ft (x
t ,xt+1, ξ
t ) + Ψt+1 (x
t+1 )–
s.t. xt+1 ∈
Xt+1t (, ξ
t ) xt g
iven
•A
SS
UM
PT
ION
S:
– C
ON
VE
XIT
Y–
EA
RL
Y A
ND
LA
TE
NE
SS
PE
NA
LT
IES
Slid
e Nu
mb
er 30
PR
OD
UC
TIO
N S
CH
ED
UL
ING
R
ES
UL
TS
•O
PT
IMA
LIT
Y:
–C
AN
DE
FIN
E O
PT
IMA
LIT
Y C
ON
DIT
ION
S–
DE
RIV
E S
UP
PO
RT
ING
PR
ICE
S
•C
YC
LIC
SC
HE
DU
LE
S:
–O
PT
IMA
L IF
ST
AT
ION
AR
Y O
R C
YC
LIC
DIS
TR
IBU
TIO
NS
–M
AY
IND
ICA
TE
KA
NB
AN
/CO
NW
IP T
YP
E O
PT
IMA
LIT
Y
•T
UR
NP
IKE
: (Birg
e/Dem
pster)
–F
RO
M O
TH
ER
DIS
RU
PT
ION
S:
– R
ET
UR
N T
O O
PT
IMA
L C
YC
LE
•L
EA
DS
TO
MA
TC
H-U
P F
RA
ME
WO
RK
Slid
e Nu
mb
er 31
MA
TC
H-U
P B
AS
ICS
•M
ET
HO
D: (B
ean,B
irge, M
ittenth
al, No
on
)•
ST
AR
T: F
IND
a PR
E-S
CH
ED
UL
E (C
YC
LIC
):–
FR
OM
FO
RE
CA
ST
S/N
OR
MA
L R
AN
DO
MN
ES
S
•M
AT
CH
-UP
PR
OC
ES
S:
–W
HE
N D
ISR
UP
TIO
NS
OC
CU
R, R
EC
OG
NIZ
E T
HE
M–
TO
DE
VE
LO
P R
ES
PO
NS
E, C
ON
ST
RU
CT
A P
LA
N T
O
MA
TC
H U
P W
ITH
TH
E P
RE
-SC
HE
DU
LE
IN T
HE
FU
TU
RE
–O
VE
RA
LL
PA
TT
ER
N R
EP
RE
SE
NT
S S
ET
TIN
G G
OA
LS
A
ND
RE
AC
TIN
G
–M
AY
AL
SO
US
E T
O IM
PR
OV
E IN
SH
OR
T R
UN
Slid
e Nu
mb
er 32
MA
TC
H-U
P P
RO
BL
EM
•G
OA
L: F
IND
A P
ER
IOD
OV
ER
WH
ICH
TO
C
HA
NG
E S
CH
ED
UL
E–
DE
FIN
E H
OR
IZO
N–
DE
FIN
E S
CE
NA
RIO
S–
DE
FIN
E P
AT
TE
RN
S
MA
CH
INE
ABC
TIM
E
DIS
RU
PT
ION
MA
TC
H-U
P
HO
RIZ
ON
Slid
e Nu
mb
er 33
HO
RIZ
ON
DE
FIN
ITIO
N
•IS
SU
ES
:–
LO
NG
EN
OU
GH
TO
:»
SM
OO
TH
OU
T R
ES
PO
NS
E»
MA
INT
AIN
LO
NG
-TE
RM
GO
AL
S»
MA
KE
EC
ON
OM
IC C
HO
ICE
–S
HO
RT
EN
OU
GH
TO
:»
AL
LO
W R
AP
ID R
ES
PO
NS
E»
CO
MP
AR
E M
AN
Y A
LT
ER
NA
TIV
ES
»N
OT
UN
DO
OP
TIM
AL
ITY
IN P
RE
-SC
HE
DU
LE
•R
ES
OL
UT
ION
–D
AIL
Y F
OR
SH
OR
T-T
ER
M
Slid
e Nu
mb
er 34
SC
EN
AR
IO D
EF
INIT
ION
•IS
SU
ES
:–
NE
ED
TO
CA
PT
UR
E P
OS
SIB
LE
FU
TU
RE
OU
TC
OM
ES
–M
US
T M
OD
EL
»D
EM
AN
D V
AR
IAT
ION
»P
RO
CE
SS
ING
INT
ER
RU
PT
ION
S–
DIF
FIC
UL
TIE
S»
INF
INIT
E N
UM
BE
RS
OF
PO
SS
IBIL
ITIE
S»
LIM
ITE
D K
NO
WL
ED
GE
BA
SE
S E
XIS
TIN
G
•A
PP
RO
AC
H–
ST
AR
T W
ITH
INIT
IAL
KN
OW
LE
DG
E–
US
E A
LL
INF
OR
MA
TIO
N T
O A
CH
IEV
E B
ES
T M
AT
CH
Slid
e Nu
mb
er 35
OU
TL
INE
•Mo
tivation
- Sh
ort an
d L
on
g T
erm F
ramew
ork
•Lo
ng
-Term
: Fin
ance/cap
acity decisio
ns
–Pro
blem
s of u
ncertain
ty–G
eneral ap
pro
ach to
ward
risk - op
tion
s
•Sh
ort-T
erm: P
rod
uctio
n sch
edu
ling
–Typ
es of u
ncertain
ty–R
esults o
n cycles an
d m
atchin
g u
p–D
ifferent ro
le of risk
·Gen
eral Mo
del A
pp
roxim
ation
s•C
om
pu
tation
•Su
mm
ary
Slid
e Nu
mb
er 36
Fu
nd
amen
tal Qu
estion
s
•D
P P
roced
ure:
–E
valuate valu
e from
each state/stag
e–
Use recu
rsion
•V
AL
UE
FU
NC
TIO
N:
∠Ψt (x
t ) = E[ψ
t (xt ,ξ
t )] wh
ere∠ξ
t is the ran
do
m elem
ent an
d∠ψ
t (xt ,ξ
t ) = min
ft (xt ,x
t+1, ξt ) + Ψ
t+1 (xt+1 )
– s.t. x
t+1 ∈ X
t+1t (, ξt ) x
t given
•S
OL
VE
: iterate from
T to
1•
PR
OB
LE
M: H
ow
to fin
d E
[ψt (x
t ,ξt )]?
∠ξt m
ay have h
igh
dim
ensio
n
Slid
e Nu
mb
er 37
AL
TE
RN
AT
IVE
S F
OR
FIN
DIN
G Ψ
t •
DIR
EC
T N
UM
ER
ICA
L IN
TE
GR
AT
ION
–P
ossib
le on
ly if very small o
r special stru
cture
–N
ot ap
plicab
le to g
eneral, larg
e pro
blem
s
•S
IMU
LA
TIO
N–
Lim
ited co
nverg
ence rate (1/ √n
error fo
r n sam
ples)
–D
ifficult estim
ates of co
nfid
ence in
tervals on
solu
tion
s
•B
OU
ND
ING
AP
PR
OX
IMA
TIO
NS
–F
ind
Ψt l,k an
d Ψ
t u,k su
ch th
at:
∠ Ψt l,k≤
Ψt ≤
Ψt u
,k
– limk Ψ
t l,k = Ψ
t = lim
k Ψt u
,k
–wh
ere limit is “ep
igrap
hical”
Slid
e Nu
mb
er 38
BO
UN
DIN
G A
PP
RO
XIM
AT
ION
S
•G
OA
LS
–M
AIN
TA
IN S
OL
VA
BL
E S
YS
TE
M–
EN
SU
RE
SO
LU
TIO
N V
AL
UE
WIT
HIN
BO
UN
DS
–C
ON
VE
RG
EN
CE
OF
BO
UN
DS
•B
AS
IC ID
EA
–U
SE
CO
NV
EX
ITY
/DU
AL
ITY
–C
ON
ST
RU
CT
FE
AS
IBL
E:
»D
UA
L S
OL
UT
ION
S•
LO
WE
R B
OU
ND
S
»P
RIM
AL
SO
LU
TIO
NS
•U
PP
ER
RO
UN
DS
•C
ON
VE
RG
EN
CE
–N
O D
UA
LIT
Y G
AP
–IM
PR
OV
ING
RE
FIN
EM
EN
TS
Slid
e Nu
mb
er 39
DIS
CR
ET
IZA
TIO
NS
•S
IMP
LIF
Y T
HE
DIS
TR
IBU
TIO
N–
RE
PL
AC
E P
BY
PK
WH
ICH
HA
S F
INIT
E S
UP
PO
RT
:
PP
K
ΞΞ
MIA
IN P
RO
CE
DU
RE
S:
LO
WE
R: JE
NS
EN
(ME
AN
) U
PP
ER
: ED
MU
ND
SO
N-M
AD
AN
SK
Y (E
XT
RE
ME
PO
INT
S)
Slid
e Nu
mb
er 40
BO
UN
D IM
PR
OV
EM
EN
TS
•P
AR
TIT
ION
ING
–S
PL
IT Ξ (S
UP
PO
RT
OF
RA
ND
OM
VE
CT
OR
) INT
O
SU
BR
EG
ION
S–
MA
KE
FU
NC
TIO
N Ψ
AS
LIN
EA
R A
S P
OS
SIB
LE
ON
EA
CH
S
UB
RE
GIO
NOR
IG. M
EA
N (JE
NS
EN
)
OR
IGIN
AL
EM
SU
B - 1
SU
B -2
NE
W E
M
NE
W JE
NS
EN
EN
FO
RC
E S
EP
AR
AB
ILIT
Y:
- FIN
D S
EP
AR
AB
LE
RE
SP
ON
SE
S T
O A
LL
RA
ND
OM
PA
RA
ME
TE
R C
HA
NG
ES
Slid
e Nu
mb
er 41
Bo
un
ds acro
ss Perio
ds
•C
om
plicatio
ns o
f man
y perio
ds
–E
xpo
nen
tial gro
wth
in d
ecision
tree in n
o. o
f perio
ds
–E
nd
effects
•M
etho
ds:
–S
tation
ary/cyclic po
licies»
Just so
lve for th
e cycle leng
th–
Ag
greg
ation
»C
ollap
se variables an
d co
nstrain
ts across p
eriod
s»
Ob
tain b
ou
nd
s from
du
ality/con
vexity–
Resp
on
se fun
ction
s»
Fin
d resp
on
se that ap
ply w
ithin
a perio
d»
Sep
arate perio
d effects
Slid
e Nu
mb
er 42
OU
TL
INE
•Mo
tivation
- Sh
ort an
d L
on
g T
erm F
ramew
ork
•Lo
ng
-Term
: Fin
ance/cap
acity decisio
ns
–Pro
blem
s of u
ncertain
ty–G
eneral ap
pro
ach to
ward
risk - op
tion
s
•Sh
ort-T
erm: P
rod
uctio
n sch
edu
ling
–Typ
es of u
ncertain
ty–R
esults o
n cycles an
d m
atchin
g u
p–D
ifferent ro
le of risk
·Gen
eral Mo
del A
pp
roxim
ation
s•C
om
pu
tation
•Su
mm
ary
Slid
e Nu
mb
er 43
SO
LV
ING
AS
LA
RG
E-S
CA
LE
M
AT
HE
MA
TIC
AL
PR
OG
RA
MS
•O
RIG
IN:
–D
ISC
RE
TIZ
AT
ION
LE
AD
S T
O M
AT
HE
MA
TIC
AL
P
RO
GR
AM
BU
T L
AR
GE
-SC
AL
E–
US
E S
TA
ND
AR
D M
ET
HO
DS
BU
T E
XP
LO
IT S
TR
UC
TU
RE
•D
IRE
CT
ME
TH
OD
S–
TA
KE
AD
VA
NT
AG
E O
F S
PA
RS
ITY
ST
RU
CT
UR
E»
SO
ME
EF
FIC
IEN
CIE
S–
US
E S
IMIL
AR
SU
BP
RO
BL
EM
ST
RU
CT
UR
E»
GR
EA
TE
R E
FF
ICIE
NC
Y - D
EC
OM
PO
SIT
ION
•S
IZE
–U
NL
IMIT
ED
(INF
INIT
E N
UM
BE
RS
OF
VA
RIA
BL
ES
)–
ST
ILL
SO
LV
AB
LE
(CA
UT
ION
ON
CL
AIM
S)
Slid
e Nu
mb
er 44
ST
AN
DA
RD
AP
PR
OA
CH
ES
•P
AR
TIT
ION
ING
•B
AS
IS F
AC
TO
RIZ
AT
ION
•
INT
ER
IOR
PO
INT
FA
CT
OR
IZA
TIO
N•
LA
GR
AN
GIA
N B
AS
ED
•M
ON
TE
CA
RL
O A
PP
RO
AC
HE
S•
DE
CO
MP
OS
ITIO
N–
BE
ND
ER
S, L
-SH
AP
ED
(VA
N S
LY
KE
- WE
TS
0–
DA
NT
ZIG
-WO
LF
E (P
RIM
AL
VE
RS
ION
)–
RE
GU
LA
RIZ
ED
(RU
SZ
CZ
YN
SK
I)
Slid
e Nu
mb
er 45
LP
-BA
SE
D M
ET
HO
DS
•U
SIN
G B
AS
IS S
TR
UC
TU
RE
PE
RIO
D 1
PE
RIO
D 2
• MO
DE
ST
GA
INS
FO
R S
IMP
LE
X•IN
TE
RIO
R P
OIN
T M
AT
RIX
ST
RU
CT
UR
E
= A
AD
2AT
=C
OM
PL
ET
E F
ILL
-IN
Slid
e Nu
mb
er 46
AL
TE
RN
AT
IVE
S F
OR
INT
ER
IOR
P
OIN
TS
•V
AR
IAB
LE
SP
LIT
TIN
G (M
UL
VE
Y E
T A
L.)
–P
UT
IN E
XP
LIC
IT N
ON
AN
TIC
IPA
TIV
ITY
CO
NT
RA
INT
S
= A
NE
W
•RE
SU
LT
•RE
DU
CE
D F
ILL
-IN B
UT
LA
RG
ER
MA
TR
IX
Slid
e Nu
mb
er 47
OT
HE
R IN
TE
RIO
R P
OIN
T
AP
PR
OA
CH
ES
•U
SE
OF
DU
AL
FA
CT
OR
IZA
TIO
N O
R
MO
DIF
IED
SC
HU
R C
OM
PL
EM
EN
T
AT D
2 A=
=
RE
SU
LT
S:
• SP
EE
DU
PS
OF
2 TO
20 • S
OM
E IN
ST
AB
ILIT
Y => IN
DE
FIN
ITE
SY
ST
EM
(VA
ND
ER
BE
I ET
AL
. C
ZY
ZY
K E
T A
L.)
• MU
LT
IST
AG
E IM
PL
EM
EN
TA
TIO
NS
US
ING
LIN
KS
(BE
RG
ER
, M
UL
VE
Y)
Slid
e Nu
mb
er 48
Lag
rang
ian-b
ased A
pp
roach
es
•G
eneral id
ea:–
Relax n
on
anticip
ativity–
Place in
ob
jective–
Sep
arable p
rob
lems
MIN
E [ Σ
t=1 T ft (xt ,x
t+1 ) ]s.t. x
t ∈ X
t x
t no
nan
ticipative
MIN
E [ Σ
t=1 T ft (xt ,x
t+1 ) ]x
t ∈ X
t + E
[w, x] + r/2||x-x|| 2
Up
date: w
t ; Pro
ject: x into
N - n
on
anticip
ative space
Co
nverg
ence: C
on
vex pro
blem
s - Pro
gressive H
edg
ing
Alg
. (R
ockafellar an
d W
ets)A
dvan
tage: M
aintain
pro
blem
structu
re (netw
orks)
Slid
e Nu
mb
er 49
Lag
rang
ian M
etho
ds an
d
Integ
er Variab
les
•Id
ea: Lag
rang
ian d
ual p
rovid
es bo
un
d fo
r p
rimal b
ut
–D
uality g
ap–
PH
A m
ay no
t con
verge
•A
lternative: stan
dard
aug
men
ted L
agrag
ian–
Co
nverg
ence to
du
al solu
tion
–L
ess separab
ility–
Du
ality gap
decreases to
zero as n
um
ber o
f scenario
s in
creases
•P
rob
lem stru
cture: P
ow
er gen
eration
p
rob
lems
–E
specially efficien
t on
parallel p
rocesso
rs
Slid
e Nu
mb
er 50
DE
CO
MP
OS
ITIO
N M
ET
HO
DS
•B
EN
DE
RS
IDE
A–
FO
RM
AN
OU
TE
R L
INE
AR
IZA
TIO
N O
F Ψ
t
–A
DD
CU
TS
ON
FU
NC
TIO
N :
– Ψt
LIN
EA
RIZ
AT
ION
AT
ITE
RA
TIO
N k
min
at k : < Ψt
new
cut
US
E A
T E
AC
H S
TA
GE
TO
AP
PR
OX
IMA
TE
VA
LU
E F
UN
CT
ION
• ITE
RA
TE
BE
TW
EE
N S
TA
GE
S U
NT
IL A
LL
MIN
= Ψt
Slid
e Nu
mb
er 51
DE
CO
MP
OS
ITIO
N
IMP
LE
ME
NT
AT
ION
•N
ES
TE
D D
EC
OM
PO
SIT
ION
–L
INE
AR
IZA
TIO
N O
F V
AL
UE
FU
NC
TIO
N A
T E
AC
H S
TA
GE
–D
EC
ISIO
NS
ON
WH
ICH
ST
AG
E T
O S
OL
VE
, WH
ICH
P
RO
BL
EM
S A
T E
AC
H S
TA
GE
•L
INE
AR
PR
OG
RA
MM
ING
SO
LU
TIO
NS
–U
SE
OS
L F
OR
LIN
EA
R S
UB
PR
OB
LE
MS
–U
SE
MIN
OS
FO
R N
ON
LIN
EA
R P
RO
BL
EM
S
•P
AR
AL
LE
L IM
PL
EM
EN
TA
TIO
N–
US
E N
ET
WO
RK
OF
RS
6000S
–P
VM
PR
OT
OC
OL
Slid
e Nu
mb
er 52
RE
SU
LT
S
•S
CA
GR
7 PR
OB
LE
M S
ET
LO
G (N
O. O
F V
AR
IAB
LE
S)
LO
G (C
PU
S)
34
56
71 2 3 4
OS
L
NE
ST
ED
DE
CO
MP
.
PA
RA
LL
EL
: 60-80% E
FF
ICIE
NC
Y IN
SP
EE
DU
P
OT
HE
R P
RO
BL
EM
S: S
IMIL
AR
RE
SU
LT
S • O
NL
Y < O
RD
ER
OF
MA
GN
ITU
DE
SP
EE
DU
P W
ITH
ST
OR
M
- TW
O-S
TA
GE
S - L
ITT
LE
CO
MM
ON
AL
ITY
IN S
UB
PR
OB
LE
MS
- ST
ILL
AB
LE
TO
SO
LV
E O
RD
ER
OF
MA
GN
ITU
DE
LA
RG
ER
PR
OB
LE
MS
Slid
e Nu
mb
er 53
SO
ME
OP
EN
ISS
UE
S
•M
OD
EL
S–
IM P
AC
T O
N M
ET
HO
DS
–R
EL
AT
ION
TO
OT
HE
R A
RE
AS
•A
PP
RO
XIM
AT
ION
S–
US
E W
ITH
SA
MP
LIN
G M
ET
HO
DS
–C
OM
PU
TA
TIO
N C
ON
ST
RA
INE
D B
OU
ND
S–
SO
LU
TIO
N B
OU
ND
S
•S
OL
UT
ION
ME
TH
OD
S–
EX
PL
OIT
SP
EC
IFIC
ST
RU
CT
UR
E–
MA
SS
IVE
LY
PA
RA
LL
EL
AR
CH
ITE
CT
UR
ES
–L
INK
S T
O A
PP
RO
XIM
AT
ION
S
Slid
e Nu
mb
er 54
CR
ITIC
ISM
S
•U
NK
NO
WN
CO
ST
S O
R D
IST
RIB
UT
ION
S–
FIN
D A
LL
AV
AIL
AB
LE
INF
OR
MA
TIO
N–
CA
N C
ON
ST
RU
CT
BO
UN
DS
OV
ER
AL
L D
IST
RIB
UT
ION
S»
FIT
TIN
G T
HE
INF
OR
MA
TIO
N–
ST
ILL
HA
VE
KN
OW
N E
RR
OR
S B
UT
AL
TE
RN
AT
IVE
S
OL
UT
ION
S
•C
OM
PU
TA
TIO
NA
L D
IFF
ICU
LT
Y–
FIT
MO
DE
L T
O S
OL
UT
ION
AB
ILIT
Y–
SIZ
E O
F P
RO
BL
EM
S IN
CR
EA
SIN
G R
AP
IDL
Y (M
OR
E
TH
AN
10 MIL
LIO
N V
AR
IAB
LE
S)
Slid
e Nu
mb
er 55
CO
NC
LU
SIO
NS
•L
ON
G A
ND
SH
OR
T T
ER
M H
OR
IZO
NS
–L
ON
G - N
EE
D F
OR
RIS
K A
VE
RS
ION
; OP
TIO
NS
–
SH
OR
T - R
ISK
MO
RE
UN
IQU
E; N
EE
D F
OR
EF
FIC
IEN
CY
–C
OO
RD
INA
TIO
N W
ITH
LO
NG
-TE
RM
: MA
TC
H-U
P
•A
PP
RO
XIM
AT
ION
S
–S
TA
TE
EX
PL
OS
ION
AC
RO
SS
ST
AG
ES
–B
OU
ND
S O
N V
AL
UE
FU
NC
TIO
N–
US
ES
OF
PR
OB
LE
M S
TR
UC
TU
RE
•S
OL
UT
ION
S–
ST
RU
CT
UR
E F
OR
DIR
EC
T M
ET
HO
DS
- INT
ER
IOR
–V
AN
ISH
ING
DU
AL
ITY
GA
PS
WIT
H IN
CR
EA
SIN
G S
IZE
–A
DV
AN
TA
GE
S IN
DE
CO
MP
OS
ITIO
N–
PR
OB
LE
M S
IZE
S IN
MIL
LIO
NS
OF
VA
RIA
BL
ES
Slid
e Nu
mb
er 56
Wh
at Next?
•In
teger variab
les - across stag
es•
Co
ntin
uo
us tim
e mo
dels
•C
om
plexity th
eory
•D
ynam
ic samp
ling
statistics•
Path
integ
ral app
roach
es from
qu
antu
m
mech
anics
•P
rob
lem stru
cture exp
loitatio
n
•D
etermin
istic samp
ling
theo
ry•
Real-tim
e app
lication
s - imp
lemen
tation
s•
Inco
rpo
rate learnin
g/B
ayesian typ
e mo
dels
•M
ultip
le agen
ts/distrib
uted
/com
petitio
n
(A B
iased P
artial List)
Slid
e Nu
mb
er 57
Mo
re Info
rmatio
n?
http
://ww
w-p
erson
al.um
ich.ed
u/~jrb
irge
