recent applications of linear programming in memory of george dantzig yinyu ye department if...
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Recent Applications of Recent Applications of Linear ProgrammingLinear Programmingin Memory of George Dantzigin Memory of George Dantzig
Yinyu YeYinyu YeDepartment if Management Science and Department if Management Science and
EngineeringEngineeringStanford UniversityStanford University
ISMP 2006ISMP 2006
OutlineOutline
• LP in Auction PricingLP in Auction Pricing– Parimutuel Call AuctionParimutuel Call Auction
• Proving Theorems using LPProving Theorems using LP– Uncapacitated Facility LocationUncapacitated Facility Location– Core of Cooperative GameCore of Cooperative Game
• Applications of LP AlgorithmsApplications of LP Algorithms– Walras-Arrow-Debreu EquilibriumWalras-Arrow-Debreu Equilibrium– Linear Conic ProgrammingLinear Conic Programming
• Photo Album of GeorgePhoto Album of George(Applications presented here are by no means complete)(Applications presented here are by no means complete)
OutlineOutline
• LP in Auction PricingLP in Auction Pricing– Parimutuel Call AuctionParimutuel Call Auction
• Proving Theorems using LPProving Theorems using LP– Uncapacitated Facility LocationUncapacitated Facility Location– Core of Cooperative GameCore of Cooperative Game
• Applications of LP AlgorithmsApplications of LP Algorithms– Walras-Arrow-Debreu equilibriumWalras-Arrow-Debreu equilibrium– Linear Conic ProgrammingLinear Conic Programming
• Photo Album of GeorgePhoto Album of George
World Cup Betting ExampleWorld Cup Betting Example
• Market for World Cup WinnerMarket for World Cup Winner– Assume 5 teams have a chance to win the 2006 World Assume 5 teams have a chance to win the 2006 World
Cup: Cup:
Argentina, Brazil, Italy, Germany and FranceArgentina, Brazil, Italy, Germany and France– We’d like to have a standard payout of $1 if a We’d like to have a standard payout of $1 if a
participant has a claim where his selected team wonparticipant has a claim where his selected team won
• Sample OrdersSample OrdersOrder Order
NumberNumberPrice Price
Limit Limit Quantity Quantity Limit q Limit q
ArgentiArgentinana
BrazilBrazil ItalyItaly GermanGermanyy
FranceFrance
11 0.750.75 1010 11 11 11
22 0.350.35 55 11
33 0.400.40 1010 11 11 11
44 0.950.95 1010 11 11 11 11
55 0.750.75 55 11 11
Markets for Contingent ClaimsMarkets for Contingent Claims
• A Contingent Claim MarketA Contingent Claim Market– S possible states of the world (one will be realized).S possible states of the world (one will be realized).– N participants who (say N participants who (say jj), submit orders to a market ), submit orders to a market
organizer containing the following information:organizer containing the following information:• aai,ji,j - State bid (either 1 or 0) - State bid (either 1 or 0)• qqjj – Limit contract quantity– Limit contract quantity• ππjj – Limit price per contract– Limit price per contract
– Call auction mechanism is used by one market Call auction mechanism is used by one market organizer.organizer.
– If orders are filled and correct state is realized, the If orders are filled and correct state is realized, the organizer will pay the participant a fixed amount organizer will pay the participant a fixed amount w w for each winning contract.for each winning contract.
– The organizer would like to determine the following:The organizer would like to determine the following:• ppii – State price – State price• xxjj – Order fill– Order fill
Central Organization of the MarketCentral Organization of the Market
• Belief-basedBelief-based• Central organizer will determine prices for each Central organizer will determine prices for each
state based on his beliefs of their likelihoodstate based on his beliefs of their likelihood• This is similar to the manner in which fixed odds This is similar to the manner in which fixed odds
bookmakers operate in the betting worldbookmakers operate in the betting world• Generally not self-fundingGenerally not self-funding
• ParimutuelParimutuel• A self-funding technique popular in horseracing A self-funding technique popular in horseracing
bettingbetting
Parimutuel MethodsParimutuel Methods• DefinitionDefinition
– Etymology: French Etymology: French pari mutuel, pari mutuel, literally, mutual stakeliterally, mutual stakeA system of betting on races whereby the winners divide the total amount A system of betting on races whereby the winners divide the total amount bet, after deducting management expenses, in proportion to the sums they bet, after deducting management expenses, in proportion to the sums they have wagered individually.have wagered individually.
• Example: Parimutuel Horseracing BettingExample: Parimutuel Horseracing Betting
Horse 1 Horse 2 Horse 3
Two winners earn $2 per bet plus stake back: Winners have stake returned then divide the winnings among themselves
Bets
Total Amount Bet = $6
Outcome: Horse 2 wins
Parimutuel Market MicrostructureParimutuel Market MicrostructureBoosaerts et al. [2001], Lange and Economides [2001],
Fortnow et al. [2003], Yang and Ng [2003], Peters et al. [2005], etc
Njx
Njqx
Siyxats
ywxMax
j
jj
jjij
jjj
0
..
LP pricing for the contingent claim market
World Cup Betting ResultsWorld Cup Betting Results
Orders FilledOrders FilledOrdOrderer
Price Price LimitLimit
Quantity Quantity LimitLimit
FilleFilledd
ArgentiArgentinana
BraziBrazill
ItalyItaly GermanGermanyy
FranceFrance
11 0.750.75 1010 55 11 11 11
22 0.350.35 55 55 11
33 0.400.40 1010 55 11 11 11
44 0.950.95 1010 00 11 11 11 11
55 0.750.75 55 55 11 11
ArgentinaArgentina BrazilBrazil ItalyItaly GermanyGermany FranceFrance
PricePrice 0.200.20 0.350.35 0.200.20 0.250.25 0.000.00
State PricesState Prices
OutlineOutline
• LP in Auction PricingLP in Auction Pricing– Parimutuel Call AuctionParimutuel Call Auction
• Proving Theorems using LPProving Theorems using LP– Uncapacitated Facility LocationUncapacitated Facility Location– Core of Cooperative GameCore of Cooperative Game
• Applications of LP AlgorithmsApplications of LP Algorithms– Walras-Arrow-Debreu equilibriumWalras-Arrow-Debreu equilibrium– Linear Conic ProgrammingLinear Conic Programming
• Photo Album of GeorgePhoto Album of George
InputInput• A set of clients A set of clients or citiesor cities D D
• A set of facilities A set of facilities F F withwith facility cost facility cost ffii
• Connection cost Connection cost CCijij, , (obey triangle (obey triangle inequality)inequality)
Output• A subset of facilities F’
• An assignment of clients to facilities in F’
Objective• Minimize the total cost (facility + connection)
Facility Location ProblemFacility Location Problem
Facility Location ProblemFacility Location Problem
location of a potential facility
client
(opening cost)
(connection cost)
Facility Location ProblemFacility Location Problem
location of a potential facility
client
(opening cost)
(connection cost)
cost connectioncost openingmin
R-Approximate Solution and Algorithm
:following thesatisfies that , cost, totalwith the
UFLP,ofsolution (integral) feasible a found algorithmAn
Cost
.1constant somefor
*
R
CostRCost
Hardness Hardness ResultsResults
NP-hard. Cornuejols, Nemhauser & Wolsey [1990].
1.463 polynomial approximation algorithm implies NP =P. Guha & Khuller [1998], Sviridenko [1998].
ILP Formulation
FiDjyx
FiDjyx
Djxts
yfxCMin
iij
iij
Fiij
Fi Dj Fiiiijij
,}1,0{,
,
1..
•Each client should be assigned to one facility.
•Clients can only be assigned to open facilities.
FiDjx
FiDjyx
Djxts
yfxCMin
ij
iij
Fiij
Fi Dj Fiiiijij
,0
,
1..
LP Relaxation and its Dual
FiDj
Fif
FiDjcts
Max
ij
iDj
ij
ijijj
Djj
,0
,..
Interpretation: clients share the cost to open a facility, and pay the connection cost.
.facility toclient ofon contributi theis },0max{ ijcijjij
Bi-Factor Dual Fitting
:following thesatisfies where,cost totalwith the
FLP, ofsolution (integral) feasible a found algorithman Suppose
jDj
j
FifR
FiDjcR
ifDj
ij
ijcijj
(2)
, )1(
.
: have then we0, and 1,constant somefor ** CRFRCF
RR
cfDj
j
ijfc
A bi-factor (Rf,Rc)-approximate algorithm is a max(Rf,Rc)-approximate algorithm
Simple Greedy Algorithm
Introduce a notion of time, such that each event can be associated with the time at which it happened. The algorithm start at time 0. Initially, all facilities are closed; all clients are unconnected; all set to 0. Let C=D
While , increase simultaneously for all , until one of the following events occurs:
(1). For some client , and a open facility , then connect client j to facility i and remove j from C;
(2). For some closed facility i, , then open
facility i, and connect client with to facility i, and remove j from C.
j
C j Cj
Cj ijj ci such that
Cj
iijj fc ),0max(
Cj ijj c
Jain et al [2003]
Time = 0Time = 0
F1=3 F2=4
3 5 4 3 6 4
Time = 1Time = 1
F1=3 F2=4
3 5 4 3 6 4
Time = 2Time = 2
F1=3 F2=4
3 5 4 3 6 4
Time = 3Time = 3
F1=3 F2=4
3 5 4 3 6 4
Time = 4Time = 4
F1=3 F2=4
3 5 4 3 6 4
Time = 5Time = 5
F1=3 F2=4
3 5 4 3 6 4
Time = 5Time = 5
F1=3 F2=4
3 5 4 3 6 4
Open the facility on left, and connect clients “green” and “red” to it.
Open the facility on left, and connect clients “green” and “red” to it.
Time = 6Time = 6
F1=3 F2=4
3 5 4 3 6 4
Continue increase the budget of client “blue”
Continue increase the budget of client “blue”
Time = 6Time = 6
The budget of “blue” now covers its connection cost to an opened facility; connect blue to it.
The budget of “blue” now covers its connection cost to an opened facility; connect blue to it.
F1=3 F2=4
3 5 4 3 6 4
5 5 6
The Bi-Factor Revealing LP
Given , is bounded above by
Subject to:
c
fR
k
jij
if
k
jj
1
1max
jl
iilj fc ),0max( ||21 D
ilijlj cc
cRfR
Jain et al [2003], Mahdian et al [2006]
alg. appr.-1.861 agot We.861.1 then ,861.1 cf RR
In particular, if
Approximation ResultsApproximation Results
Ratio Reference Algorithm1+ln(|D|) Hochbaum (1982) Greedy algorithm3.16 Shmoys et.al (1997) LP rounding2.408 Guha and Kuller (1998) LP rounding + Greedy augmentation1.736 Chudak (1998) LP rounding1.728 Charika and Guha (1999) LP + P-dual + Greedy augmentation1.61 Jain et.al (2003) Greedy algorithm1.517 Mahdian et.al (2006) Revised Greedy algorithm
Other Other Revealing LP Examples Examples
• N. Bansal et al. on “Further N. Bansal et al. on “Further improvements in competitive improvements in competitive guarantees for QoS buffering,” 2004.guarantees for QoS buffering,” 2004.
• Mehta et al on “Adwords and Mehta et al on “Adwords and Generalized Online Matching,” 2005 Generalized Online Matching,” 2005
• A set of alliance-proof allocations of profit A set of alliance-proof allocations of profit (Scarf [1967])(Scarf [1967])
• Deterministic game (Deterministic game (using linear programming using linear programming
duality, Dantzig/Von Neumannduality, Dantzig/Von Neumann [1948])[1948])– Linear Production, MST, flow game, some location
games (Owen [1975]), Owen [1975]), Samet and Zemel [1984], [1984], Tamir [1991], Deng et al. [1994], Feigle et al. [1997], Goemans and Skutella [2004], etc.)
• Stochastic game (using stochastic linear programming duality, Rockafellar and Wets [1976])– Inventory game, Newsvendor (Anupindi et al. [2001],
Muller et al. [2002], Slikker et al. [2005], Chen and Zhang [2006], etc. )
Core of Cooperative GameCore of Cooperative Game
OutlineOutline
• LP in Auction PricingLP in Auction Pricing– Parimutuel Call AuctionParimutuel Call Auction
• Proving Theorems using LPProving Theorems using LP– Uncapacitated Facility LocationUncapacitated Facility Location– Core of AllianceCore of Alliance
• Applications of LP AlgorithmsApplications of LP Algorithms– Walras-Arrow-Debreu equilibriumWalras-Arrow-Debreu equilibrium– Linear Conic ProgrammingLinear Conic Programming
• Photo Album of GeorgePhoto Album of George
Walras-Arrow-Debreu EquilibriumWalras-Arrow-Debreu Equilibrium
The problem was first formulated by Leon Walras in 1874, Elements of Pure Economics, or the Theory of Social Wealth
n players, each with• an initial endowment of a divisible good • utility function for consuming all goods—own and others.Every player1. sells the entire initial endowment2. uses the revenue to buy a bundle of goods such that his or her
utility function is maximized.Walras asked:
Can prices be set for all the goods such that the market clears?Answer by Arrow and Debreu in 1954:
yes, under mild conditions if the utility functions are concave.
Walras-Arrow-Debreu EquilibriumWalras-Arrow-Debreu Equilibrium
Goods Traders
U1(.)
U2(.)
Un(.)
1 unit
1 unit
1 unit
P1
P2
Pn
........ ........
1
2
n
1
2
n
P1
P2
Pn
Fisher EquilibriumFisher EquilibriumP1 w1
Goods Buyers
U1(.)
U2(.)
Un(.)
1 unit
1 unit
1 unit
P2
Pn
........ ........
1
2
n
1
2
n
w2
wn
Utility FunctionsUtility Functions
}{min)( : UtilityLeontief
)( :UtilityLinear
ij
ijjii
ijj
ijii
a
xxu
xaxu
Equilibrium ComputationEquilibrium Computation
UtilityUtility\\ModelModel Fisher WAD
Linear Convex Opt. LCP
Leontief Convex Opt.
Eisenberg and Gale [1959] , Scarf [1973], Eaves [1976,1985]
Equilibrium ComputationEquilibrium Computation
UtilityUtility\\ModelModel Fisher WAD
Linear Convex Opt. Convex Opt
Leontief Convex Opt.
Nenakhov and Primak [1983], Jain [2004]
Equilibrium ComputationEquilibrium Computation
UtilityUtility\\ModelModel Fisher WAD
Linear LP-class LP-class
Leontief LP-class*
[2004, 2005]
Equilibrium ComputationEquilibrium Computation
UtilityUtility\\ModelModel Fisher WAD
Linear LP-class LP-class
Leontief LP-class* NP-Hard
Codenotti et al. [2005],Chen and Deng [2005, 2006],
Linear Conic ProgrammingLinear Conic Programming
*
iii
iii
ii
KS
C,SAyts
yb
K X
,...,m,,ibXAts
X C
..
Maxmize
1 ..
Minimize
Many excellent sessions in ISMP 2006 …
OutlineOutline
• LP in Auction PricingLP in Auction Pricing– Parimutuel Call AuctionParimutuel Call Auction– Core of AllianceCore of Alliance
• Proving Theorems using LPProving Theorems using LP– Uncapacitated Facility LocationUncapacitated Facility Location
• Applications of LP AlgorithmsApplications of LP Algorithms– Walras-Arrow-Debreu equilibriumWalras-Arrow-Debreu equilibrium– Linear Conic ProgrammingLinear Conic Programming
• Photo Album of GeorgePhoto Album of George
Childhood YearsChildhood Years
University Student YearsUniversity Student Years
1967 Stanford OR1967 Stanford OR
1975 National Medal of Science1975 National Medal of Science
1975 Nobel Laureate1975 Nobel Laureate
1987 Student Graduation1987 Student Graduation
2003 Science Fiction2003 Science FictionCOMP
IN OUR OWN IMAGE- a computer science odyssey -
byGeorge B. Dantzig
Nach, pale and shaking, rushed in to tell Adam, Skylab’s Captain,
that a biogerm plague is sweeping the Earth, killing
millions like flies.
COMPIn Our Own Image
Copyright © 2003 by George Bernard Dantzig
All rights reserved
2004 902004 90thth Birthday Party Birthday Party
Organized by MS&E, Stanford, November 12, 2004 (Lustig, Thapa, etc)
2004 902004 90thth Birthday Party Birthday Party
2004 902004 90thth Birthday Party Birthday Party
LP/LP/DantzigDantzig Legacy Continues Legacy Continues ……
THE DANTZIG-LIEBERMANTHE DANTZIG-LIEBERMANOPERATIONS RESEARCH OPERATIONS RESEARCH
FELLOWSHIP FUNDFELLOWSHIP FUND
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