recent applications of linear programming in memory of george dantzig yinyu ye department if...

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



• 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





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


location of a potential facility

client

(opening cost)

(connection cost)


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


• Proving Theorems using LPProving Theorems using LP– Uncapacitated Facility LocationUncapacitated Facility Location– Core of AllianceCore of Alliance



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]



Linear Convex Opt. Convex Opt

Leontief Convex Opt.

Nenakhov and Primak [1983], Jain [2004]



Linear LP-class LP-class

Leontief LP-class*

[2004, 2005]



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



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

LP/LP/DantzigDantzig Legacy Continues Legacy Continues ……

THE DANTZIG-LIEBERMANTHE DANTZIG-LIEBERMANOPERATIONS RESEARCH OPERATIONS RESEARCH

FELLOWSHIP FUNDFELLOWSHIP FUND

recent applications of linear programming in memory of george dantzig yinyu ye department if...

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