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Recent Applications of Linear Programming in Memory of George Dantzig

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### Recent Applications of Linear Programmingin Memory of George Dantzig

Yinyu Ye

Department if Management Science and Engineering

Stanford University

ISMP 2006

Outline

- LP in Auction Pricing
- Parimutuel Call Auction
- Proving Theorems using LP
- Uncapacitated Facility Location
- Core of Cooperative Game
- Applications of LP Algorithms
- Walras-Arrow-Debreu Equilibrium
- Linear Conic Programming
- Photo Album of George

(Applications presented here are by no means complete)

Outline

- LP in Auction Pricing
- Parimutuel Call Auction
- Proving Theorems using LP
- Uncapacitated Facility Location
- Core of Cooperative Game
- Applications of LP Algorithms
- Walras-Arrow-Debreu equilibrium
- Linear Conic Programming
- Photo Album of George

World Cup Betting Example

- Market for World Cup Winner
- Assume 5 teams have a chance to win the 2006 World Cup:

Argentina, Brazil, Italy, Germany and France

- We’d like to have a standard payout of $1 if a participant has a claim where his selected team won
- Sample Orders

Markets for Contingent Claims

- A Contingent Claim Market
- S possible states of the world (one will be realized).
- N participants who (say j), submit orders to a market organizer containing the following information:
- ai,j - State bid (either 1 or 0)
- qj– Limit contract quantity
- πj– Limit price per contract
- Call auction mechanism is used by one market organizer.
- If orders are filled and correct state is realized, the organizer will pay the participant a fixed amount w for each winning contract.
- The organizer would like to determine the following:
- pi – State price
- xj– Order fill

Central Organization of the Market

- Belief-based
- Central organizer will determine prices for each state based on his beliefs of their likelihood
- This is similar to the manner in which fixed odds bookmakers operate in the betting world
- Generally not self-funding
- Parimutuel
- A self-funding technique popular in horseracing betting

Horse 2

Horse 3

Bets

Total Amount Bet = $6

Outcome: Horse 2 wins

Two winners earn $2 per bet plus stake back: Winners have stake returned then divide the winnings among themselves

Parimutuel Methods- Definition
- Etymology: French pari mutuel, literally, mutual stakeA system of betting on races whereby the winners divide the total amount bet, after deducting management expenses, in proportion to the sums they have wagered individually.
- Example: Parimutuel Horseracing Betting

LP pricing for the contingent claim market

Parimutuel Market MicrostructureBoosaerts et al. [2001], Lange and Economides [2001],

Fortnow et al. [2003], Yang and Ng [2003], Peters et al. [2005], etc

Outline

- LP in Auction Pricing
- Parimutuel Call Auction
- Proving Theorems using LP
- Uncapacitated Facility Location
- Core of Cooperative Game
- Applications of LP Algorithms
- Walras-Arrow-Debreu equilibrium
- Linear Conic Programming
- Photo Album of George

Facility Location Problem

Input

- A set of clients or cities D
- A set of facilities F withfacility cost fi
- Connection cost Cij, (obey triangle inequality)

Output

- A subset of facilities F’
- An assignment of clients to facilities in F’

Objective

- Minimize the total cost (facility + connection)

Facility Location Problem

- location of a potential facility

client

(opening cost)

(connection cost)

Facility Location Problem

- location of a potential facility

client

(opening cost)

(connection cost)

and Algorithm

Hardness Results

- NP-hard.

Cornuejols, Nemhauser & Wolsey [1990].

- 1.463 polynomial approximation algorithm implies NP =P.

Guha & Khuller [1998], Sviridenko [1998].

- Each client should be assigned to one facility.
- Clients can only be assigned to open facilities.

LP Relaxation and its Dual

Interpretation:clients share the cost to open a facility, and pay the connection cost.

A bi-factor (Rf,Rc)-approximate algorithm is a max(Rf,Rc)-approximate algorithm

Jain et al [2003]

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.

The Bi-Factor Revealing LP

Jain et al [2003], Mahdian et al [2006]

Given , is bounded above by

Subject to:

Other Revealing LP Examples

- N. Bansal et al. on “Further improvements in competitive guarantees for QoS buffering,” 2004.
- Mehta et al on “Adwords and Generalized Online Matching,” 2005

Core of Cooperative Game

- A set of alliance-proof allocations of profit (Scarf [1967])
- Deterministic game (using linear programming duality, Dantzig/Von Neumann[1948])
- Linear Production, MST, flow game, some location games (Owen [1975]), Samet and Zemel [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. )

Outline

- LP in Auction Pricing
- Parimutuel Call Auction
- Proving Theorems using LP
- Uncapacitated Facility Location
- Core of Alliance
- Applications of LP Algorithms
- Walras-Arrow-Debreu equilibrium
- Linear Conic Programming
- Photo Album of George

Walras-Arrow-Debreu Equilibrium

The problem was first formulated by Leon Walrasin 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 player

- sells the entire initial endowment
- 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 Equilibrium

P1

P1

1 unit

1

1

U1(.)

P2

P2

U2(.)

1 unit

2

2

........

........

Pn

Pn

1 unit

n

n

Un(.)

Goods

Traders

Equilibrium Computation

Eisenberg and Gale[1959] , Scarf [1973], Eaves [1976,1985]

Equilibrium Computation

Nenakhov and Primak [1983], Jain [2004]

Equilibrium Computation

[2004, 2005]

Linear Conic Programming

Many excellent sessions in ISMP 2006 …

Outline

- LP in Auction Pricing
- Parimutuel Call Auction
- Core of Alliance
- Proving Theorems using LP
- Uncapacitated Facility Location
- Applications of LP Algorithms
- Walras-Arrow-Debreu equilibrium
- Linear Conic Programming
- Photo Album of George

2003 Science Fiction

COMP

IN OUR OWN IMAGE

- a computer science odyssey -

by

George 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.

COMP

In Our Own Image

Copyright © 2003 by

George Bernard Dantzig

All rights reserved

2004 90th Birthday Party

Organized by MS&E, Stanford, November 12, 2004 (Lustig, Thapa, etc)

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