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Lifting Procedures

Houston Chapter of INFORMS

30 May 2002

Maarten Oosten

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Outline

• Introduction

• Lifting Procedures: Review

• Generalization of the Lifting Procedures

• Summary

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Example: Vending Machine

• A Swiss Roll costs 40 cents

• No change light blinks

• We have 3 quarters, 5 dimes, and 10 cents

• We prefer to use as few coins as possible

How many of each type of coins should we use?

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Decision variables:

Vending Machine (2)

}3,2,1,0{quartersX

}5,4,3,2,1,0{dim esX

}10,...,1,0{centsX

Payment equation:

Objective function:

401025 dim centsesquarters XXX

}{ dim centsesquarters XXXMIN

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LP Relaxation Pcoin

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Projection Qcoin

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Cutting Planes•We will use at most one quarter

•We will use at least one dime

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Convex Hull Hcoin

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Definitions: Polyhedra

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Definitions: Faces

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Definitions: Cones

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Outline

• Introduction

• Lifting Procedures: Review

• Generalization of the Lifting Procedures

• Summary

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• Consider

where Bn is the space of n-dimensional binary vectors

• Define

• Define S1 as S\S0

• Let P be the convex hull of S

• Let P0 be the convex hull of S0

• Let P1 be the convex hull of S1

Definitions

}|max{ nT BSxxc

}0|{0 nxxSS

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• Let be a valid inequality for P0 • Then for some is

called a lifting from P0 to P of the inequality if it is valid for P

• It is valid if and only if the coefficient

satisfies:

Traditional Lifting

0axaT

0axaT

0axxa nT R

}|max{ 10 Sxxaa T

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Literature Review

• Wolsey, 1976• Zemel, 1978• Balas & Zemel, 1984• Nemhauser & Wolsey, 1988• Boyd & Pulleyblank, 1991• Gu et al, 1995

No guarantee that a facet defining inequality of P0 lifts to a facet defining inequality of P if the

dimension gap is larger than 1

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Example

Let S = {(0,0,0), (1,1,0), (1,0,1), (0,1,1)}

0

0

0

2

,,

zyx

zyx

zyx

zyx

RzyxP

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Example Polytope

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Example Traditional Lifting

• is a facet defining inequality for P0 • Then for some is a lifting from P0 to P of the inequality if it is valid for P

• It is valid if and only if the coefficient satisfies:

• Strongest lifted inequality is

1y

1 zy R

0}1:),,(|max{1 zSzyxy

1y

1y

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Example Traditional Lifting (2)

Due to the symmetry of the polytope, no matter in which order the variables are

lifted, the resulting lifted inequalities are always trivial inequalities

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Outline

• Introduction

• Lifting Procedures: Review

• Generalization of the Lifting Procedures

• Summary

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• Take into account all equalities that hold for P0 but not for P

and should satisfy the solutions of S1:

for (x,y,z) = (0,1,1)

for (x,y,z) = (1,0,1)

Example Extended lifting

1)( yxzy R ,

10

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Extreme point: = = ½

Corresponding inequality: 2 zyx

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Extended Lifting

• For every facet defining inequality of P0, we can construct at least one facet defining inequality of P.

• We do need a minimal representation of all equations that hold for P0 but not for P.

• We do need to find the extreme points of the lifting polyhedron of the inequality

},|{:),( 100

0

SxdaDxxaRaa TTTppT

‘extended lifting of the inequality aTx a0’

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Lift all equalities that hold for P0 but not for P

and should satisfy the solutions of S1:

for (x,y,z) = (0,1,1)

for (x,y,z) = (1,0,1)

Example Equality lifting

1)( yxz R ,

1

1

Two extreme rays: (,) = (-1,1) and (,) = (-1,-1)

Corresponding inequalities: 0 zyx0 zyx

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Equality Lifting

• With a minimal representation of all equations (‘equality set of P0’) that hold for P0 but not for P, we can construct at least one facet of P.

• We do need to find the extreme rays of the lifting cone of the equality set of P0.

},|{: 10

SxdDxR TTpp

‘extended lifting of the equality system’

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Complete Lifting

• The other way around: for every facet of P is the lifting of at least one face of P0.

• We do need to find the extreme rays of the complete lifting cone of the polytope P0.

},|,{: 10 0

SxdaaDxAxaRR TTTTppm

‘complete lifting of the minimal

facial description of P0’

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Outline

• Introduction

• Lifting Procedures: Review

• Generalization of the Lifting Procedures

• Summary

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Summary

• Every facet can be lifted to a facet

• Equalities can be lifted to a facet

• There are complete descriptions of the set of solutions that are partly a facial description, partly a listing of solutions. Lifting procedures describe the relations between these descriptions.

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Polarity context

• Suppose P0 is the empty set.• We do need a minimal representation of all

equations that hold for P0 but not for P, for example: x1=0, x2=0, … xn=0, and 0=1.

• The lifting cone of the equality set of P0

reduces to the polar cone of P:

)(},0|{

},|{0

1

10

PSxxR

SxdDxR

nTp

TTpp

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Duality context

0..

)(

Sxts

xcT

0

0

x

exE

dDx

aAx

RxP n

0..

)(

EDActs

edaTTTT

TTT

If is an extreme ray of this cone, your inequality defines a facet of P0

or 1Sx and 0),(

If (,) is an extreme ray of this cone, your inequality defines a facet of P

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Outline

• Introduction

• Lifting Procedures: Review

• Generalization of the Lifting Procedures

• Summary