1 lifting procedures houston chapter of informs 30 may 2002 maarten oosten
TRANSCRIPT
![Page 1: 1 Lifting Procedures Houston Chapter of INFORMS 30 May 2002 Maarten Oosten](https://reader031.vdocuments.site/reader031/viewer/2022032607/56649ec55503460f94bd03ab/html5/thumbnails/1.jpg)
1
Lifting Procedures
Houston Chapter of INFORMS
30 May 2002
Maarten Oosten
![Page 2: 1 Lifting Procedures Houston Chapter of INFORMS 30 May 2002 Maarten Oosten](https://reader031.vdocuments.site/reader031/viewer/2022032607/56649ec55503460f94bd03ab/html5/thumbnails/2.jpg)
2
Outline
• Introduction
• Lifting Procedures: Review
• Generalization of the Lifting Procedures
• Summary
![Page 3: 1 Lifting Procedures Houston Chapter of INFORMS 30 May 2002 Maarten Oosten](https://reader031.vdocuments.site/reader031/viewer/2022032607/56649ec55503460f94bd03ab/html5/thumbnails/3.jpg)
3
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?
![Page 4: 1 Lifting Procedures Houston Chapter of INFORMS 30 May 2002 Maarten Oosten](https://reader031.vdocuments.site/reader031/viewer/2022032607/56649ec55503460f94bd03ab/html5/thumbnails/4.jpg)
4
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
![Page 5: 1 Lifting Procedures Houston Chapter of INFORMS 30 May 2002 Maarten Oosten](https://reader031.vdocuments.site/reader031/viewer/2022032607/56649ec55503460f94bd03ab/html5/thumbnails/5.jpg)
5
LP Relaxation Pcoin
![Page 6: 1 Lifting Procedures Houston Chapter of INFORMS 30 May 2002 Maarten Oosten](https://reader031.vdocuments.site/reader031/viewer/2022032607/56649ec55503460f94bd03ab/html5/thumbnails/6.jpg)
6
Projection Qcoin
![Page 7: 1 Lifting Procedures Houston Chapter of INFORMS 30 May 2002 Maarten Oosten](https://reader031.vdocuments.site/reader031/viewer/2022032607/56649ec55503460f94bd03ab/html5/thumbnails/7.jpg)
7
Cutting Planes•We will use at most one quarter
•We will use at least one dime
![Page 8: 1 Lifting Procedures Houston Chapter of INFORMS 30 May 2002 Maarten Oosten](https://reader031.vdocuments.site/reader031/viewer/2022032607/56649ec55503460f94bd03ab/html5/thumbnails/8.jpg)
8
Convex Hull Hcoin
![Page 9: 1 Lifting Procedures Houston Chapter of INFORMS 30 May 2002 Maarten Oosten](https://reader031.vdocuments.site/reader031/viewer/2022032607/56649ec55503460f94bd03ab/html5/thumbnails/9.jpg)
9
Definitions: Polyhedra
![Page 10: 1 Lifting Procedures Houston Chapter of INFORMS 30 May 2002 Maarten Oosten](https://reader031.vdocuments.site/reader031/viewer/2022032607/56649ec55503460f94bd03ab/html5/thumbnails/10.jpg)
10
Definitions: Faces
![Page 11: 1 Lifting Procedures Houston Chapter of INFORMS 30 May 2002 Maarten Oosten](https://reader031.vdocuments.site/reader031/viewer/2022032607/56649ec55503460f94bd03ab/html5/thumbnails/11.jpg)
11
Definitions: Cones
![Page 12: 1 Lifting Procedures Houston Chapter of INFORMS 30 May 2002 Maarten Oosten](https://reader031.vdocuments.site/reader031/viewer/2022032607/56649ec55503460f94bd03ab/html5/thumbnails/12.jpg)
12
Outline
• Introduction
• Lifting Procedures: Review
• Generalization of the Lifting Procedures
• Summary
![Page 13: 1 Lifting Procedures Houston Chapter of INFORMS 30 May 2002 Maarten Oosten](https://reader031.vdocuments.site/reader031/viewer/2022032607/56649ec55503460f94bd03ab/html5/thumbnails/13.jpg)
14
• 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
![Page 14: 1 Lifting Procedures Houston Chapter of INFORMS 30 May 2002 Maarten Oosten](https://reader031.vdocuments.site/reader031/viewer/2022032607/56649ec55503460f94bd03ab/html5/thumbnails/14.jpg)
15
• 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
![Page 15: 1 Lifting Procedures Houston Chapter of INFORMS 30 May 2002 Maarten Oosten](https://reader031.vdocuments.site/reader031/viewer/2022032607/56649ec55503460f94bd03ab/html5/thumbnails/15.jpg)
16
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
![Page 16: 1 Lifting Procedures Houston Chapter of INFORMS 30 May 2002 Maarten Oosten](https://reader031.vdocuments.site/reader031/viewer/2022032607/56649ec55503460f94bd03ab/html5/thumbnails/16.jpg)
17
Example
Let S = {(0,0,0), (1,1,0), (1,0,1), (0,1,1)}
0
0
0
2
,,
zyx
zyx
zyx
zyx
RzyxP
![Page 17: 1 Lifting Procedures Houston Chapter of INFORMS 30 May 2002 Maarten Oosten](https://reader031.vdocuments.site/reader031/viewer/2022032607/56649ec55503460f94bd03ab/html5/thumbnails/17.jpg)
18
Example Polytope
![Page 18: 1 Lifting Procedures Houston Chapter of INFORMS 30 May 2002 Maarten Oosten](https://reader031.vdocuments.site/reader031/viewer/2022032607/56649ec55503460f94bd03ab/html5/thumbnails/18.jpg)
19
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
![Page 19: 1 Lifting Procedures Houston Chapter of INFORMS 30 May 2002 Maarten Oosten](https://reader031.vdocuments.site/reader031/viewer/2022032607/56649ec55503460f94bd03ab/html5/thumbnails/19.jpg)
20
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
![Page 20: 1 Lifting Procedures Houston Chapter of INFORMS 30 May 2002 Maarten Oosten](https://reader031.vdocuments.site/reader031/viewer/2022032607/56649ec55503460f94bd03ab/html5/thumbnails/20.jpg)
21
Outline
• Introduction
• Lifting Procedures: Review
• Generalization of the Lifting Procedures
• Summary
![Page 21: 1 Lifting Procedures Houston Chapter of INFORMS 30 May 2002 Maarten Oosten](https://reader031.vdocuments.site/reader031/viewer/2022032607/56649ec55503460f94bd03ab/html5/thumbnails/21.jpg)
22
• 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
11
Extreme point: = = ½
Corresponding inequality: 2 zyx
![Page 22: 1 Lifting Procedures Houston Chapter of INFORMS 30 May 2002 Maarten Oosten](https://reader031.vdocuments.site/reader031/viewer/2022032607/56649ec55503460f94bd03ab/html5/thumbnails/22.jpg)
23
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’
![Page 23: 1 Lifting Procedures Houston Chapter of INFORMS 30 May 2002 Maarten Oosten](https://reader031.vdocuments.site/reader031/viewer/2022032607/56649ec55503460f94bd03ab/html5/thumbnails/23.jpg)
24
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
![Page 24: 1 Lifting Procedures Houston Chapter of INFORMS 30 May 2002 Maarten Oosten](https://reader031.vdocuments.site/reader031/viewer/2022032607/56649ec55503460f94bd03ab/html5/thumbnails/24.jpg)
25
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’
![Page 25: 1 Lifting Procedures Houston Chapter of INFORMS 30 May 2002 Maarten Oosten](https://reader031.vdocuments.site/reader031/viewer/2022032607/56649ec55503460f94bd03ab/html5/thumbnails/25.jpg)
26
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’
![Page 26: 1 Lifting Procedures Houston Chapter of INFORMS 30 May 2002 Maarten Oosten](https://reader031.vdocuments.site/reader031/viewer/2022032607/56649ec55503460f94bd03ab/html5/thumbnails/26.jpg)
27
Outline
• Introduction
• Lifting Procedures: Review
• Generalization of the Lifting Procedures
• Summary
![Page 27: 1 Lifting Procedures Houston Chapter of INFORMS 30 May 2002 Maarten Oosten](https://reader031.vdocuments.site/reader031/viewer/2022032607/56649ec55503460f94bd03ab/html5/thumbnails/27.jpg)
28
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.
![Page 28: 1 Lifting Procedures Houston Chapter of INFORMS 30 May 2002 Maarten Oosten](https://reader031.vdocuments.site/reader031/viewer/2022032607/56649ec55503460f94bd03ab/html5/thumbnails/28.jpg)
29
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
![Page 29: 1 Lifting Procedures Houston Chapter of INFORMS 30 May 2002 Maarten Oosten](https://reader031.vdocuments.site/reader031/viewer/2022032607/56649ec55503460f94bd03ab/html5/thumbnails/29.jpg)
30
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
![Page 30: 1 Lifting Procedures Houston Chapter of INFORMS 30 May 2002 Maarten Oosten](https://reader031.vdocuments.site/reader031/viewer/2022032607/56649ec55503460f94bd03ab/html5/thumbnails/30.jpg)
31
Outline
• Introduction
• Lifting Procedures: Review
• Generalization of the Lifting Procedures
• Summary