Forty Years of Corner Polyhedra. Two Types of I.P. All Variables (x,t) and data (B,N) integer....
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Transcript of Forty Years of Corner Polyhedra. Two Types of I.P. All Variables (x,t) and data (B,N) integer....
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Forty Years of Corner Polyhedra
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Two Types of I.P.
• All Variables (x,t) and data (B,N) integer. Example: Traveling Salesman
• Some Variables (x,t) Integer, some continuous, data continuous. Example: Scheduling,Economies of scale.
• Corner Polyhedra relevant to both
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Corner Polyhedra Origins Stock Cutting
• Computing Lots of Knapsacks
• Periodicity observed
• Gomory-Gilmore 1966 "The Theory and Computation of Knapsack Functions“
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1 1
1 1
Integer Programming Equations
Corner Polyhedr
(Mod 1)
at basis B
on Relaxation
Variables x Integer
Non-negativity Relaxed on x
Bx Nt b
Ix B Nt B b
B Nt B b
Equations
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V
L.P., I.P and Corner Polyhedron
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Another View - T-Space
2 4 6 8 10t1
1
2
3
4
5
6
t2
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Cutting Planes for Corner Polyhedra are Cutting Planes for
General I.P.
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Valid, Minimal, Facet
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T-Space View
2 4 6 8 10t1
1
2
3
4
5
6
t2
FMV
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Cutting Planes for Corner Polyhedra
1 1
1 1
i
(Mod 1)
{ } and
a solution
Valid Cutting Plane; non-negative scalar ( )
( ) 1
subaadditive, normalized
i g
i i gi
i
i i g i ii
B Nt B b
B N v B b v
t v v
v
if t v v then t v
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Structure Theorem- 1969
o
( ) / ( )
is a facet of the corner polyhedron
produced by G if and only if it is a basic feasible
solution of this list of equations and inequalities
(g)+ (g-g ) 1 (all g)
(g)+ (g') ( ') (all g
G M I M B
g g
, g')
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Typical Structured Facescomputed using Balinski program
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Size Problem :Shooting Geometry
2 4 6 8 10t1
1
2
3
4
5
6
t2
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Size Problem -Shooting Theorem
0
the Facet solving the L.P.
min v
(g)+ (g -g) 1 (all g)
(g)+ (g') ( ') (all g, g')
Is the Facet first hit by the random direction v
g g
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Concentration of HitsEllis Johnson and Lisa Evans
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Much More to be Learned
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ComparingInteger Programs and Corner
Polyhedron• General Integer Programs – Complex, no obvious
structure• Corner Polyhedra – Highly structured, but
complexity increases rapidly with group size.• Next Step: Making this supply of cutting planes
available for non-integer data and continuous variables. Gomory-Johnson 1970
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Cutting Planes for Type Two
• Example: Gomory Mixed Integer Cut
• Variables ti Integer
• Variables t+, t- Non-Integer
• Valid subadditive function
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Typical Structured Faces
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Interpolating to get cutting plane function on the real line
2 4 6 8 100
0.2
0.4
0.6
0.8
1
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Interpolating
2 4 6 8 100
0.5
1
1.5
2
2.5
3
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Interpolating
2 4 6 8 100
1
2
3
4
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Gomory-Johnson Theorem
If (x) has only two slopes and satisfies
the minimality condition (x)+ (1-x)=1
then it is a facet.
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Integer Variables Example 2
0.2 0.4 0.6 0.8 1
0.2
0.4
0.6
0.8
1
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Integer Based Cuts
• A great variety of cutting planes generated from Integer Theory
• But more developed cutting planes weaker than the Gomory Mixed Integer Cut for their continuous variables
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Comparing
0.2 0.4 0.6 0.8 1
0.2
0.4
0.6
0.8
1
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( ) Gomory Mixed Integer Cut
Integer Variables
x
-2 -1.5 -1 -0.5 0.5 1 1.5 2
0.5
1
1.5
2
2.5
3
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Integer Cuts lead to Cuts for the Continuous Variables
-2 -1.5 -1 -0.5 0.5 1 1.5 2
0.5
1
1.5
2
2.5
3
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-2 -1.5 -1 -0.5 0.5 1 1.5 2
0.5
1
1.5
2
2.5
3
( )x
Gomory Mixed Integer CutContinuous Variables
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New Direction
• Reverse the present Direction
• Create facets for continous variables
• Turn them into facets for the integer problem
• Montreal January 2007, Georgia Tech August 2007
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-2 -1.5 -1 -0.5 0.5 1 1.5 2
0.5
1
1.5
2
2.5
3
Start With Continuous x
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-2 -1.5 -1 -0.5 0.5 1 1.5 2
0.5
1
1.5
2
2.5
3
Create Integer Cut: Shifting and Intersecting
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Shifting and Intersecting
i
i i
i
Cutting Plane; non-negative scalar ( )
( ) 1
If a t is integer, v can be changed by
an integer . So (v ) min ( )
shifting + intersecting
i
i i g i ii
i
v
if t v v then t v
v v
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One Dimension Continuous Problem
1 1
All t continuous
Theorem: The Gomory Mixed Integer cut is the only
(Mod 1)
cutting plane that is a facet for both the pure integer and the
pure continuous one di
B Nt B b
mensional problems.
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Direction
• Move on to More Dimensions
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Helper Theorem
Theorem If is a facet of the continous problem, then (kv)=k (v).
This will enable us to create 2-dimensional facets for the continuous problem.
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Creating 2D facets
-1.5 -1 -0.5 0.5 1 1.5 2
-1.5
-1
-0.5
0.5
1
1.5
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The triopoly figure
0 1 2
-0.5
0
0.5
00.250.50.751
-0.5
0
0.5
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This corresponds to
-2 -1.5 -1 -0.5 0.5 1 1.5 2
0.5
1
1.5
2
2.5
3
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The periodic figure
-2 -1.5 -1 -0.5 0.5 1 1.5 2
0.5
1
1.5
2
2.5
3
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Two Dimensional Periodic Figure
-1
0
1
2
XXX
-1
0
1
2
YYY
00.250.50.751ZZZ
-1
0
1
2
YYY
00.250.50.751ZZZ
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One Periodic Unit
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Creating Another Facet
-1 1 2 3
-1.5
-1
-0.5
0.5
1
1.5
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The Periodic Figure - Another Facet
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More
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But there are four sided figures too
Corneujois and Margot have given a complete characterization of the two dimensional cutting planes for the pure continuous problem.
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All of the three sided polygons create Facets
• For the continuous problem
• For the Integer Problem
• For the General problem
• Two Dimensional analog of Gomory Mixed Integer Cut
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xi Integer ti Continuous
1 1
2 2
x 0.34, 1.12 -0.11, 1.01 1.10+
-0.35, 0.44 0.70, -0.44 0.14
Bx+Nt=b
t
x t
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Basis B
1 1
1 1
2 2 2
1 0 0.75, 0.15 0.6
0 1 0,35, 0.55 0.8
Ix B N B b
x t
x t
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Corner Polyhedron Equations
1
2 2
1 1
0.75, 0.15 0.6
0.35, 0.55 0.8
t
t
B Nt B b
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T-SpaceGomory Mixed Integer Cuts
1 2 3 4t1
1
2
3
4
t2
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T- Space – some 2D Cuts Added
1 2 3 4t1
1
2
3
4
t2
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Summary
• Corner Polyhedra are very structured
• The structure can be exploited to create the 2D facets analogous to the Gomory Mixed Integer Cut
• There is much more to learn about Corner Polyhedra and it is learnable
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Challenges
• Generalize cuts from 2D to n dimensions
• Work with families of cutting planes (like stock cutting)
• Introduce data fuzziness to exploit large facets and ignore small ones
• Clarify issues about functions that are not piecewise linear.
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END
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Backup Slides
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Thoughts About Integer Programming
University of Montreal, January 26, 2007 40th Birthday Celebration of the
Department of Computer Science and Operations Research
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Corner Polyhedra and
2-Dimensional Cuttimg Planes
George Nemhauser Symposium
June 26-27 2007
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11 1 1
2 2 2 2
3 3 3 3
4 4 4 4
i
fc n f
c n f fv
c n f f
c n f f
Mod(1) B-1N has exactly Det(B) distinct
Columns vi
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One Periodic Unit
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Why π(x) Produces the Inequality• It is subadditive π(x) + π(y) π(x+y) on the
unit interval (Mod 1)
• It has π(x) =1 at the goal point x=f0
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Origin of Continuous Variables Procedure
0 0i
i
i
If for some t then ( / )( )
For large apply ; the result is (( / )) ( ) 1
( ) ) 1
( ) 0 ( ) 0.
i i i i i ii
i i i i i
i i
i i
c t c c k k t c
k c k k t
s c t
where s c s c for x and s x s x for x
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Shifting
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References• “Some Polyhedra Related to Combinatorial Problems,”
Journal of Linear Algebra and Its Applications, Vol. 2, No. 4, October 1969, pp.451-558
• “Some Continuous Functions Related to Corner Polyhedra, Part I” with Ellis L. Johnson, Mathematical Programming, Vol. 3, No. 1, North-Holland, August, 1972, pp. 23-85.
• “Some Continuous Functions Related to Corner Polyhedra, Part II” with Ellis L. Johnson, Mathematical Programming, Vol. 3, No. 3, North-Holland, December 1972, pp. 359-389.
• “T-space and Cutting Planes” Paper, with Ellis L. Johnson, Mathematical Programming, Ser. B 96: Springer-Verlag, pp 341-375 (2003).