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....
Forty Years of Corner Polyhedra
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
Corner Polyhedra Origins Stock Cutting
• Computing Lots of Knapsacks
• Periodicity observed
• Gomory-Gilmore 1966 "The Theory and Computation of Knapsack Functions“
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
V
L.P., I.P and Corner Polyhedron
Another View - T-Space
2 4 6 8 10t1
1
2
3
4
5
6
t2
Cutting Planes for Corner Polyhedra are Cutting Planes for
General I.P.
Valid, Minimal, Facet
T-Space View
2 4 6 8 10t1
1
2
3
4
5
6
t2
FMV
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
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')
Typical Structured Facescomputed using Balinski program
Size Problem :Shooting Geometry
2 4 6 8 10t1
1
2
3
4
5
6
t2
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
Concentration of HitsEllis Johnson and Lisa Evans
Much More to be Learned
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
Cutting Planes for Type Two
• Example: Gomory Mixed Integer Cut
• Variables ti Integer
• Variables t+, t- Non-Integer
• Valid subadditive function
Typical Structured Faces
Interpolating to get cutting plane function on the real line
2 4 6 8 100
0.2
0.4
0.6
0.8
1
Interpolating
2 4 6 8 100
0.5
1
1.5
2
2.5
3
Interpolating
2 4 6 8 100
1
2
3
4
Gomory-Johnson Theorem
If (x) has only two slopes and satisfies
the minimality condition (x)+ (1-x)=1
then it is a facet.
Integer Variables Example 2
0.2 0.4 0.6 0.8 1
0.2
0.4
0.6
0.8
1
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
Comparing
0.2 0.4 0.6 0.8 1
0.2
0.4
0.6
0.8
1
( ) 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
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
-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
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
-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
-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
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
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.
Direction
• Move on to More Dimensions
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.
Creating 2D facets
-1.5 -1 -0.5 0.5 1 1.5 2
-1.5
-1
-0.5
0.5
1
1.5
The triopoly figure
0 1 2
-0.5
0
0.5
00.250.50.751
-0.5
0
0.5
This corresponds to
-2 -1.5 -1 -0.5 0.5 1 1.5 2
0.5
1
1.5
2
2.5
3
The periodic figure
-2 -1.5 -1 -0.5 0.5 1 1.5 2
0.5
1
1.5
2
2.5
3
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
One Periodic Unit
Creating Another Facet
-1 1 2 3
-1.5
-1
-0.5
0.5
1
1.5
The Periodic Figure - Another Facet
More
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.
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
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
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
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
T-SpaceGomory Mixed Integer Cuts
1 2 3 4t1
1
2
3
4
t2
T- Space – some 2D Cuts Added
1 2 3 4t1
1
2
3
4
t2
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
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.
END
Backup Slides
Thoughts About Integer Programming
University of Montreal, January 26, 2007 40th Birthday Celebration of the
Department of Computer Science and Operations Research
Corner Polyhedra and
2-Dimensional Cuttimg Planes
George Nemhauser Symposium
June 26-27 2007
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
One Periodic Unit
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
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
Shifting
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).