Download - Corner Polyhedra and 2-Dimensional Cuttimg Planes George Nemhauser Symposium June 26-27 2007
![Page 1: Corner Polyhedra and 2-Dimensional Cuttimg Planes George Nemhauser Symposium June 26-27 2007](https://reader034.vdocument.in/reader034/viewer/2022051819/5514092f550346ec488b4ce1/html5/thumbnails/1.jpg)
Corner Polyhedra and
2-Dimensional Cuttimg Planes
George Nemhauser Symposium
June 26-27 2007
![Page 2: Corner Polyhedra and 2-Dimensional Cuttimg Planes George Nemhauser Symposium June 26-27 2007](https://reader034.vdocument.in/reader034/viewer/2022051819/5514092f550346ec488b4ce1/html5/thumbnails/2.jpg)
Integer Programming - Notation
Some or all of (x,t) Integer
(x,t) Non-Negative
Max cx
Bx Nt b
![Page 3: Corner Polyhedra and 2-Dimensional Cuttimg Planes George Nemhauser Symposium June 26-27 2007](https://reader034.vdocument.in/reader034/viewer/2022051819/5514092f550346ec488b4ce1/html5/thumbnails/3.jpg)
V
L.P., I.P and Corner Polyhedron
![Page 4: Corner Polyhedra and 2-Dimensional Cuttimg Planes George Nemhauser Symposium June 26-27 2007](https://reader034.vdocument.in/reader034/viewer/2022051819/5514092f550346ec488b4ce1/html5/thumbnails/4.jpg)
1 1
1 1
Corner Polyhedr
Integer Programming
(Mod 1)
on at basis B
Variables x Integer
Non-negativity Relaxed on
;
at ba
x
sis B
Bx Nt b
Ix
B Nt B
B Nt B b
b
Equations
![Page 5: Corner Polyhedra and 2-Dimensional Cuttimg Planes George Nemhauser Symposium June 26-27 2007](https://reader034.vdocument.in/reader034/viewer/2022051819/5514092f550346ec488b4ce1/html5/thumbnails/5.jpg)
V
L.P., I.P and Corner Polyhedron
![Page 6: Corner Polyhedra and 2-Dimensional Cuttimg Planes George Nemhauser Symposium June 26-27 2007](https://reader034.vdocument.in/reader034/viewer/2022051819/5514092f550346ec488b4ce1/html5/thumbnails/6.jpg)
ComparingInteger Programs and Corner
Polyhedron• General Integer Programs – Complex, no
obvious structure
• Corner Polyhedra – Highly structured
![Page 7: Corner Polyhedra and 2-Dimensional Cuttimg Planes George Nemhauser Symposium June 26-27 2007](https://reader034.vdocument.in/reader034/viewer/2022051819/5514092f550346ec488b4ce1/html5/thumbnails/7.jpg)
Cutting Planes for Corner Polyhedra are Cutting Planes for
General I.P.
![Page 8: Corner Polyhedra and 2-Dimensional Cuttimg Planes George Nemhauser Symposium June 26-27 2007](https://reader034.vdocument.in/reader034/viewer/2022051819/5514092f550346ec488b4ce1/html5/thumbnails/8.jpg)
Valid, Minimal, Facet
![Page 9: Corner Polyhedra and 2-Dimensional Cuttimg Planes George Nemhauser Symposium June 26-27 2007](https://reader034.vdocument.in/reader034/viewer/2022051819/5514092f550346ec488b4ce1/html5/thumbnails/9.jpg)
Cutting Planes
1 1
1 1
i
(Mod 1)
{ } and
Cutting Plane; non-negative scalar ( )
( ) 1
i g
i
i i g i ii
B Nt B b
B N v B b v
v
if t v v then t v
![Page 10: Corner Polyhedra and 2-Dimensional Cuttimg Planes George Nemhauser Symposium June 26-27 2007](https://reader034.vdocument.in/reader034/viewer/2022051819/5514092f550346ec488b4ce1/html5/thumbnails/10.jpg)
General Cutting Planes
i
Additive group G (ususally N space)
with elements v
Non-Negative ( ) such that
For any {t } from the origin to
the path ( ) 1.
g
i i gi
i ii
v
path v
t v v
length t v
![Page 11: Corner Polyhedra and 2-Dimensional Cuttimg Planes George Nemhauser Symposium June 26-27 2007](https://reader034.vdocument.in/reader034/viewer/2022051819/5514092f550346ec488b4ce1/html5/thumbnails/11.jpg)
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.
![Page 12: Corner Polyhedra and 2-Dimensional Cuttimg Planes George Nemhauser Symposium June 26-27 2007](https://reader034.vdocument.in/reader034/viewer/2022051819/5514092f550346ec488b4ce1/html5/thumbnails/12.jpg)
1 1
1 1
-1
Equations for Corner Polyhedr
(Mod 1)
(Mod 1)
Add and Subtract Columns of
This forms group G
on
B N
Ix B Nt B b
B Nt B b
First Type Data and Variables Integer
![Page 13: Corner Polyhedra and 2-Dimensional Cuttimg Planes George Nemhauser Symposium June 26-27 2007](https://reader034.vdocument.in/reader034/viewer/2022051819/5514092f550346ec488b4ce1/html5/thumbnails/13.jpg)
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
![Page 14: Corner Polyhedra and 2-Dimensional Cuttimg Planes George Nemhauser Symposium June 26-27 2007](https://reader034.vdocument.in/reader034/viewer/2022051819/5514092f550346ec488b4ce1/html5/thumbnails/14.jpg)
Structure Theorem
o
is a facet 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')g g
![Page 15: Corner Polyhedra and 2-Dimensional Cuttimg Planes George Nemhauser Symposium June 26-27 2007](https://reader034.vdocument.in/reader034/viewer/2022051819/5514092f550346ec488b4ce1/html5/thumbnails/15.jpg)
Typical Structured Faces
![Page 16: Corner Polyhedra and 2-Dimensional Cuttimg Planes George Nemhauser Symposium June 26-27 2007](https://reader034.vdocument.in/reader034/viewer/2022051819/5514092f550346ec488b4ce1/html5/thumbnails/16.jpg)
Shooting Theorem
0
The Facet first hit by the random direction v
is the Facet solving the L.P.
min vg
(g)+ (g -g) 1 (all g)
(g)+ (g') ( ') (all g, g')g g
![Page 17: Corner Polyhedra and 2-Dimensional Cuttimg Planes George Nemhauser Symposium June 26-27 2007](https://reader034.vdocument.in/reader034/viewer/2022051819/5514092f550346ec488b4ce1/html5/thumbnails/17.jpg)
Concentration of HitsEllis Johnson and Lisa Evans
![Page 18: Corner Polyhedra and 2-Dimensional Cuttimg Planes George Nemhauser Symposium June 26-27 2007](https://reader034.vdocument.in/reader034/viewer/2022051819/5514092f550346ec488b4ce1/html5/thumbnails/18.jpg)
Second Type: Data non-integer , some Variables Continuous
G is n-space, elements v are n-vectors
Cutting Plane is Non-Negative ( ) such that
For any
( ) 1.
i i gi
i ii
v
path t v v
t v
![Page 19: Corner Polyhedra and 2-Dimensional Cuttimg Planes George Nemhauser Symposium June 26-27 2007](https://reader034.vdocument.in/reader034/viewer/2022051819/5514092f550346ec488b4ce1/html5/thumbnails/19.jpg)
Cutting Planes Must Be Created
,1
,2
i ,3 ,
,
Usually only one equation is used
From the n dimensional equation
If v ; ( ) ( )
.
.
i i gi
i
i
i i i j
i n
t v v
v
v
v v v
v
![Page 20: Corner Polyhedra and 2-Dimensional Cuttimg Planes George Nemhauser Symposium June 26-27 2007](https://reader034.vdocument.in/reader034/viewer/2022051819/5514092f550346ec488b4ce1/html5/thumbnails/20.jpg)
Cutting Planes Direct Construction
• Example: Gomory Mixed Integer Cut
• Variables ti Integer
• Variables t+, t- Non-Integer
![Page 21: Corner Polyhedra and 2-Dimensional Cuttimg Planes George Nemhauser Symposium June 26-27 2007](https://reader034.vdocument.in/reader034/viewer/2022051819/5514092f550346ec488b4ce1/html5/thumbnails/21.jpg)
![Page 22: Corner Polyhedra and 2-Dimensional Cuttimg Planes George Nemhauser Symposium June 26-27 2007](https://reader034.vdocument.in/reader034/viewer/2022051819/5514092f550346ec488b4ce1/html5/thumbnails/22.jpg)
( ) 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
![Page 23: Corner Polyhedra and 2-Dimensional Cuttimg Planes George Nemhauser Symposium June 26-27 2007](https://reader034.vdocument.in/reader034/viewer/2022051819/5514092f550346ec488b4ce1/html5/thumbnails/23.jpg)
( ) Gomory Mixed Integer Cut
Continuous Variables
x
-2 -1.5 -1 -0.5 0.5 1 1.5 2
0.5
1
1.5
2
2.5
3
![Page 24: Corner Polyhedra and 2-Dimensional Cuttimg Planes George Nemhauser Symposium June 26-27 2007](https://reader034.vdocument.in/reader034/viewer/2022051819/5514092f550346ec488b4ce1/html5/thumbnails/24.jpg)
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
![Page 25: Corner Polyhedra and 2-Dimensional Cuttimg Planes George Nemhauser Symposium June 26-27 2007](https://reader034.vdocument.in/reader034/viewer/2022051819/5514092f550346ec488b4ce1/html5/thumbnails/25.jpg)
Two Integer Variables Examples: Both are Facets
0.2 0.4 0.6 0.8 1
0.2
0.4
0.6
0.8
1
![Page 26: Corner Polyhedra and 2-Dimensional Cuttimg Planes George Nemhauser Symposium June 26-27 2007](https://reader034.vdocument.in/reader034/viewer/2022051819/5514092f550346ec488b4ce1/html5/thumbnails/26.jpg)
Integer Variables Example 2
0.2 0.4 0.6 0.8 1
0.2
0.4
0.6
0.8
1
![Page 27: Corner Polyhedra and 2-Dimensional Cuttimg Planes George Nemhauser Symposium June 26-27 2007](https://reader034.vdocument.in/reader034/viewer/2022051819/5514092f550346ec488b4ce1/html5/thumbnails/27.jpg)
Gomory-Johnson Theorem
If (x) has only two slopes and satisfies
the minimality condition (x)+ (1-x)=1
then it is a facet.
![Page 28: Corner Polyhedra and 2-Dimensional Cuttimg Planes George Nemhauser Symposium June 26-27 2007](https://reader034.vdocument.in/reader034/viewer/2022051819/5514092f550346ec488b4ce1/html5/thumbnails/28.jpg)
Integer versus Continuous
• Integer Theory More Developed
• But more developed cutting planes weaker than the Gomory Mixed Integer Cut for continuous variables
![Page 29: Corner Polyhedra and 2-Dimensional Cuttimg Planes George Nemhauser Symposium June 26-27 2007](https://reader034.vdocument.in/reader034/viewer/2022051819/5514092f550346ec488b4ce1/html5/thumbnails/29.jpg)
Comparing
0.2 0.4 0.6 0.8 1
0.2
0.4
0.6
0.8
1
![Page 30: Corner Polyhedra and 2-Dimensional Cuttimg Planes George Nemhauser Symposium June 26-27 2007](https://reader034.vdocument.in/reader034/viewer/2022051819/5514092f550346ec488b4ce1/html5/thumbnails/30.jpg)
New Direction
• Reverse the present Direction
• Create continuous facets
• Turn them into facets for the integer problem
![Page 31: Corner Polyhedra and 2-Dimensional Cuttimg Planes George Nemhauser Symposium June 26-27 2007](https://reader034.vdocument.in/reader034/viewer/2022051819/5514092f550346ec488b4ce1/html5/thumbnails/31.jpg)
-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
![Page 32: Corner Polyhedra and 2-Dimensional Cuttimg Planes George Nemhauser Symposium June 26-27 2007](https://reader034.vdocument.in/reader034/viewer/2022051819/5514092f550346ec488b4ce1/html5/thumbnails/32.jpg)
-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 Minimizing
![Page 33: Corner Polyhedra and 2-Dimensional Cuttimg Planes George Nemhauser Symposium June 26-27 2007](https://reader034.vdocument.in/reader034/viewer/2022051819/5514092f550346ec488b4ce1/html5/thumbnails/33.jpg)
The Continuous Problem and A Theorem
1 1
Pure Continuous Problem: All t continuous
:The Gomory
(Mo
Mixed
d 1)
Theor Integer em Cut is the only
cutting plane that is a facet for both the pure integer and the
B Nt B b
pure continuous one dimensional problems.
![Page 34: Corner Polyhedra and 2-Dimensional Cuttimg Planes George Nemhauser Symposium June 26-27 2007](https://reader034.vdocument.in/reader034/viewer/2022051819/5514092f550346ec488b4ce1/html5/thumbnails/34.jpg)
Direction
• Move on to More Dimensions
![Page 35: Corner Polyhedra and 2-Dimensional Cuttimg Planes George Nemhauser Symposium June 26-27 2007](https://reader034.vdocument.in/reader034/viewer/2022051819/5514092f550346ec488b4ce1/html5/thumbnails/35.jpg)
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.
![Page 36: Corner Polyhedra and 2-Dimensional Cuttimg Planes George Nemhauser Symposium June 26-27 2007](https://reader034.vdocument.in/reader034/viewer/2022051819/5514092f550346ec488b4ce1/html5/thumbnails/36.jpg)
Creating 2D facets
-1.5 -1 -0.5 0.5 1 1.5 2
-1.5
-1
-0.5
0.5
1
1.5
![Page 37: Corner Polyhedra and 2-Dimensional Cuttimg Planes George Nemhauser Symposium June 26-27 2007](https://reader034.vdocument.in/reader034/viewer/2022051819/5514092f550346ec488b4ce1/html5/thumbnails/37.jpg)
The triopoly figure
0 1 2
-0.5
0
0.5
00.250.50.751
-0.5
0
0.5
![Page 38: Corner Polyhedra and 2-Dimensional Cuttimg Planes George Nemhauser Symposium June 26-27 2007](https://reader034.vdocument.in/reader034/viewer/2022051819/5514092f550346ec488b4ce1/html5/thumbnails/38.jpg)
This corresponds to
-2 -1.5 -1 -0.5 0.5 1 1.5 2
0.5
1
1.5
2
2.5
3
![Page 39: Corner Polyhedra and 2-Dimensional Cuttimg Planes George Nemhauser Symposium June 26-27 2007](https://reader034.vdocument.in/reader034/viewer/2022051819/5514092f550346ec488b4ce1/html5/thumbnails/39.jpg)
The periodic figure
-2 -1.5 -1 -0.5 0.5 1 1.5 2
0.5
1
1.5
2
2.5
3
![Page 40: Corner Polyhedra and 2-Dimensional Cuttimg Planes George Nemhauser Symposium June 26-27 2007](https://reader034.vdocument.in/reader034/viewer/2022051819/5514092f550346ec488b4ce1/html5/thumbnails/40.jpg)
The 2D Periodic figure- a facet-1
0
1
2
XXX
-1
0
1
2
YYY
00.250.50.751ZZZ
-1
0
1
2
YYY
00.250.50.751ZZZ
![Page 41: Corner Polyhedra and 2-Dimensional Cuttimg Planes George Nemhauser Symposium June 26-27 2007](https://reader034.vdocument.in/reader034/viewer/2022051819/5514092f550346ec488b4ce1/html5/thumbnails/41.jpg)
One Periodic Unit
![Page 42: Corner Polyhedra and 2-Dimensional Cuttimg Planes George Nemhauser Symposium June 26-27 2007](https://reader034.vdocument.in/reader034/viewer/2022051819/5514092f550346ec488b4ce1/html5/thumbnails/42.jpg)
Creating Another Facet
-1 1 2 3
-1.5
-1
-0.5
0.5
1
1.5
![Page 43: Corner Polyhedra and 2-Dimensional Cuttimg Planes George Nemhauser Symposium June 26-27 2007](https://reader034.vdocument.in/reader034/viewer/2022051819/5514092f550346ec488b4ce1/html5/thumbnails/43.jpg)
The Periodic Figure - Another Facet
![Page 44: Corner Polyhedra and 2-Dimensional Cuttimg Planes George Nemhauser Symposium June 26-27 2007](https://reader034.vdocument.in/reader034/viewer/2022051819/5514092f550346ec488b4ce1/html5/thumbnails/44.jpg)
More
![Page 45: Corner Polyhedra and 2-Dimensional Cuttimg Planes George Nemhauser Symposium June 26-27 2007](https://reader034.vdocument.in/reader034/viewer/2022051819/5514092f550346ec488b4ce1/html5/thumbnails/45.jpg)
These are all Facets
• For the continuous problem (all the facets)
• For the Integer Problem
• For the General problem
• Two Dimensional analog of Gomory Mixed Integer Cut
![Page 46: Corner Polyhedra and 2-Dimensional Cuttimg Planes George Nemhauser Symposium June 26-27 2007](https://reader034.vdocument.in/reader034/viewer/2022051819/5514092f550346ec488b4ce1/html5/thumbnails/46.jpg)
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
![Page 47: Corner Polyhedra and 2-Dimensional Cuttimg Planes George Nemhauser Symposium June 26-27 2007](https://reader034.vdocument.in/reader034/viewer/2022051819/5514092f550346ec488b4ce1/html5/thumbnails/47.jpg)
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
![Page 48: Corner Polyhedra and 2-Dimensional Cuttimg Planes George Nemhauser Symposium June 26-27 2007](https://reader034.vdocument.in/reader034/viewer/2022051819/5514092f550346ec488b4ce1/html5/thumbnails/48.jpg)
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
![Page 49: Corner Polyhedra and 2-Dimensional Cuttimg Planes George Nemhauser Symposium June 26-27 2007](https://reader034.vdocument.in/reader034/viewer/2022051819/5514092f550346ec488b4ce1/html5/thumbnails/49.jpg)
T-SpaceGomory Mixed Integer Cuts
1 2 3 4t1
1
2
3
4
t2
![Page 50: Corner Polyhedra and 2-Dimensional Cuttimg Planes George Nemhauser Symposium June 26-27 2007](https://reader034.vdocument.in/reader034/viewer/2022051819/5514092f550346ec488b4ce1/html5/thumbnails/50.jpg)
T- Space – some 2D Cuts Added
1 2 3 4t1
1
2
3
4
t2
![Page 51: Corner Polyhedra and 2-Dimensional Cuttimg Planes George Nemhauser Symposium June 26-27 2007](https://reader034.vdocument.in/reader034/viewer/2022051819/5514092f550346ec488b4ce1/html5/thumbnails/51.jpg)
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
![Page 52: Corner Polyhedra and 2-Dimensional Cuttimg Planes George Nemhauser Symposium June 26-27 2007](https://reader034.vdocument.in/reader034/viewer/2022051819/5514092f550346ec488b4ce1/html5/thumbnails/52.jpg)
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.
![Page 53: Corner Polyhedra and 2-Dimensional Cuttimg Planes George Nemhauser Symposium June 26-27 2007](https://reader034.vdocument.in/reader034/viewer/2022051819/5514092f550346ec488b4ce1/html5/thumbnails/53.jpg)
END
![Page 54: Corner Polyhedra and 2-Dimensional Cuttimg Planes George Nemhauser Symposium June 26-27 2007](https://reader034.vdocument.in/reader034/viewer/2022051819/5514092f550346ec488b4ce1/html5/thumbnails/54.jpg)
Backup Slides
![Page 55: Corner Polyhedra and 2-Dimensional Cuttimg Planes George Nemhauser Symposium June 26-27 2007](https://reader034.vdocument.in/reader034/viewer/2022051819/5514092f550346ec488b4ce1/html5/thumbnails/55.jpg)
One Periodic Unit
![Page 56: Corner Polyhedra and 2-Dimensional Cuttimg Planes George Nemhauser Symposium June 26-27 2007](https://reader034.vdocument.in/reader034/viewer/2022051819/5514092f550346ec488b4ce1/html5/thumbnails/56.jpg)
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
![Page 57: Corner Polyhedra and 2-Dimensional Cuttimg Planes George Nemhauser Symposium June 26-27 2007](https://reader034.vdocument.in/reader034/viewer/2022051819/5514092f550346ec488b4ce1/html5/thumbnails/57.jpg)
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
![Page 58: Corner Polyhedra and 2-Dimensional Cuttimg Planes George Nemhauser Symposium June 26-27 2007](https://reader034.vdocument.in/reader034/viewer/2022051819/5514092f550346ec488b4ce1/html5/thumbnails/58.jpg)
Shifting
![Page 59: Corner Polyhedra and 2-Dimensional Cuttimg Planes George Nemhauser Symposium June 26-27 2007](https://reader034.vdocument.in/reader034/viewer/2022051819/5514092f550346ec488b4ce1/html5/thumbnails/59.jpg)
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).