Download presentation

Presentation is loading. Please wait.

Published byVanessa Reeves Modified over 2 years ago

1
Forty Years of Corner Polyhedra

2
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

3
Corner Polyhedra Origins Stock Cutting Computing Lots of Knapsacks Periodicity observed Gomory-Gilmore 1966 "The Theory and Computation of Knapsack Functions

4
Equations

5

6
L.P., I.P and Corner Polyhedron

7
Another View - T-Space

8
Cutting Planes for Corner Polyhedra are Cutting Planes for General I.P.

9
Valid, Minimal, Facet

10
T-Space View

11
Cutting Planes for Corner Polyhedra

12
Structure Theorem- 1969

13
Typical Structured Faces computed using Balinski program

14
Size Problem : Shooting Geometry

15
Size Problem -Shooting Theorem

16
Concentration of Hits Ellis Johnson and Lisa Evans

17
Much More to be Learned

18
Comparing Integer 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

19
Cutting Planes for Type Two Example: Gomory Mixed Integer Cut Variables t i Integer Variables t +, t - Non-Integer Valid subadditive function

20

21
Typical Structured Faces

22
Interpolating to get cutting plane function on the real line

23
Interpolating

24

25
Gomory-Johnson Theorem

26
Integer Variables Example 2

27
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

28

29
Comparing

30

31

32
Integer Cuts lead to Cuts for the Continuous Variables

33
Gomory Mixed Integer Cut Continuous Variables

34
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

35
Start With Continuous x

36
Create Integer Cut: Shifting and Intersecting

37
Shifting and Intersecting

38
One Dimension Continuous Problem

39
Direction Move on to More Dimensions

40
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.

41
Creating 2D facets

42
The triopoly figure

43
This corresponds to

44
The periodic figure

45
Two Dimensional Periodic Figure

46
One Periodic Unit

47
Creating Another Facet

48
The Periodic Figure - Another Facet

49
More

50
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.

51
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

52
x i Integer t i Continuous

53
Basis B

54
Corner Polyhedron Equations

55
T-Space Gomory Mixed Integer Cuts

56
T- Space – some 2D Cuts Added

57
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

58
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.

59
END

60
Backup Slides

61
Thoughts About Integer Programming University of Montreal, January 26, th Birthday Celebration of the Department of Computer Science and Operations Research

62
Corner Polyhedra and 2-Dimensional Cuttimg Planes George Nemhauser Symposium June

63
Mod(1) B -1 N has exactly Det(B) distinct Columns v i

64
One Periodic Unit

65
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=f 0

66
Origin of Continuous Variables Procedure

67
Shifting

68
References Some Polyhedra Related to Combinatorial Problems, Journal of Linear Algebra and Its Applications, Vol. 2, No. 4, October 1969, pp Some Continuous Functions Related to Corner Polyhedra, Part I with Ellis L. Johnson, Mathematical Programming, Vol. 3, No. 1, North-Holland, August, 1972, pp Some Continuous Functions Related to Corner Polyhedra, Part II with Ellis L. Johnson, Mathematical Programming, Vol. 3, No. 3, North-Holland, December 1972, pp T-space and Cutting Planes Paper, with Ellis L. Johnson, Mathematical Programming, Ser. B 96: Springer-Verlag, pp (2003).

Similar presentations

© 2016 SlidePlayer.com Inc.

All rights reserved.

Ads by Google