Download presentation

Presentation is loading. Please wait.

Published byLillian Coughlin Modified over 4 years ago

1
Corner Polyhedra and 2-Dimensional Cuttimg Planes George Nemhauser Symposium June 26-27 2007

2
Integer Programming - Notation

3
L.P., I.P and Corner Polyhedron

4
Equations

5
L.P., I.P and Corner Polyhedron

6
Comparing Integer Programs and Corner Polyhedron General Integer Programs – Complex, no obvious structure Corner Polyhedra – Highly structured

7
Cutting Planes for Corner Polyhedra are Cutting Planes for General I.P.

8
Valid, Minimal, Facet

9
Cutting Planes

10
General Cutting Planes

11
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.

12
First Type Data and Variables Integer

13
Mod(1) B -1 N has exactly Det(B) distinct Columns v i

14
Structure Theorem

15
Typical Structured Faces

16
Shooting Theorem

17
Concentration of Hits Ellis Johnson and Lisa Evans

18
Second Type: Data non-integer, some Variables Continuous

19
Cutting Planes Must Be Created

20
Cutting Planes Direct Construction Example: Gomory Mixed Integer Cut Variables t i Integer Variables t +, t - Non-Integer

24
Integer Cuts lead to Cuts for the Continuous Variables

25
Two Integer Variables Examples: Both are Facets

26
Integer Variables Example 2

27
Gomory-Johnson Theorem

28
Integer versus Continuous Integer Theory More Developed But more developed cutting planes weaker than the Gomory Mixed Integer Cut for continuous variables

29
Comparing

30
New Direction Reverse the present Direction Create continuous facets Turn them into facets for the integer problem

31
Start With Continuous x

32
Create Integer Cut: Shifting and Minimizing

33
The Continuous Problem and A Theorem

34
Direction Move on to More Dimensions

35
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.

36
Creating 2D facets

37
The triopoly figure

38
This corresponds to

39
The periodic figure

40
The 2D Periodic figure- a facet

41
One Periodic Unit

42
Creating Another Facet

43
The Periodic Figure - Another Facet

44
More

45
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

46
x i Integer t i Continuous

47
Basis B

48
Corner Polyhedron Equations

49
T-Space Gomory Mixed Integer Cuts

50
T- Space – some 2D Cuts Added

51
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

52
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.

53
END

54
Backup Slides

55
One Periodic Unit

56
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

57
Origin of Continuous Variables Procedure

58
Shifting

59
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).

Similar presentations

OK

Computer Algorithms Integer Programming ECE 665 Professor Maciej Ciesielski By DFG.

Computer Algorithms Integer Programming ECE 665 Professor Maciej Ciesielski By DFG.

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on conservation of momentum ppt Ppt on business etiquette in india Ppt on event handling in javascript what is a number Raster scan display ppt on tv Ppt on basic computer organisation Ppt on tropical evergreen forest Ppt on 3 idiots movie songs Ppt on event driven programming disadvantages Ppt on sobel edge detection Ppt on networking related topics such