Download presentation

Presentation is loading. Please wait.

Published byLucas Pereira Modified over 3 years ago

1
Thoughts About Integer Programming University of Montreal, January 26, 2007

2
Integer Programming Max c x Ax=b Some or All x Integer

3
Why Does Integer Prrogramming Matter? Navy Task Force Patterns in Stock Cutting Economies of Scale in Industries Trade Theory – Conflicting National Interests

4
CUTS WASTE Roll of Paper at Mill

5
The Effect of the Number of Industries (8)

6
The Effect of the Number of Industries (3)

7
The Effect of the Number of Industries (2)

8
How Do You Solve I.Ps? Branch and Bound, Cutting Planes

9
L.P.,I.P.and Corner Polyhedron

10
I.P. and Corner Polyhedron Integer Programs – Complex, no obvious structure Corner Polyhedra – Highly Structured We use Corner Polyhedra to generate cutting planes

11
Equations

12
T-Space

13
Corner Polyhedra and Groups

14
Structure of Corner Polyhedra I

15
Structure of Corner Polyhedra II

16
Shooting Theorem:

17
Concentration of Hits Ellis Johnson and Lisa Evans

18
Cutting Planes From Corner Polyhedra

20
Why Does this Work?

21
Equations 2

22
Why π(x) Produces the Equality It is subadditive: π(x) + π(y) π(x+y) It has π(x) =1 at the goal point x=f 0

23
Cutting Planes are Plentiful Hierarchy: Valid, Minimal, Facet

24
Hierarchy

25
Example: Two Facets

26
Low is Good - High is Bad

27
Example 3

28
Gomory-Johnson Theorem

29
3-Slope Example

30
Continuous Variables t

32
Origin of Continuous Variables Procedure

33
Integer versus Continuous Integer Variables Case More Developed But all of the more developed cutting planes are weaker than the Gomory Mixed Integer Cut with respect to continuous variables

34
Comparing

35
The Continuous Problem and A Theorem

36
Cuts Provide Two Different Functions on the Real Line

37
Start with Continuous Case

38
Direction Create continuous facets Turn them into facets for the integer problem

39
Helpful 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.

40
Creating 2D facets

41
The triopoly figure

42
This corresponds to

43
The related periodic figure

44
This Corresponds To

45
Results for a Very Small Problem

46
Gomory Mixed Integer Cuts

47
2D Cuts Added

48
Summary Corner Polyhedra are very structured There is much to learn about them It seems likely that that structure can can be exploited to produce better computations

49
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.

50
END

51
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

Presentation is loading. Please wait....

OK

Learning Objectives for Section 5.3

Learning Objectives for Section 5.3

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on acid base and salt Ppt on indian politics quoted Ppt on bluetooth communication code Ppt on railway track laying Ppt on amplitude shift keying modem Ppt on operating system memory management Ppt on education in india during british rule Converter pdf to ppt online Ppt on gunn diode oscillators Share ppt online