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).