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).