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

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

21. Typical Structured Faces

22. Interpolating to get cutting plane function on the real line

23. Interpolating

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

29. Comparing

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, 2007 40 th Birthday Celebration of the Department of Computer Science and Operations Research

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

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.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).