1. Polytopes & Arrangements: Diameter & Curvature
Yuriy Zinchenko, Calgary
Antoine Deza, McMaster
Tamás Terlaky, Lehigh

2. Outline
Definitions
Connections
Present work

3. Diameter & Curvature

4. Motivation Diameter (of a polytope) :
lower bound for the number of iterations
for the simplex method (pivoting methods)
Curvature (of the central path associated to a polytope) :
large curvature indicates large number of iterations
for (central path following) interior point methods

5. We are not alone… To mention just a few
Megiddo-Shub simplex of high-curvature
Vavasis-Ye O(n2) bound on the number of relatively straight segments in the path
recently announced work by DeLoera, Vinzant, Strumfels on central curve (aslo, see the poster)

6. Definitions

7. The (discrete) alphabet

8. The (discrete) alphabet

9. The (discrete) alphabet

10. The (discrete) alphabet

11. The (discrete) alphabet

12. The (discrete) alphabet

13. The (discrete) alphabet

14. The (discrete) alphabet

15. The (discrete) alphabet

16. The (discrete) alphabet

17. The (continuous) alphabet

18. The (continuous) alphabet

19. The (continuous) alphabet

20. An interlude on curvature

21. An interlude on curvature

22. An interlude on curvature

23. An interlude on curvature

24. An interlude on curvature. r

25. An interlude on curvature

26. An interlude on curvature

27. An interlude on curvature
What does it mean? .
r = 1
r = 1 g

28. An interlude on curvature
What does it mean? . r g

29. An interlude on curvature

30. An interlude on curvature

31. An interlude on curvature

32. The (continuous) alphabet

33. The (continuous) alphabet

34. The (continuous) alphabet

35. The (continuous) alphabet

36. The (continuous) alphabet

37. The (continuous) alphabet

38. Connections

39. Diameter & Curvature

40. Diameter & Curvature

41. Diameter & Curvature

42. Diameter & Curvature

43. Diameter & Curvature

44. Diameter & Curvature

45. Diameter & Curvature

46. Substantiation

47. Substantiation

48. Substantiation

49. Substantiation

50. Substantiation

51. Substantiation

52. Substantiation

53. More about central path

54. More about central path

55. More about central path

56. More about central path
N:S
S:N

57. Substantiation

58. Substantiation

59. Substantiation

60. Substantiation

61. Substantiation

62. Substantiation

63. Substantiation

64. Substantiation

65. Substantiation

66. Substantiation

67. Substantiation c

68. Substantiation c A4

69. Substantiation c A4

70. Substantiation c

71. Substantiation ~ c c c

72. Substantiation  c

73. Present work

74. Diameter & Curvature

75. Where do we go from here? In a sense, P – to – P is almost one-to-one
does the worst edge-path say anything about P ?
P “has no inflection points”
Can we answer d = 2,3 ?
Anything beyond that ?
note, d-step may be used with any O( d k (n-d) m )
Suspect true bound to be d (n-d) /2 for diameter & d (n-d) /2 for curvature

76. Motivation Thank you! Diameter (of a polytope) :
lower bound for the number of iterations
for the simplex method (pivoting methods)
Curvature (of the central path associated to a polytope) :
large curvature indicates large number of iterations
for (central path following) interior point methods
Thank you!