1 / 55

Antoine Deza, McMaster Tamás Terlaky, Lehigh Yuriy Zinchenko, Calgary Feng Xie, McMaster

Antoine Deza, McMaster Tamás Terlaky, Lehigh Yuriy Zinchenko, Calgary Feng Xie, McMaster. Hyperplane arrangements with large average diameter. P 2. P 1. P 4. P 6. P 5. P 3. Polytopes & Arrangements : Diameter & Curvature. Motivations and algorithmic issues.

aelwen
Download Presentation

Antoine Deza, McMaster Tamás Terlaky, Lehigh Yuriy Zinchenko, Calgary Feng Xie, McMaster

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Antoine Deza, McMaster Tamás Terlaky, Lehigh Yuriy Zinchenko, Calgary Feng Xie, McMaster Hyperplane arrangements with large average diameter P2 P1 P4 P6 P5 P3

  2. Polytopes& Arrangements : Diameter & Curvature Motivations and algorithmic issues • 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 (path following) interior point methods

  3. Polytopes& Arrangements : Diameter & Curvature n = 5 : inequalities d = 2 : dimension PolytopePdefined byn inequalities in dimension d • polytope : bounded polyhedron

  4. Polytopes& Arrangements : Diameter & Curvature n = 5 : inequalities d = 2 : dimension PolytopePdefined byn inequalities in dimension d

  5. Polytopes& Arrangements : Diameter & Curvature n = 5 : inequalities d = 2 : dimension P PolytopePdefined byn inequalities in dimension d

  6. Polytopes& Arrangements :Diameter& Curvature n = 5 : inequalities d = 2 : dimension (P) = 2 : diameter vertex c1 P vertex c2 Diameter(P): smallest number such that any two vertices (c1,c2) can be connected by a path with at most(P) edges

  7. Polytopes& Arrangements :Diameter& Curvature n = 5 : inequalities d = 2 : dimension (P) = 2 : diameter P Diameter (P): smallest number such that any two vertices can be connected by a path with at most(P) edges Hirsch Conjecture(1957): (P) ≤ n - d

  8. c Polytopes& Arrangements : Diameter &Curvature n = 5 : inequalities d = 2 : dimension P c(P): total curvatureof the primal central path of max{cTx : xP}

  9. c Polytopes& Arrangements : Diameter &Curvature n = 5 : inequalities d = 2 : dimension P c(P): total curvatureof the primal central path of max{cTx : xP} • c(P): redundant inequalities count

  10. Polytopes& Arrangements : Diameter &Curvature c n = 5 : inequalities d = 2 : dimension P c(P): total curvature of the primal central path of max{cTx : xP} (P): largest total curvaturec(P) over of all possiblec

  11. Polytopes& Arrangements : Diameter &Curvature c n = 5 : inequalities d = 2 : dimension P c(P): total curvature of the primal central path of max{cTx : xP} (P): largest total curvature c(P) over of all possiblec Continuous analogue of Hirsch Conjecture: (P) = O(n)

  12. Polytopes& Arrangements : Diameter &Curvature c n = 5 : inequalities d = 2 : dimension P c(P): total curvature of the primal central path of max{cTx : xP} (P): largest total curvature c(P) over of all possiblec Continuous analogue of Hirsch Conjecture: (P) = O(n) • Dedieu-Shub hypothesis (2005): (P) = O(d)

  13. Polytopes& Arrangements : Diameter &Curvature c n = 5 : inequalities d = 2 : dimension P c(P): total curvature of the primal central path of max{cTx : xP} (P): largest total curvature c(P) over of all possiblec Continuous analogue of Hirsch Conjecture: (P) = O(n) • Dedieu-Shub hypothesis (2005): (P) = O(d) • D.-T.-Z. (2008) polytope C such that: (C) ≥ (1.5)d

  14. c Polytopes& Arrangements : Diameter &Curvature Central path following interior point methods • start from the analytic center • follow the central path • converge to an optimal solution • polynomial time algorithms for linear optimization • : number of iterations • n: number of inequalities • L : input-data bit-length analytic center central path optimal solution •  : central path parameter • xP :Ax ≥ b

  15. total curvature Polytopes& Arrangements : Diameter &Curvature total curvature C2curve :[a, b] Rd parameterized by its arc lengtht(note: ║(t)║ = 1) curvature at t: y = sin (x)

  16. total curvature Polytopes& Arrangements : Diameter &Curvature total curvature C2curve :[a, b] Rn parameterized by its arc lengtht(note: ║(t)║ = 1) curvature at t: total curvature: y = sin (x)

  17. Polytopes & Arrangements : Diameter & Curvature n = 5 : hyperplanes d = 2 : dimension ArrangementAdefined byn hyperplanes in dimension d

  18. Polytopes & Arrangements : Diameter & Curvature n = 5 : hyperplanes d = 2 : dimension Simple arrangement: n dand anydhyperplanes intersect at a unique distinct point

  19. Polytopes & Arrangements : Diameter & Curvature P2 n = 5 : hyperplanes d = 2 : dimension I = 6: bounded cells P1 P4 P6 P5 P3 For a simple arrangement, thenumber of bounded cellsI=

  20. c Polytopes &Arrangements: Diameter & Curvature P2 n = 5 : hyperplanes d = 2 : dimension I = 6: bounded cells P1 P4 P6 P5 P3 c(A) : average value of c(Pi) over the bounded cells Piof A: c(A) = with I= • c(Pi): redundant inequalities count

  21. c Polytopes &Arrangements: Diameter &Curvature P2 n = 5 : hyperplanes d = 2 : dimension I = 6: bounded cells P1 P4 P6 P5 P3 c(A) : average value of c(Pi) over the bounded cells Pi of A: (A) : largest value of c(A) over all possiblec

  22. c Polytopes &Arrangements: Diameter &Curvature P2 n = 5 : hyperplanes d = 2 : dimension I = 6: bounded cells P1 P4 P6 P5 P3 c(A) : average value of c(Pi) over the bounded cells Pi of A: (A) : largest value of c(A) over all possiblec Dedieu-Malajovich-Shub(2005): (A) ≤ 2 d

  23. Polytopes &Arrangements : Diameter& Curvature P2 n = 5 : hyperplanes d = 2 : dimension I = 6: bounded cells P1 P4 P6 P5 P3 (A) : average diameter of a bounded cell of A: • A: simple arrangement

  24. Polytopes &Arrangements : Diameter& Curvature P2 n = 5 : hyperplanes d = 2 : dimension I = 6: bounded cells P1 P4 P6 P5 P3 (A) : average diameter of a bounded cell of A: (A) = with I = • (A): average diameter ≠ diameter of A ex: (A)= 1.333…

  25. Polytopes &Arrangements : Diameter& Curvature P2 n = 5 : hyperplanes d = 2 : dimension I = 6: bounded cells P1 P4 P6 P5 P3 (A) : average diameter of a bounded cell of A: (A) = with I = • (Pi): only active inequalities count

  26. Polytopes & Arrangements : Diameter & Curvature P2 n = 5 : hyperplanes d = 2 : dimension I = 6: bounded cells P1 P4 P6 P5 P3 (A) : average diameter of a bounded cell of A: Conjecture [D.-T.-Z.]: (A) ≤ d (discrete analogue of Dedieu-Malajovich-Shub result)

  27. Polytopes & Arrangements : Diameter & Curvature (P) ≤ n - d(Hirsch conjecture 1957) (P) ≤ 2 n(conjecture D.-T.-Z. 2008) (A) ≤ d(conjecture D.-T.-Z. 2008) (A) ≤ 2 d(Dedieu-Malajovich-Shub 2005)

  28. Polytopes & Arrangements : Diameter & Curvature Links, low dimensions and substantiation • for unbounded polyhedra • Hirsch conjecture is not valid • central path may not be defined

  29. Polytopes & Arrangements : Diameter & Curvature Links, low dimensions and substantiation • for unbounded polyhedra • Hirsch conjecture is not valid • central path may not be defined • Hirsch conjecture (P) ≤ n - dimplies that (A) ≤ d • Hirsch conjecture holds for d = 2, (A) ≤ 2 • Hirsch conjecture holds for d = 3, (A) ≤ 3 • Barnette (1974) and Kalai-Kleitman (1992) bounds imply • (A) ≤ • (A) ≤

  30. Polytopes & Arrangements : Diameter & Curvature Links, low dimensions and substantiation n = 6 d = 2 • bounded cells ofA*: ( n – 2) (n – 1) / 2 4-gons and n - 2 triangles

  31. Polytopes & Arrangements : Diameter & Curvature Links, low dimensions and substantiation dimension2 (A) ≥ 2 -

  32. Polytopes & Arrangements : Diameter & Curvature Links, low dimensions and substantiation dimension2 (A) = 2 - maximize(diameter) amounts to minimize( #external edge + #odd-gons)

  33. Polytopes & Arrangements : Diameter & Curvature Links, low dimensions and substantiation dimension3 (A) ≤ 3 + and A* satisfies (A*) = 3 - A*

  34. Polytopes & Arrangements : Diameter & Curvature Links, low dimensions and substantiation dimensiond A* satisfies (A*) ≥ d A* cyclic arrangement • A* mainly consists of cubical cells

  35. Polytopes & Arrangements : Diameter & Curvature Links, low dimensions and substantiation dimensiond A* satisfies (A*) ≥ n • Haimovich’s probabilistic analysis of shadow-vertex simplex method - Borgwardt (1987) • Ziegler (1992) combinatorics of (pseudo) hyperplane addition to A* (Bruhat order) • Forge – Ramírez Alfonsín (2001) counting k-face cells of A*

  36. Polytopes & Arrangements : Diameter & Curvature Links, low dimensions and substantiation dimension2(A) = dimension3 (A) asympotically equal to 3 dimensiond d ≤ (A) ≤ d ? [D.-Xie 2008] • Haimovich’s probabilistic analysis of shadow-vertex simplex method - Borgwardt (1987) • Ziegler (1992) combinatorics of (pseudo) hyperplane addition to A* (Bruhat order) • Forge – Ramírez Alfonsín (2001) counting k-face cells of A*

  37. Polytopes & Arrangements : Diameter & Curvature Is it the right setting? (forget optimization and the vector c) bounded / unbounded cells (sphere arrangements)? simple / degenerate arrangements? pseudo-hyperplanes / underlying oriented matroid ?

  38. Polytopes & Arrangements : Diameter & Curvature Links, low dimensions and substantiation • Hirsch conjecture is tight • for n > d ≥ 7 P such that (P) ≥ n - dHolt-Klee (1998) • Fritzsche-Holt (1999) • Holt (2004)

  39. Polytopes & Arrangements : Diameter & Curvature Links, low dimensions and substantiation • Hirsch conjecture is tight • for n > d ≥ 7 P such that (P) ≥ n - dHolt-Klee (1998) • Fritzsche-Holt (1999) • Holt (2004) • Is the continuous analogue true?

  40. Polytopes & Arrangements : Diameter & Curvature Links, low dimensions and substantiation • Hirsch conjecture is tight • for n > d ≥ 7 P such that (P) ≥ n - dHolt-Klee (1998) • Fritzsche-Holt (1999) • Holt (2004) • Is the continuous analogue true? yes

  41. Polytopes & Arrangements : Diameter & Curvature Links, low dimensions and substantiation • Hirsch conjecture is tight • for n > d ≥ 7 P such that (P) ≥ n - dHolt-Klee (1998) • Fritzsche-Holt (1999) • Holt (2004) • continuous analogue of tightness: • for n ≥ d ≥ 2 D.-T.-Z. (2008)

  42. Polytopes & Arrangements : Diameter & Curvature • slope: careful and geometric decrease

  43. Polytopes & Arrangements : Diameter & Curvature • slope: careful and geometric decrease

  44. Polytopes & Arrangements : Diameter & Curvature Links, low dimensions and substantiation • Hirsch conjecture is tight • for n > d ≥ 7 P such that (P) ≥ n - d • continuous analogue oftightness: • for n ≥ d ≥ 2 P such that (P) = Ω(n) D.-T.-Z. (2008)

  45. Polytopes & Arrangements : Diameter & Curvature Links, low dimensions and substantiation • Hirsch conjecture is tight • for n > d ≥ 7 P such that (P) ≥ n - d • continuous analogue of tightness: • for n ≥ d ≥ 2 P such that (P) = Ω(n) D.-T.-Z. (2008) • Hirsch special casen = 2d (for all d) is equivalentto the general case • (d-step conjecture) Klee-Walkup (1967)

  46. Polytopes & Arrangements : Diameter & Curvature Links, low dimensions and substantiation • Hirsch conjecture is tight • for n > d ≥ 7 P such that (P) ≥ n - d • continuous analogue of tightness: • for n ≥ d ≥ 2 P such that (P) = Ω(n) D.-T.-Z. (2008) • Hirsch special casen = 2d (for all d) is equivalentto the general case • (d-step conjecture) Klee-Walkup (1967) • Is the continuous analogue true?

  47. Polytopes & Arrangements : Diameter & Curvature Links, low dimensions and substantiation • Hirsch conjecture is tight • for n > d ≥ 7 P such that (P) ≥ n - d • continuous analogue of tightness: • for n ≥ d ≥ 2 P such that (P) = Ω(n) D.-T.-Z. (2008) • Hirsch special casen = 2d (for all d) is equivalentto the general case • (d-step conjecture) Klee-Walkup (1967) • Is the continuous analogue true? yes

  48. Polytopes & Arrangements : Diameter & Curvature Links, low dimensions and substantiation • Hirsch conjecture is tight • for n > d ≥ 7 P such that (P) ≥ n - d • continuous analogue of tightness: • for n ≥ d ≥ 2 P such that (P) = Ω(n) D.-T.-Z. (2008) • Hirsch special casen = 2d (for all d) is equivalentto the general case • (d-step conjecture) Klee-Walkup (1967) • continuous analogue of d-step equivalence: • (P) = O(n)(P) = O(d) for n = 2dD.-T.-Z. (2008)

  49. Polytopes & Arrangements : Diameter & Curvature holds for n= 2 andn= 3 (P) ≤ n - dHirsch conjecture (1957) (P) < n – d Holt-Klee (1998) (P) ≤ 2 nconjecture D.-T.-Z. (2008) (P)=Ω(n) D.-T.-Z (2008) holds for n= 2 andn= 3 redundant inequalities counts for(P) (A) ≤ dconjecture D.-T.-Z. (2008) (A) ≤ 2 d Dedieu-Malajovich-Shub (2005) (P) ≤ n – d (P) ≤ d for n = 2d Klee-Walkup (1967) (P) = O(n)(P) = O(d) for n = 2d D.-T.-Z. (2008)

  50. Polytopes & Arrangements : Diameter & Curvature holds for n= 2 andn= 3 (P) ≤ n - dHirsch conjecture (1957) (P) < n – d Holt-Klee (1998) (P) ≤ 2 nconjecture D.-T.-Z. (2008) (P)=Ω(n) D.-T.-Z (2008) holds for n= 2 andn= 3 redundant inequalities counts for(P) (A) ≤ dconjecture D.-T.-Z. (2008) (A) ≤ 2 d Dedieu-Malajovich-Shub (2005) (P) ≤ n – d (P) ≤ d for n = 2d Klee-Walkup (1967) (P) = O(n)(P) = O(d) for n = 2d D.-T.-Z. (2008)

More Related