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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.

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

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  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)

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