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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. - PowerPoint PPT Presentation
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Antoine Deza, McMasterTamás Terlaky, Lehigh
Yuriy Zinchenko, CalgaryFeng Xie, McMaster
Hyperplane arrangements with large average diameter
P1
P2
P3
P4
P6 P5
Polytopes & Arrangements : Diameter & Curvature
Motivations and algorithmic issues
Diameter (of a polytope) :
lower bound for the number of iterationsfor the simplex method (pivoting methods)
Curvature (of the central path associated to a polytope) :
large curvature indicates large number of iterationsfor (path following) interior point methods
Polytopes & Arrangements : Diameter & Curvature
Polytope P defined by n inequalities in dimension d
n = 5 : inequalities d = 2 : dimension
polytope : bounded polyhedron
Polytopes & Arrangements : Diameter & Curvature
Polytope P defined by n inequalities in dimension d
n = 5 : inequalities d = 2 : dimension
Polytopes & Arrangements : Diameter & Curvature
P
Polytope P defined by n inequalities in dimension d
n = 5 : inequalities d = 2 : dimension
Polytopes & Arrangements : Diameter & Curvature
vertex c1
vertex c2
Diameter (P): smallest number such that any two vertices (c1,c2) can be connected by a path with at most (P) edges
P
n = 5 : inequalities d = 2 : dimension(P) = 2 : diameter
Polytopes & Arrangements : Diameter & Curvature
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
P
n = 5 : inequalities d = 2 : dimension(P) = 2 : diameter
Polytopes & Arrangements : Diameter & Curvature
c(P): total curvature of the primal central path of maxcTx : xP
c
P
n = 5 : inequalities d = 2 : dimension
Polytopes & Arrangements : Diameter & Curvature
c(P): total curvature of the primal central path of maxcTx : xP
c
P
n = 5 : inequalities d = 2 : dimension
c(P): redundant inequalities count
Polytopes & Arrangements : Diameter & Curvature
c(P): total curvature of the primal central path of maxcTx : xP
(P): largest total curvature c(P) over of all possible c
c
P
n = 5 : inequalities d = 2 : dimension
Polytopes & Arrangements : Diameter & Curvature
c(P): total curvature of the primal central path of maxcTx : xP
(P): largest total curvature c(P) over of all possible c
Continuous analogue of Hirsch Conjecture: (P) = O(n)
c
P
n = 5 : inequalities d = 2 : dimension
Polytopes & Arrangements : Diameter & Curvature
c(P): total curvature of the primal central path of maxcTx : xP
(P): largest total curvature c(P) over of all possible c
Continuous analogue of Hirsch Conjecture: (P) = O(n)
c
P
n = 5 : inequalities d = 2 : dimension
Dedieu-Shub hypothesis (2005): (P) = O(d)
Polytopes & Arrangements : Diameter & Curvature
c(P): total curvature of the primal central path of maxcTx : xP
(P): largest total curvature c(P) over of all possible c
Continuous analogue of Hirsch Conjecture: (P) = O(n)
c
P
n = 5 : inequalities d = 2 : dimension
Dedieu-Shub hypothesis (2005): (P) = O(d) D.-T.-Z. (2008) polytope C such that: (C) ≥ (1.5)d
• 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
( )O nL
analytic center
centralpathoptimal
solution
Polytopes & Arrangements : Diameter & CurvatureCentral path following interior point methods
: central path parameter xP : Ax ≥ b
max ln( )ii
x Ax b cc
total curvature
C2 curve : [a, b] Rd parameterized by its arc length t (note: (t) = 1)
curvature at t : )()( 2
2
tt
t
y = sin (x)
total curvature
Polytopes & Arrangements : Diameter & Curvature
C2 curve : [a, b] Rn parameterized by its arc length t (note: (t) = 1)
curvature at t :
total curvature :
total curvature
)()( 2
2
tt
t
b
a
dttK )(
y = sin (x)
total curvature
Polytopes & Arrangements : Diameter & Curvature
Polytopes & Arrangements : Diameter & Curvature
Arrangement A defined by n hyperplanes in dimension d
n = 5 : hyperplanes d = 2 : dimension
Polytopes & Arrangements : Diameter & Curvature
Simple arrangement: n d and any d hyperplanes intersect at a unique distinct point
n = 5 : hyperplanes d = 2 : dimension
Polytopes & Arrangements : Diameter & Curvature
For a simple arrangement, the number of bounded cells I =1
nd
n = 5 : hyperplanes d = 2 : dimension I = 6 : bounded cells
P1
P2
P3
P4
P6 P5
Polytopes & Arrangements : Diameter & Curvature
c(A) : average value of c(Pi) over the bounded cells Pi of A:
c(A) = with I =1
( )i
ii
P
I
c
I1
nd
c
P1
P3
P2
P6
n = 5 : hyperplanes d = 2 : dimension I = 6 : bounded cells
c(Pi): redundant inequalities count
P5
P4
Polytopes & Arrangements : Diameter & Curvature
c(A) : average value of c(Pi) over the bounded cells Pi of A:
(A) : largest value of c(A) over all possible c
P1
P3
P2
P6
c
n = 5 : hyperplanes d = 2 : dimension I = 6 : bounded cells
P5
P4
Polytopes & Arrangements : Diameter & Curvature
c(A) : average value of c(Pi) over the bounded cells Pi of A:
(A) : largest value of c(A) over all possible c
Dedieu-Malajovich-Shub (2005): (A) ≤ 2 d
P1
P3
P2
P6
c
n = 5 : hyperplanes d = 2 : dimension I = 6 : bounded cells
P5
P4
Polytopes & Arrangements : Diameter & Curvature
(A) : average diameter of a bounded cell of A:
P1
P3
P2
P6
P4
P5
n = 5 : hyperplanes d = 2 : dimension I = 6 : bounded cells
A : simple arrangement
Polytopes & Arrangements : Diameter & Curvature
(A) : average diameter of a bounded cell of A:
(A) = with I =1
( )i
ii
P
I
I
P1
P3
P2
P6
1
nd
P4
P5
n = 5 : hyperplanes d = 2 : dimension I = 6 : bounded cells
(A): average diameter ≠ diameter of A ex: (A)= 1.333…
Polytopes & Arrangements : Diameter & Curvature
(A) : average diameter of a bounded cell of A:
(A) = with I =1
( )i
ii
P
I
I
P1
P3
P2
P6
1
nd
P4
P5
n = 5 : hyperplanes d = 2 : dimension I = 6 : bounded cells
(Pi): only active inequalities count
Polytopes & Arrangements : Diameter & Curvature
P1
P3
P2
P6
P4
P5
n = 5 : hyperplanes d = 2 : dimension I = 6 : bounded cells
(A) : average diameter of a bounded cell of A:
Conjecture [D.-T.-Z.]: (A) ≤ d (discrete analogue of Dedieu-Malajovich-Shub result)
Polytopes & Arrangements : Diameter & Curvature
(P) ≤ n - d (Hirsch conjecture 1957)
(A) ≤ d (conjecture D.-T.-Z. 2008) (A) ≤ 2 d (Dedieu-Malajovich-Shub 2005)
(P) ≤ 2 n (conjecture D.-T.-Z. 2008)
Polytopes & Arrangements : Diameter & CurvatureLinks, low dimensions and substantiation
• for unbounded polyhedra• Hirsch conjecture is not valid• central path may not be defined
Polytopes & Arrangements : Diameter & CurvatureLinks, low dimensions and substantiation
• for unbounded polyhedra• Hirsch conjecture is not valid• central path may not be defined
• Hirsch conjecture (P) ≤ n - d implies 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) ≤
11
nn
1nn
11
nn
log2
4 21
1
dn
nd ndd
n
21 5 21 2 3
dndn d
• bounded cells of A* : ( n – 2) (n – 1) / 2 4-gons and n - 2 triangles
n = 6 d = 2
Polytopes & Arrangements : Diameter & CurvatureLinks, low dimensions and substantiation
Polytopes & Arrangements : Diameter & CurvatureLinks, low dimensions and substantiation
dimension 2
(A) ≥ 2 -
2 / 2( 1)( 2)
n
n n
Polytopes & Arrangements : Diameter & CurvatureLinks, low dimensions and substantiation
dimension 2
(A) = 2 -
maximize(diameter) amounts tominimize( #external edge + #odd-gons)
2 / 2( 1)( 2)
n
n n
Polytopes & Arrangements : Diameter & CurvatureLinks, low dimensions and substantiation
dimension 3
(A) ≤ 3 +
and A* satisfies
(A*) = 3 -
24(2 16 21)3( 1)( 2)( 3)
n n
n n n
6 / 2 261 ( 1)( 2)( 3)
nn n n n
A*
Polytopes & Arrangements : Diameter & CurvatureLinks, low dimensions and substantiation
dimension d
A* satisfies
(A*) ≥ d
A* cyclic arrangement
- -1
/n d nd d
A* mainly consists of cubical cells
Polytopes & Arrangements : Diameter & CurvatureLinks, low dimensions and substantiation
dimension d
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*
- -1
/n d nd d
Polytopes & Arrangements : Diameter & CurvatureLinks, low dimensions and substantiation
dimension 2 (A) =
dimension 3 (A) asympotically equal to 3 dimension d d ≤ (A) ≤ d ?
[D.-Xie 2008]
- -1
/n d nd d
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*
2 / 2( 1)( 2)
n
n n
Polytopes & Arrangements : Diameter & Curvature
bounded / unbounded cells (sphere arrangements)?
simple / degenerate arrangements? pseudo-hyperplanes / underlying oriented matroid ?
Is it the right setting?(forget optimization and the vector c)
Polytopes & Arrangements : Diameter & Curvature
• Hirsch conjecture is tight • for n > d ≥ 7 P such that (P) ≥ n - d Holt-Klee (1998)
Fritzsche-Holt (1999) Holt (2004)
Links, low dimensions and substantiation
Polytopes & Arrangements : Diameter & Curvature
Links, low dimensions and substantiation
• Hirsch conjecture is tight • for n > d ≥ 7 P such that (P) ≥ n - d Holt-Klee (1998)
Fritzsche-Holt (1999) Holt (2004)
Is the continuous analogue true?
Polytopes & Arrangements : Diameter & Curvature
Links, low dimensions and substantiation
• Hirsch conjecture is tight • for n > d ≥ 7 P such that (P) ≥ n - d Holt-Klee (1998)
Fritzsche-Holt (1999) Holt (2004)
Is the continuous analogue true? yes
Polytopes & Arrangements : Diameter & Curvature
Links, low dimensions and substantiation
( ) such that lim inf
c
nP
nP
• Hirsch conjecture is tight • for n > d ≥ 7 P such that (P) ≥ n - d Holt-Klee (1998)
Fritzsche-Holt (1999) Holt (2004)
•continuous analogue of tightness:• for n ≥ d ≥ 2 D.-T.-Z. (2008)
Polytopes & Arrangements : Diameter & Curvature
slope: careful and geometric decrease
( ) such that liminf
c
nP
nP
Polytopes & Arrangements : Diameter & Curvature
slope: careful and geometric decrease
( ) such that liminf
c
nP
nP
Polytopes & Arrangements : Diameter & Curvature
• 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)
Links, low dimensions and substantiation
Polytopes & Arrangements : Diameter & Curvature
• 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 case n = 2d (for all d) is equivalent to the general case (d-step conjecture) Klee-Walkup (1967)
Links, low dimensions and substantiation
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 case n = 2d (for all d) is equivalent to the general case (d-step conjecture) Klee-Walkup (1967)
Is the continuous analogue true?
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 case n = 2d (for all d) is equivalent to the general case (d-step conjecture) Klee-Walkup (1967)
Is the continuous analogue true? yes
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 case n = 2d (for all d) is equivalent to the general case (d-step conjecture) Klee-Walkup (1967)
•continuous analogue of d-step equivalence:• (P) = O(n) (P) = O(d) for n = 2d D.-T.-Z. (2008)
Polytopes & Arrangements : Diameter & Curvature
(P) ≤ n - d Hirsch conjecture (1957) (P) < n – d Holt-Klee (1998)
(A) ≤ d conjecture D.-T.-Z. (2008) (A) ≤ 2 d Dedieu-Malajovich-Shub (2005)
(P) ≤ 2 n conjecture D.-T.-Z. (2008) (P)=Ω(n) D.-T.-Z (2008)
holds for n = 2 and n = 3
holds for n = 2 and n = 3
(P) ≤ n – d (P) ≤ d for n = 2d Klee-Walkup (1967)(P) = O(n) (P) = O(d) for n = 2d D.-T.-Z. (2008)
redundant inequalities counts for (P)
Polytopes & Arrangements : Diameter & Curvature
(P) ≤ n - d Hirsch conjecture (1957) (P) < n – d Holt-Klee (1998)
(A) ≤ d conjecture D.-T.-Z. (2008) (A) ≤ 2 d Dedieu-Malajovich-Shub (2005)
(P) ≤ 2 n conjecture D.-T.-Z. (2008) (P)=Ω(n) D.-T.-Z (2008)
holds for n = 2 and n = 3
holds for n = 2 and n = 3
(P) ≤ n – d (P) ≤ d for n = 2d Klee-Walkup (1967)(P) = O(n) (P) = O(d) for n = 2d D.-T.-Z. (2008)
redundant inequalities counts for (P)
c
Want: (P) = O(n), n (P) = O(d) for n = 2d, d
Clearly (P) K n, n (P) K n for n = 2d, d
Need (P) K n, n (P) n for n = 2d, d
Case (i) d < n < 2d
e.g., P with n=3, d=2
P P with n=4, d=2 c(P) c(P) 2 d< 2 n
Polytopes & Arrangements : Diameter & CurvatureProof of a continuous analogue of Klee-Walkup result (d-step conjecture)
A4
Want: (P) = O(n), n (P) = O(d) for n = 2d, d
Need (P) K n, n (P) n for n = 2d, d
Case (ii) 2d < n
e.g., P with n=5, d=2
P P with n=6, d=3 c(P) (P) 2 (d + 1) … < 2 (n - d) < 2 n
Polytopes & Arrangements : Diameter & Curvature
c
c
Proof of a continuous analogue of Klee-Walkup result (d-step conjecture)
Polytopes & Arrangements : Diameter & Curvature
Motivations, algorithmic issues and analogies
Diameter (of a polytope) :
lower bound for the number of iterationsfor the simplex method (pivoting methods)
Curvature (of the central path associated to a polytope) :
large curvature indicates large number of iterationfor (central path following) interior point methods
we believe:
polynomial upper bound for (P) ≤ d (n – d) / 2
Polytopes & Arrangements : Diameter & Curvature
bounded / unbounded cells (sphere arrangements)?
simple / degenerate arrangements? pseudo-hyperplanes / underlying oriented matroid ?
Is it the right setting?(forget optimization and the vector c)
thank you
Polytopes & Arrangements : Diameter & Curvature
Motivations, algorithmic issues and analogies
Diameter (of a polytope) :
lower bound for the number of iterationsfor the simplex method (pivoting methods)
Curvature (of the central path associated to a polytope) :
large curvature indicates large number of iterationfor (central path following) interior point methods
we believe:
polynomial upper bound for (P) ≤ d (n – d) / 2
thank you