antoine deza, mcmaster tamás terlaky, lehigh yuriy zinchenko, calgary feng xie, mcmaster

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


Polytope P defined by n inequalities in dimension d

n = 5 : inequalities d = 2 : dimension

polytope : bounded polyhedron


P




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


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


c(P): total curvature of the primal central path of maxcTx : xP

c

P




c

P


c(P): redundant inequalities count



(P): largest total curvature c(P) over of all possible c

c

P





Continuous analogue of Hirsch Conjecture: (P) = O(n)

c

P






c

P


Dedieu-Shub hypothesis (2005): (P) = O(d)





c

P


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


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



Arrangement A defined by n hyperplanes in dimension d

n = 5 : hyperplanes d = 2 : dimension


Simple arrangement: n d and any d hyperplanes intersect at a unique distinct point

n = 5 : hyperplanes d = 2 : dimension


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


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


c(Pi): redundant inequalities count

P5

P4



(A) : largest value of c(A) over all possible c

P1

P3

P2

P6

c


P5

P4



(A) : largest value of c(A) over all possible c

Dedieu-Malajovich-Shub (2005): (A) ≤ 2 d

P1

P3

P2

P6

c


P5

P4


(A) : average diameter of a bounded cell of A:

P1

P3

P2

P6

P4

P5


A : simple arrangement



(A) = with I =1

( )i

ii

P

I

I

P1

P3

P2

P6

1

nd

P4

P5


(A): average diameter ≠ diameter of A ex: (A)= 1.333…



(A) = with I =1

( )i

ii

P

I

I

P1

P3

P2

P6

1

nd

P4

P5


(Pi): only active inequalities count


P1

P3

P2

P6

P4

P5



Conjecture [D.-T.-Z.]: (A) ≤ d (discrete analogue of Dedieu-Malajovich-Shub result)


(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


• 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



dimension 2

(A) ≥ 2 -

2 / 2( 1)( 2)

n

n n


dimension 2

(A) = 2 -

maximize(diameter) amounts tominimize( #external edge + #odd-gons)

2 / 2( 1)( 2)

n

n n


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*


dimension d

A* satisfies

(A*) ≥ d

A* cyclic arrangement

- -1

/n d nd d

A* mainly consists of cubical cells


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


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


bounded / unbounded cells (sphere arrangements)?

simple / degenerate arrangements? pseudo-hyperplanes / underlying oriented matroid ?

Is it the right setting?(forget optimization and the vector c)


• 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





Is the continuous analogue true?





Is the continuous analogue true? yes



( ) such that lim inf

c

nP

nP



•continuous analogue of tightness:• for n ≥ d ≥ 2 D.-T.-Z. (2008)


slope: careful and geometric decrease

( ) such that liminf

c

nP

nP


• 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)




• 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?






Is the continuous analogue true? yes






•continuous analogue of d-step equivalence:• (P) = O(n) (P) = O(d) for n = 2d D.-T.-Z. (2008)


(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


c

c

Proof of a continuous analogue of Klee-Walkup result (d-step conjecture)


Motivations, algorithmic issues and analogies




large curvature indicates large number of iterationfor (central path following) interior point methods

we believe:

polynomial upper bound for (P) ≤ d (n – d) / 2


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


Motivations, algorithmic issues and analogies




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

antoine deza, mcmaster tamás terlaky, lehigh yuriy zinchenko, calgary feng xie, mcmaster

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