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

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

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

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

n = 5 : inequalities d = 2 : dimension(P) = 2 : diameter

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

c(P): redundant inequalities count

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

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

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

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

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

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

n = 5 : hyperplanes d = 2 : dimension I = 6 : bounded cells

c(A) : average value of c(Pi) over the bounded cells Pi of A:

c(A) = with I =1

c(Pi): redundant inequalities count

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

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

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

A : simple arrangement

(A) = with I =1

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

(A) = with I =1

(Pi): only active inequalities count

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

• 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) ≤

nd ndd

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)

dimension 2

(A) = 2 -

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

2 / 2( 1)( 2)

dimension 3

(A) ≤ 3 +

and A* satisfies

(A*) = 3 -

24(2 16 21)3( 1)( 2)( 3)

6 / 2 261 ( 1)( 2)( 3)

nn n n n

dimension d

A* satisfies

(A*) ≥ d

A* cyclic arrangement

/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*

/n d nd d

dimension 2 (A) =

dimension 3 (A) asympotically equal to 3 dimension d d ≤ (A) ≤ d ?

[D.-Xie 2008]

/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)

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

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

slope: careful and geometric decrease

( ) such that liminf

slope: careful and geometric decrease

( ) such that liminf

• 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

(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)

(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

(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)

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)

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

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

n inequalities

inequalities d

dimension d n

p n dp n

diameter curvaturepolytope

p edgesp n

diameter curvaturediameter

diameter curvatureppolytope

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