Linear Symmetries in Integer Convex Optimization Achill Schrmann - - PowerPoint PPT Presentation

Linear Symmetries in Integer Convex Optimization

Achill Schürmann (University of Rostock)

( based on work with Katrin Herr, Frieder Ladisch and Thomas Rehn )

January 2017 Aussois

Polyhedral Computations

How to use linear symmetry ?

• II. Integer Linear Programming

max

II.

• I. Representation Conversion

I.

• III.Lattice Point Counting & Exact

Volumes III.

( DFG-Project SCHU 1503/6-1 )

Symmetric Integral Optimization

min ct x

• We consider problems

x ∈ Zn s.t. x ∈ F ⊆ Rn (some convex feasible set)

• with a given integral linear symmetry group

A C++ Tool

• helps to compute linear automorphism groups
• converts representations using Recursive Decompositions

also available through polymake Getting the group: Getting vertices up to symmetry :

Examples of Linear Symmetries

• Permutation matrices (permuting coordinates)
• Signed permutation matrices (hyper-octahedral group)
• Non-orthogonal linear symmetries

g =   1 1 1  

EX:

g =   ±1 ±1 ±1  

EX:

✓1 ◆ ✓0 1 ◆ ✓ −1 1 ◆ ✓−1 ◆ ✓ −1 ◆ ✓ 1 −1 ◆

EX:

g = −1 1 1

• ∈ GL2(Z)

Linear Symmetries in MIPLIB 2010

• Thomas Rehn (2014) and with Marc Pfetsch (2015+): 


At least 209 of the 357 instances in MIPLIB 2010
 have non-trivial (linear) permutation symmetries

• 6 of the 50 smallest instances (<1500 variables) have


integral linear symmetries which are not signed permutations!


Convexity and Linear Symmetries

Optimum attained within
 fixed subspace
 Optimum not necessarily 
 attained in fixed subspace 


... with integrality constraints

without integrality with integrality

Core Points

DEF: (conv Γz) ∩ Zn = Γz

x1 + x2 + x3 = 1

z ∈ Zn is a core point for Γ ≤ GLn(Z) if

fi x e d s p a c e

THM: If a Γ-invariant convex integer optimization problem has a solution, then a core point attains the optimal value. ( even a representative ) w.r.t. Γ

fixed space

( see Bödi, Herr, Joswig, Math. Program. Ser. A, 2013 for )

Γ = Sn

Core Points of Symmetric Groups

• 1. project polytope and Z onto

fixed space

• 2. enumerate projected integer

points in projected polytope

• 3. check feasibility of fibers by

core sets BÖDI, HERR, JOSWIG 2012, S

• Even naive enumeration approach beats commercial software 


fixed space

• For Γ = Sn acting on coordinates of Rn, all core points

are 0/1-vectors up to translations by multiples of I

• Core points of direct products are direct products of core points

up to translations of integral vectors from the fixed space

• For Γ = Sn1 × · · · × Snk core points are 0/1-vectors

Competing with Gurobi and CPLEX

( on some “designed symmetric IP-problems” )

Core set-V

Let , . . . , be core set representatives. Then: (Γ) ∼ =

• ζ

+ ⇤

=

ζ : ζ ∈ Z, ζ ∈ { , }, ⇤

=

ζ ≤ ⇥

• new IP-variables ζ , ζ , . . . , ζ
• for S or direct products thereof:

same number of variables, = −

• open problem from MIPLIB 2010 collection
• 2883 binary variables, 4408 constraints
• automorphism group contains (S )

as a subgroup

• after variable transformation and presolving there are 230 less variables and

460 less constraints

• transformed instance is solved by Gurobi 5.0 with 16 threads in about 18

hours

Thomas Rehn ( PhD 2014 )
 


Rehn’s reformulation idea

Toll-like receptor

(from Wikipedia)

Solves “ ”

Transitive Permutation Groups

( with all coordinates in the same orbit )

• coming with a decomposition Rn =

k

• i=1

Vi with the Vi being Γ-invariant irreducible subspaces ( V1 = I ) THM: EX: For the cyclic group Cn with n odd, there are (n − 1)/2 Cn-invariant 2-dimensional subspaces V2, . . . , Vk.

Finite vs. Infinite

( for transitive permutation groups ) COR: CONJECTURE: All other transitive permutation groups have infinitely
 many core points up to translations by multiples of

• true for all groups with irrational invariant subspaces
• true for all imprimitive groups (with rational inv. subspaces)
• true for all primitive groups up to degree

( Peter Cameron, 1972 )

= 2-homogeneous

Creating difficult IP-instances

using primitive permutation groups with infinite core sets

using Gurobi 5.5.0 on Intel Core-i7 with eight logical CPUs at 2.8GHz and 16 GB RAM

Irrational Invariant Subspaces

THM: normalizer equivalence. For Γ ≤ Sn acting transitive on coordinates of Rn with (Fix Γ)⊥ not containing rational Γ-invariant subspaces, there are only finitely many core points up to EX: Theorem applies i.e. to cyclic permutation groups Cn ≤ GLn(Z), with n prime DEF: z, w ∈ Zn are normalizer equivalent w.r.t. Γ ≤ GLn(Z) if there is a t ∈ (Fix Γ) ∩ Zn and M ∈ NGLn(Z)(Γ) with w = M · z + t

( NG(Γ) = {M ∈ G : M · Γ = Γ · M} is normalizer of Γ in G )

Normalizer Reformulations

Γ = Cn =

• · · ·

1 1 ... . . . 1

• APPL:

has infinite normalizer group in GLn(Z) for all primes n ≥ 5 ⇒ ”usually” Cn-invariant ILPs have a ”simpler reformulation” THM: Any Γ-invariant ILP min ctx s.t Ax ≤ b, x ∈ Zn, is equivalent to the Γ-invariant ILP min(ctM)x s.t (AM)x ≤ b, x ∈ Zn, for any M ∈ NGLn(Z)(Γ).

• EXTEND THEORY


classify / approximate core points for interesting groups

• NEW ALGORITHMS


create new algorithms and heuristics that exploit knowledge about core points, i.e. combine with branching, cutting, etc.

ToDo

• COMPUTE GROUPS


compute and analyze more (mixed) integer linear symmetry groups of symmetric convex integer optimization problems