Aussois January 2017 Linear Symmetries in Integer Convex Optimization Achill Schürmann (University of Rostock) ( based on work with Katrin Herr, Frieder Ladisch and Thomas Rehn )
Polyhedral Computations • I. Representation Conversion • II. Integer Linear Programming • III.Lattice Point Counting & Exact Volumes max I. II. III. How to use linear symmetry ? ( DFG-Project SCHU 1503/6-1 )
Symmetric Integral Optimization • We consider problems min c t x s.t. x ∈ F ⊆ R n (some convex feasible set) x ∈ Z n • with a given integral linear symmetry group
A C++ Tool also available through polymake • helps to compute linear automorphism groups • converts representations using Recursive Decompositions Getting the group: Getting vertices up to symmetry :
Examples of Linear Symmetries • Permutation matrices (permuting coordinates) 0 0 1 EX: g = 1 0 0 0 1 0 • Signed permutation matrices (hyper-octahedral group) 0 0 ± 1 EX: g = ± 1 0 0 0 ± 1 0 ✓ 0 ◆ ✓ ◆ − 1 1 1 • Non-orthogonal linear symmetries ✓ 1 ◆ ✓ − 1 ◆ 0 0 � 0 − 1 � EX: g = ∈ GL 2 ( Z ) 1 1 ✓ ◆ ✓ ◆ 0 1 − 1 − 1
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 without integrality with integrality Optimum attained within Optimum not necessarily fixed subspace attained in fixed subspace ... with integrality constraints
Core Points Γ = S n ( see Bödi, Herr, Joswig, Math. Program. Ser. A , 2013 for ) z ∈ Z n is a core point for Γ ≤ GL n ( Z ) if DEF: ( conv Γ z ) ∩ Z n = Γ z e c a p s d e x fi fixed space x 1 + x 2 + x 3 = 1 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
Core Points of Symmetric Groups • For Γ = S n acting on coordinates of R n , 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 • For Γ = S n 1 × · · · × S n k core points are 0/1-vectors up to translations of integral vectors from the fixed space • Even naive enumeration approach beats commercial software 1. project polytope and Z onto fixed space 2. enumerate projected integer fixed space points in projected polytope 3. check feasibility of fibers by core sets B ÖDI , H ERR , J OSWIG 2012, S
Competing with Gurobi and CPLEX ( on some “designed symmetric IP-problems” )
Rehn’s reformulation idea Core set- V Let be core set representatives. Then: , . . . , � ⇥ ⇤ ⇤ ( Γ ) ∼ + : ζ ∈ Z , ζ ∈ { , } , = ζ ζ ζ ≤ = = Thomas Rehn • new IP-variables ζ , ζ , . . . , ζ ( PhD 2014 ) • for S or direct products thereof: same number of variables, = − Solves “ ” • 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 Toll-like receptor hours (from Wikipedia)
Transitive Permutation Groups ( with all coordinates in the same orbit ) k • coming with a decomposition R n = � V i i = 1 with the V i being Γ -invariant irreducible subspaces ( V 1 = � I � ) EX: For the cyclic group C n with n odd, there are ( n − 1 ) / 2 C n -invariant 2-dimensional subspaces V 2 , . . . , V k . THM:
Finite vs. Infinite ( for transitive permutation groups ) COR: = 2-homogeneous ( Peter Cameron, 1972 ) 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
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 For Γ ≤ S n acting transitive on coordinates of R n THM: with ( Fix Γ ) ⊥ not containing rational Γ -invariant subspaces, there are only finitely many core points up to normalizer equivalence. z , w ∈ Z n are normalizer equivalent w.r.t. Γ ≤ GL n ( Z ) DEF: if there is a t ∈ ( Fix Γ ) ∩ Z n and M ∈ N GL n ( Z ) ( Γ ) with w = M · z + t ( N G ( Γ ) = { M ∈ G : M · Γ = Γ · M } is normalizer of Γ in G ) EX: Theorem applies i.e. to cyclic permutation groups C n ≤ GL n ( Z ) , with n prime
Normalizer Reformulations min c t x s.t Ax ≤ b , x ∈ Z n , THM: Any Γ -invariant ILP is equivalent to the Γ -invariant ILP min( c t M ) x s.t ( AM ) x ≤ b , x ∈ Z n , for any M ∈ N GL n ( Z ) ( Γ ) . 0 0 1 � � · · · 1 0 � � � � APPL: Γ = C n = . ... � � . � . � � � 0 1 0 has infinite normalizer group in GL n ( Z ) for all primes n ≥ 5 ⇒ ”usually” C n -invariant ILPs have a ”simpler reformulation”
ToDo • COMPUTE GROUPS compute and analyze more (mixed) integer linear symmetry groups of symmetric convex integer optimization problems • 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.
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