ICERM Conference on Computational Challenges in the Theory of Lattices Providence, April 2018
Variations and Applications of Voronoi’s algorithm
Achill Schürmann (Universität Rostock)
( based on work with Mathieu Dutour Sikiric and Frank Vallentin ) PRELUDE Voronoi’s Algorithm - classically - Lattices and Quadratic Forms Lattices and Quadratic Forms Lattices and Quadratic Forms Reduction Theory for positive definite quadratic forms
n t GLn(Z) acts on by Q U QU S>0 7! Task of a reduction theory is to provide a fundamental domain
Classical reductions were obtained by Lagrange, Gauß, Korkin and Zolotareff, Minkowski and others… All the same for n = 2 : Voronoi’s reduction idea
Georgy Voronoi (1868 – 1908)
Observation: The fundamental domain can be obtained from polyhedral cones that are spanned by rank-1 forms only
Voronoi’s reduction idea
Georgy Voronoi (1868 – 1908)
Observation: The fundamental domain can be obtained from polyhedral cones that are spanned by rank-1 forms only
Voronoi’s algorithm gives a recipe for the construction of a complete list of such polyhedral cones up to GL n ( Z ) -equivalence Perfect Forms Perfect Forms
DEF: min(Q)= min Q[x] is the arithmetical minimum x n 0 2Z \{ } Perfect Forms
DEF: min(Q)= min Q[x] is the arithmetical minimum x n 0 2Z \{ } Q is uniquely determined by min(Q) and n DEF: Q >0 perfect 2 S , n MinQ = x Z : Q[x]=min(Q) { 2 } Perfect Forms
DEF: min(Q)= min Q[x] is the arithmetical minimum x n 0 2Z \{ } Q is uniquely determined by min(Q) and n DEF: Q >0 perfect 2 S , n MinQ = x Z : Q[x]=min(Q) { 2 }
DEF: For Q n , its Voronoi cone is (Q)=cone xxt : x MinQ 2 S>0 V { 2 } Perfect Forms
DEF: min(Q)= min Q[x] is the arithmetical minimum x n 0 2Z \{ } Q is uniquely determined by min(Q) and n DEF: Q >0 perfect 2 S , n MinQ = x Z : Q[x]=min(Q) { 2 }
DEF: For Q n , its Voronoi cone is (Q)=cone xxt : x MinQ 2 S>0 V { 2 }
THM: Voronoi cones give a polyhedral tessellation of n S>0 and there are only finitely many up to GL n ( Z ) -equivalence. Perfect Forms
DEF: min(Q)= min Q[x] is the arithmetical minimum x n 0 2Z \{ } Q is uniquely determined by min(Q) and n DEF: Q >0 perfect 2 S , n MinQ = x Z : Q[x]=min(Q) { 2 }
DEF: For Q n , its Voronoi cone is (Q)=cone xxt : x MinQ 2 S>0 V { 2 }
THM: Voronoi cones give a polyhedral tessellation of n S>0 and there are only finitely many up to GL n ( Z ) -equivalence. (Voronoi cones are full dimensional if and only if Q is perfect!) Ryshkov Polyhedron
The set of all positive definite quadratic forms / matrices with arithmetical minimum at least 1 is called Ryshkov polyhedron Ryshkov Polyhedron
The set of all positive definite quadratic forms / matrices with arithmetical minimum at least 1 is called Ryshkov polyhedron
= Q n : Q[x] 1 for all x Zn 0 R 2 S>0 2 \{ } Ryshkov Polyhedron
The set of all positive definite quadratic forms / matrices with arithmetical minimum at least 1 is called Ryshkov polyhedron Ryshkov Polyhedra = Q n : Q[x] 1 for all x Zn 0 R 2 S>0 2 \{ }
is a locallyis a finitelocally polyhedron finite polyhedron • R • R
Vertices of are perfect forms • R
1 n n ↵ (det(Q + ↵Q0)) is strictly concave on • 7! S>0 Ryshkov Polyhedron
The set of all positive definite quadratic forms / matrices with arithmetical minimum at least 1 is called Ryshkov polyhedron Ryshkov Polyhedra = Q n : Q[x] 1 for all x Zn 0 R 2 S>0 2 \{ }
is a locallyis a finitelocally polyhedron finite polyhedron • R • R • Vertices of are perfect Vertices of are perfectR forms • R
1 n n ↵ (det(Q + ↵Q0)) is strictly concave on • 7! S>0 Voronoi’sVoronoi’s algorithm Algorithm
Start with a perfect form Q
1. SVP: Compute Min Q and describing inequalities of the polyhedral cone
n (Q)= Q0 : Q0[x] 1 for all x Min Q P { 2 S 2 } 2. PolyRepConv: Enumerate extreme rays R ,...,R of (Q) 1 k P 3. SVPs: Determine contiguous perfect forms Qi = Q + ↵Ri, i =1,...,k
4. ISOMs: Test if Qi is arithmetically equivalent to a known form 5. Repeat steps 1.–4. for new perfect forms Computational Results Computational Results Computational Results
5000000 Wessel van Woerden, 2018 ?! Adjacency Decomposition Method (for vertex enumeration) Adjacency Decomposition Method (for vertex enumeration)
• Find initial orbit(s) / representing vertice(s) Adjacency Decomposition Method (for vertex enumeration)
• Find initial orbit(s) / representing vertice(s) • For each new orbit representative • enumerate neighboring vertices Adjacency Decomposition Method (for vertex enumeration)
• Find initial orbit(s) / representing vertice(s) • For each new orbit representative • enumerate neighboring vertices • add as orbit representative if in a new orbit Adjacency Decomposition Method (for vertex enumeration)
• Find initial orbit(s) / representing vertice(s) • For each new orbit representative • enumerate neighboring vertices (up to symmetry) • add as orbit representative if in a new orbit Representation conversion problem Adjacency Decomposition Method (for vertex enumeration)
• Find initial orbit(s) / representing vertice(s) • For each new orbit representative • enumerate neighboring vertices (up to symmetry) • add as orbit representative if in a new orbit Representation conversion problem
BOTTLENECK: Stabilizer and In-Orbit computations
=> Need of efficient data structures and algorithms for permutation groups: BSGS, (partition) backtracking Representation Conversion in practice Representation Conversion in practice
Best known Algorithm:
Mathieu Representation Conversion in practice
Best known Algorithm:
Mathieu A C++-Tool
also available through polymake
helps to compute linear automorphism groups • Thomas Rehn • converts polyhedral representations using (Phd 2014) Recursive Decomposition Methods (Incidence/Adjacency)
http://www.geometrie.uni-rostock.de/software/ Applicaton: Lattice Sphere Packings
The lattice sphere packing problem can be phrased as: Applicaton: Lattice Sphere Packings
The lattice sphere packing problem can be phrased as: Part II: Koecher’s generalization and T-perfect forms Koecher’s generalization
1960/61 Max Koecher generalized Voronoi’s reduction theory and proofs to a setting with a self-dual cone C Max Koecher, 1924-1990 Koecher’s generalization
1960/61 Max Koecher generalized Voronoi’s reduction theory and proofs to a setting with a self-dual cone C Max Koecher, 1924-1990
Under certain conditions, he shows that C is covered by a tessellation of polyhedral Voronoi cones and “approximated from inside“ by a Ryshkov polyhedron Koecher’s generalization
1960/61 Max Koecher generalized Voronoi’s reduction theory and proofs to a setting with a self-dual cone C Max Koecher, 1924-1990
Under certain conditions, he shows that C is covered by a tessellation of polyhedral Voronoi cones and “approximated from inside“ by a Ryshkov polyhedron
Can in particular be applied to obtain reduction domains for the action of GL ( ) on suitable quadratic spaces n OK Applications in Math
Ryshkov Polyhedron
( ) symmetric O Applications in Math
Ryshkov Polyhedron Vertices / Perfect Forms: • Reduction theory Hermite constant Representation • Conversion Polyhedral complex:
• Cohomology of ( ) O Hecke operators ( ) symmetric • O • Compactifications of moduli spaces of Abelian varieties Applications in Math
Ryshkov Polyhedron Vertices / Perfect Forms: • Reduction theory Hermite constant Representation • Conversion Polyhedral complex:
• Cohomology of ( ) O Hecke operators ( ) symmetric • O • Compactifications of See Mathieu’s talk moduli spaces after the coffee break! of Abelian varieties
AIM Square group 2012: Gangl, Dutour Sikirić, Schürmann, Gunnells, Yasaki, Hanke Embedding Koecher’s theory
For practical computations: Koecher’s theory can be embedded into a linear subspace T in some higher dimensional space of symmetric matrices EmbeddingEquivariant theory Koecher’s theory
For a finite group G GLn(Z) the space of invariant forms For practical computations:⇢ Koecher’s theory can be embedded T := Q n : G Aut Q G into a {linear2 S subspace⇢ T } is ain linear some subspace higher of dimensionaln; T spacen is calledof symmetricBravais space matrices S G \ S>0
IDEA (Berge,´ Martinet, Sigrist, 1992):
Intersect Ryshkov polyhedron with a linear subspace T n R ⇢ S EmbeddingEquivariant theory Koecher’s theory
For a finite group G GLn(Z) the space of invariant forms For practical computations:⇢ Koecher’s theory can be embedded T := Q n : G Aut Q G into a {linear2 S subspace⇢ T } is ain linear some subspace higher of dimensionaln; T spacen is calledof symmetricBravais space matrices S G \ S>0
IDEA (Berge,´ Martinet, Sigrist, 1992):
Intersect Ryshkov polyhedron with a linear subspace T n R ⇢ S
T-perfect and T-extreme forms
DEF: Q T n T-perfect and T-extreme2 \ S>0 forms is T -extreme if it attains a loc. max. of within T • DEF: Q T n is T -perfect if it is a vertex of T 2 \ S>•0 R \ 1 is T -extreme if it attainsis T -eutactic a loc. max.if Q of withinT relint(T (Q) T ) • • | 2 V | is T -perfect if it is a vertex of T • R \ 1 is T -eutacticTHMif Q (BMS,T 1992):relint( (QTQ) -extremeT ) QT-perfect and T -eutactic • | 2 V | ,
THM (BMS, 1992): QT-extreme QT-perfect and T -eutactic ,n Q, Q0 T >0 are called T -equivalent, if U GLn(Z) with • 2 \ S 9 2 t t 0 n Q = U QU and T = U TU Q, Q0 T >0 are called T -equivalent, if U GLn(Z) with • 2 \ S 9 2 t t Q0 = U QU and T = U TU THM (Jaquet-Chiffelle, 1995): T -perfect Q : (Q)=1 / finite { G } ⇠TG THM (Jaquet-Chiffelle, 1995): TVoronoi’s-perfect Q algorithm: (Q)=1 can be/ appliedfinite to TG ){ G } ⇠TG R \ Voronoi’s algorithm can be applied to T ) R \ G T-Algorithm Voronoi’s Algorithm for a linear subspace T
SVPs: Obtain a T -perfect form Q
1. SVP: Compute Min Q and describing inequalities of the polyhedral cone
(Q)= Q0 T : Q0[x] 1 for all x Min Q P { 2 2 } 2. PolyRepConv: Enumerate extreme rays R ,...,R of (Q) 1 k P 3. For the indefinite Ri, i =1,...,k
SVPs: Determine contiguous perfect forms Qi = Q + ↵Ri
4. T-ISOMs: Test if Qi is T -equivalent to a known form 5. Repeat steps 1.–4. for new perfect forms T-Algorithm Voronoi’s Algorithm for a linear subspace T
SVPs: Obtain a T -perfect form Q
1. SVP: Compute Min Q and describing inequalities of the polyhedral cone
(Q)= Q0 T : Q0[x] 1 for all x Min Q P { 2 2 } 2. PolyRepConv: Enumerate extreme rays R ,...,R of (Q) 1 k P Possible 3. For the indefinite Ri, i =1,...,k existence of SVPs: Determine contiguous perfect forms Qi = Q + ↵Ri “Dead-Ends“ (for PQFs R) 4. T-ISOMs: Test if Qi is T -equivalent to a known form 5. Repeat steps 1.–4. for new perfect forms T-perfect and T-extreme forms
DEF: Q T n 2 \ S>0 is T -extreme if it attains a loc. max. of within T • is T -perfect if it is a vertex of T • R \ 1 is T -eutactic if Q T relint( (Q) T ) • T-perfect and| 2T-extremeV | forms T-perfect and T-extreme forms THMDEF: (BMS, 1992):Q T QTn -extreme QT-perfect and T -eutactic G-invariant2 \ S> 0 theory, DEF: Q T n is T -extreme if it attains a loc. max. of within T 2 \ S>0• is T -perfectn if it is a vertex of T is T -extreme ifQ, it• Q attains0 T a loc.>0 max.are called of withinT -equivalentTR \ , if U GLn(Z) with • • 2 \ S 1 9 2 is T -eutactic if Q T relint( (Q) T ) is T -perfect if it is a vertex of T t t • • R \Q0 = U| QU2 and VT = |U TU Equivariant1 theory is T -eutactic if Q T relint( (Q) T ) • THM (BMS,| 2 1992):V QT|-extreme QT-perfect and T -eutactic , THM (BMS, 1992):THMFor (Jaquet-Chiffelle,QT a finite-extreme group G 1995):GLQTn(TZ-perfect)-perfectthe space andQ T: of-eutactic invariant(Q)=1 forms/ finite ,⇢ { G } ⇠TG n n Q, Q0 T >T0 are:= called TQ-equivalent: G, if AutU QGLn(Z) with • 2 \ S G { 2 S ⇢9 2 } n n t n t Q, Q0 T is> a0 are linear called subspaceT -equivalent ofQ0 =;, ifU QUUT GLandn(ZTis) with= calledU TUBravais space • 2 \ S S 9 G2\ S>0 t t Q0 = U QU and T = U TU IDEA (Berge,´ Martinet, Sigrist, 1992): THM (Jaquet-Chiffelle, 1995): T -perfect Q : (Q)=1 / finite { G } ⇠TG Intersect Ryshkov polyhedron with a linear subspace T n THM (Jaquet-Chiffelle, 1995): TG-perfect Q : (Q)=1R / TG finite ⇢ S { } ⇠ Applicaton: Lattice Sphere Packings Examples/Applicationswith prescribed symmetry
n 2 4 6 8 10 12 # -perfect 1 1 2 5 1628 ? maximumE 0.9069 ... 0.6168 ... 0.3729 ... 0.2536 ... 0.0360 ... Perfect Eisenstein forms
n 2 4 6 8 10 12 # -perfect 1 1 1 2 8192 ? maximumG 0.7853 ... 0.6168 ... 0.3229 ... 0.2536 ... Perfect Gaussian forms
n 4 8 12 16 # -perfect 1 1 8 ? maximumQ 0.6168 ... 0.2536 ... 0.03125 ... Perfect Quaternion forms PART III: A new Generalization Further Generalization? … and application!
IDEA: Generalize Voronoi’s theory to other convex cones and their duals Further Generalization? … and application!
IDEA: Generalize Voronoi’s theory to other convex cones and their duals
In particular to the completely positive cone 2M.DutourSikiri´c,A.Sch¨urmann,andF.Vallentin
From the algorithmic side some successful ideas have been proposed by Jarre A SIMPLEXand Schmallowsky ALGORITHM [14], Nie [17], FOR Sponsel RATIONAL and D¨ur [20], Groetzner and D¨ur [11], and Anstreicher,CP-FACTORIZATION Burer, and Dickinson [4, Section 3.3]. In this paper we describe a new procedure that works for all matrices in the rational closure ´ ¨ ˜ T n MATHIEU DUTOUR SIKIRIC, ACHILL SCHURMANN,n = cone ANDxx FRANK: x Q VALLENTIN0 CP { 2 } 2M.DutourSikiri´c,A.Sch¨urmann,andF.Vallentinof the cone of completely positive matrices. Moreover, it also computes certificates in case A . Abstract. FromWe describe the algorithmic62 andCP analyzen side an algorithmic some successful procedure, ideas similar have to been the proposed by Jarre simplex algorithmIn for contrast linear programming, to most of the which other tests approaches whether mentioned or not a given above (the exception being and Schmallowsky [14], Nie [17], Sponsel and D¨ur [20], Groetzner and D¨ur [11], symmetric matrixthe approach is completely by Anstreicher, positive. For matrices Burer and in the Dickinson rational whose closure factorization method is and Anstreicher, Burer, and Dickinson [4, Section 3.3]. of the cone ofbased completely on the positive ellipsoid matrices method), our procedure our algorithm computes is a exact rational in the sense that it uses In this paper we describe a new procedure that works for all matrices in the cp-factorization.rational For matrices numbers which only are if the not input completely matrix positive is rational a certificate and so does not have to cope in form ofrational a separating closure hyperplane is computed, whenever our procedure ter- with numerical instabilities.˜ In particularT it findsn a rational cp-factorization if it minates. We conjecture that there existsn a= pivot cone rulexx so: thatx ourQ 0 procedure exists. However, questionsCP of convergence{ remain,2 } in particular for all A which are always terminatesof the cone for of rational completely input matrices. positive matrices. Moreover, it also computes certificates not contained in the “irrational boundary part” (bd n) ˜ n. in case A . CP \ CP 62 CPn In contrast2. The to most copositive of the other minimum approaches and mentioned copositive above perfect (the exception matrices being the approach by Anstreicher, Burer and Dickinson whose factorization method is n basedBy on thewe ellipsoid denote method), the Euclidean our algorithm vector space is exact of symmetric in the sensen thatn matrices it uses with inner productS 1. A,Introduction B = Trace(AB)= n A B .Withrespecttothisinner⇥ rational numbers onlyh ifi the input matrix is rationali,j=1 ij andij so does not have to cope product we have the following duality relations between the cone of copositive Copositive programmingwith numericalFurther gives instabilities. a common Generalization? In framework particular to it formulatefinds a rational… manyand cp-factorizationapplication! di cult if it matrices and and the cone of completelyP positive matrices optimization problemsexists. However, as convex questions conic ofones. convergence In fact, remain, many in NP-hard particular problems for all A which are not contained in the “irrational boundary part”n (bd ) ˜ . =( )⇤ = B : A, B 0n for allnA , are known to have suchIDEA: reformulations COP Generalizen CP (see Voronoi’sn for{ example theory2 S to theh surveysi CP \ [2,CP 8]).2 AllCPn} the di culty of theseand2. problems The other copositive appears convex minimum tocones be and “converted” and their copositive duals into the perfect di culty matrices of understanding the conen of copositive matrices n whichn =( consistsn)⇤. of all symmetric By n we denoteT the EuclideanCOP vectorCPn spaceCOP of symmetric n n matrices with n n matrices B So,S inwithIn order particularx toBx show to0 that forthe all acompletely givenx R symmetric0 positive.n Its dual matrix cone cone A isis notthe⇥ completely cone positive, it ⇥ inner2 S product A, B = Trace(AB2)= AijBij.Withrespecttothisinner h i i,j=1 productsu ces we to have find the a copositive followingT matrix dualitynB relationsn betweenwith B,A the< cone0. We of copositive call B a sepa- n = cone xx : x R P0 2andCOP its dual, theh copositivei cone matricesrating and witnessCP and thefor coneA { of completelyn in2 this case,} positive because matrices the linear hyperplane orthogonal to B separates A and62 CP . of completely positive n n matrices. Therefore,n itn seems no surprise that many Using⇥ then notation=( nBCP)⇤[x=] forB xTBx:=A,B,xx B T0, for we all obtainA n , basic questions about this coneCOP are stillCP open{ and2 appearS h h toi be dii cult. 2 CP } and n n One important problem is to find an algorithmicn = B test: B[x deciding] 0 for all whetherx R or0 . not COP { 2n S=( n) ⇤. 2 } a given symmetric matrix A is completely positive.CP IfCOP possiblen one would like to So,Definition in order to show 2.1. thatFor a a symmetric given symmetric matrix matrixB Aweis not define completely the copositive positive, minimum it obtain a certificateas for either A n or A n n. In terms2 ofS the definitions the su ces to find a2 copositiveCPn matrix62 >CP0 B n withn B,A < 0. We call B a sepa- most natural certificate for A CPn is giving⇢ S a cp-factorization2⇢COPCOP h n i rating witness for A minn in this(B)=inf case, becauseB[v]: thev linearZ 0 hyperplane0 , orthogonal 2 CP62 CP COP 2 \{ } to Bandseparates we denotem A and the setn of. vectors attaining it by A, B = (A B) n h i T CP· T nT S UsingA = the notationxixi Bwith[x] forxx1,...,xBx =mB,xxnR 0, we obtain Min (B)= v h Z2 0 : iB[v]=min (B) . i=1 COP n 2 COPn X n = B : B[x] 0 for all x R 0 . The followingCOP proposition{ 2 S shows that matrices2 in the } interior of the cone of and for A giving a separating hyperplanes defined byn B so that Definitionn copositive 2.1. matricesFor a symmetric attain their matrix copositiveB minimum.we define thencopositive minimum B,A < 0.62 DickinsonCP and Gijben [5] showed that this (strong)2 S membership2 COP prob- h i as lem is NP-hard. Lemma 2.2. Let B be a matrix in the interiorn of the cone of copositive matrices. min (B)=inf B[v]:v Z 0 0 , Then, the copositiveCOP minimum of B is strictly2 positive\{ } and it is attained by only and we denote the set of vectors attaining it by finitely many vectors. Date:April15,2018. n Min (B)= v Z 0 : B[v]=min (B) . 2010 Mathematics SubjectProof. Classification.Since B isCOP copositive,90C20. we2 have the inequalityCOP min (B) 0. Suppose that Key words and phrases.The followingcopositive proposition programming, shows complete that positivity,matrices in matrixn the interior factorization,COP of the cone of min (B) = 0. Then there is a sequence vi Z 0 0 of pairwise distinct lattice copositive minimum.copositiveCOP matrices attain their copositive minimum.2 \{ } points such that B[vi]tendstozerowheni tends to infinity. From the sequence The first author was supported by the Humboldt foundation. The third author is partially n 1 supported by the SFB/TRRLemmavi we 2.2. 191 construct “SymplecticLet B abe new a Structures matrix sequence in in theu Geometry,i interiorof points of Algebra on the the cone and unit of Dynamics”, copositive sphere S matrices.by setting funded by the DFG.Then, the copositive minimum of B is strictly positive and it is attained by only finitely many vectors. 1 Proof. Since B is copositive, we have the inequality min (B) 0. Suppose that n COP min (B) = 0. Then there is a sequence vi Z 0 0 of pairwise distinct lattice COP 2 \{ } points such that B[vi]tendstozerowheni tends to infinity. From the sequence n 1 vi we construct a new sequence ui of points on the unit sphere S by setting Application: Copositive Optimization Application: Copositive Optimization
• Copositive optimization problems are convex conic problems
min C, Q such that Q, A = b , i = 1,...,m h i h ii i and Q CONE 2 Application: Copositive Optimization
• Copositive optimization problems are convex conic problems
min C, Q such that Q, A = b , i = 1,...,m h i h ii i and Q CONE 2
n CONE = R 0 Linear Programming (LP) Application: Copositive Optimization
• Copositive optimization problems are convex conic problems
min C, Q such that Q, A = b , i = 1,...,m h i h ii i and Q CONE 2
n CONE = R 0 Linear Programming (LP)
n CONE = 0 Semidefinite ProgrammingS (SDP) Application: Copositive Optimization
• Copositive optimization problems are convex conic problems
min C, Q such that Q, A = b , i = 1,...,m h i h ii i and Q CONE 2
n CONE = R 0 CONE = n or n Linear Programming (LP) Copositive ProgrammingCP COP (CP)
n CONE = 0 Semidefinite ProgrammingS (SDP) Application: Copositive Optimization
• Copositive optimization problems are convex conic problems
min C, Q such that Q, A = b , i = 1,...,m h i h ii i and Q CONE 2
n CONE = R 0 CONE = n or n Linear Programming (LP) Copositive ProgrammingCP COP (CP) NP-hard (2000) n CONE = 0 Semidefinite ProgrammingS (SDP) Application: Copositive Optimization
• Copositive optimization problems are convex conic problems
min C, Q such that Q, A = b , i = 1,...,m h i h ii i and Q CONE 2
n CONE = R 0 CONE = n or n Linear Programming (LP) Copositive ProgrammingCP COP (CP) NP-hard (2000) n CONE = 0 Semidefinite ProgrammingS (SDP)
Such problems have a duality theory and allow certificates for solutions! cp-factorizations and certificates
n DEF: A finite set X R 0 is called a certificate ⇢ for Q n being completely positive, 2 S if it gives a cp-factorization Q = xx> x X X2 cp-factorizations and certificates
n DEF: A finite set X R 0 is called a certificate ⇢ for Q n being completely positive, 2 S if it gives a cp-factorization Q = xx> x X X2 PROBLEM: How to find a cp-factorization for a given Q ? cp-factorizations and certificates
n DEF: A finite set X R 0 is called a certificate ⇢ for Q n being completely positive, 2 S if it gives a cp-factorization Q = xx> x X X2 PROBLEM: How to find a cp-factorization for a given Q ?
Known approaches so far: • Anstreicher, Burer and Dickinson (in Dickinson’s thesis 2013) give an algorithm only for matrices in interior based on ellipsoid method • Numerical heuristics have been proposed by Jarre, Schmallowsky (2009), Nie (2014), Sponsel and Dür (2014), Groetzner and Dür (preprint 2018) cp-factorizations and certificates
n DEF: A finite set X R 0 is called a certificate ⇢ for Q n being completely positive, 2 S if it gives a cp-factorization Q = xx> x X X2 PROBLEM: How to find a cp-factorization for a given Q ?
Known approaches so far: • Anstreicher, Burer and Dickinson (in Dickinson’s thesis 2013) give an algorithm only for matrices in interior based on ellipsoid method • Numerical heuristics have been proposed by Jarre, Schmallowsky (2009), Nie (2014), Sponsel and Dür (2014), Groetzner and Dür (preprint 2018)
Non of these approaches is exact and latter do not even guarantee to find solutions! Copositive minimum (COP-SVP)
DEF: min Q =minQ[x] is the copositive minimum n COP x Z 0 0 2 \{ } Copositive minimum (COP-SVP)
DEF: min Q =minQ[x] is the copositive minimum n COP x Z 0 0 2 \{ } Difficult to compute! Copositive minimum (COP-SVP)
DEF: min Q =minQ[x] is the copositive minimum n COP x Z 0 0 2 \{ } Difficult to compute!
THM: (Bundfuss and Dür, 2008) For Q int we can construct a family of simplices k 2 COPn n in the standard simplex = x R 0 : x1 + ...xn = 1 2 k such that each has vertices v1,...vn with vi>Qvj > 0 Copositive minimum (COP-SVP)
DEF: min Q =minQ[x] is the copositive minimum n COP x Z 0 0 2 \{ } Difficult to compute!
THM: (Bundfuss and Dür, 2008) For Q int we can construct a family of simplices k 2 COPn n in the standard simplex = x R 0 : x1 + ...xn = 1 2 k such that each has vertices v1,...vn with vi>Qvj > 0
Computation in practice: ”Fincke-Pohst strategy” to compute min Q in each cone k COP Generalized Ryshkov polyhedron Generalized Ryshkov polyhedron
The set of all copositive quadratic forms / matrices with copositive minimum at least 1 is called Ryshkov polyhedron n = Q n : Q[x] 1 for all x Z 0 0 R 2 COP 2 \{ } Generalized Ryshkov polyhedron
The set of all copositive quadratic forms / matrices with copositive minimum at least 1 is called Ryshkov polyhedron n = Q n : Q[x] 1 for all x Z 0 0 R 2 COP 2 \{ } Generalized Ryshkov polyhedron
The set of all copositive quadratic forms / matrices with copositive minimum at least 1 is called Ryshkov polyhedron n = Q n : Q[x] 1 for all x Z 0 0 R 2 COP 2 \{ } DEF: Q int is called -perfect if and only if 2 COPn COP Q is uniquely determined by min Q and COP n Min Q = x Z 0 : Q[x]=min Q COP 2 COP Generalized Ryshkov polyhedron
The set of all copositive quadratic forms / matrices with copositive minimum at least 1 is called Ryshkov polyhedron n = Q n : Q[x] 1 for all x Z 0 0 R 2 COP 2 \{ } DEF: Q int is called -perfect if and only if 2 COPn COP Q is uniquely determined by min Q and COP n Min Q = x Z 0 : Q[x]=min Q COP 2 COP is a locally finite polyhedron (with dead-ends / rays) • R Generalized Ryshkov polyhedron
The set of all copositive quadratic forms / matrices with copositive minimum at least 1 is called Ryshkov polyhedron n = Q n : Q[x] 1 for all x Z 0 0 R 2 COP 2 \{ } DEF: Q int is called -perfect if and only if 2 COPn COP Q is uniquely determined by min Q and COP n Min Q = x Z 0 : Q[x]=min Q COP 2 COP is a locally finite polyhedron (with dead-ends / rays) • R Vertices of are -perfect • R COP Voronoi-type simplex algorithm Input: A n 2 S>0 Voronoi-type simplex algorithm Input: A n 2 S>0 COP-SVPs: Obtain an initial -perfect matrix B COP P Voronoi-type simplex algorithm Input: A n 2 S>0 COP-SVPs: Obtain an initial -perfect matrix B COP P 1. BP, A < 0 A n (with witness BP) h i 62 CP 2. LP: A cone xx> : x Min BP A ˜ n 2 2 COP 2 CP 3. COP-SVP: Compute Min BP and the polyhedral cone COP n (BP)= B : B[x] 1 for all x Min BP P { 2 S 2 COP }
4. PolyRepConv: Determine a generator R of an extreme ray of (BP) P with A, R < 0. h i 5. LPs: R n A n (with witness R) 2 COP 62 CP 6. COP-SVPs: Determine the contiguous -perfect matrix COP BN := BP + R with > 0 and min BN = 1 COP 7. Set BP := BN and 1. Voronoi-type simplex algorithm
n ˜ n Input: A >0 n = cone xx> : x Q 2 S CP 2 COP-SVPs: Obtain an initial -perfect matrix B COP P 1. BP, A < 0 A n (with witness BP) h i 62 CP 2. LP: A cone xx> : x Min BP A ˜ n 2 2 COP 2 CP 3. COP-SVP: Compute Min BP and the polyhedral cone COP n (BP)= B : B[x] 1 for all x Min BP P { 2 S 2 COP }
4. PolyRepConv: Determine a generator R of an extreme ray of (BP) P with A, R < 0. h i 5. LPs: R n A n (with witness R) 2 COP 62 CP 6. COP-SVPs: Determine the contiguous -perfect matrix COP BN := BP + R with > 0 and min BN = 1 COP 7. Set BP := BN and 1. Voronoi-type simplex algorithm
n ˜ n Input: A >0 n = cone xx> : x Q 2 S CP 2 COP-SVPs: Obtain an initial -perfect matrix B COP P 1. BP, A < 0 A n (with witness BP) h i 62 CP 2. LP: A cone xx> : x Min BP A ˜ n 2 2 COP 2 CP 3. COP-SVP: Compute Min BP and the polyhedral cone COP n (BP)= B : B[x] 1 for all x Min BP P { 2 S 2 COP }
4. PolyRepConv: Determine a generator R of an extreme ray of (BP) P with A, R < 0. ( flexible ”pivot-rule” ) h i 5. LPs: R n A n (with witness R) 2 COP 62 CP 6. COP-SVPs: Determine the contiguous -perfect matrix COP BN := BP + R with > 0 and min BN = 1 COP 7. Set BP := BN and 1. 4M.DutourSikiri´c,A.Sch¨urmann,andF.Vallentin
Definition 2.3. A copositive matrix B int( n) is called -perfect if it is uniquely determined by its copositive minimum2 COPmin B andCOP the set Min B attaining it. COP COP
In other words, B int( n)is -perfect if and only if it is the unique solution of the system2 of linearCOP equationsCOP B,vvT =min B, for all v Min B. h i COP 2 COP In particular all vertices of are -perfect. R COP Lemma 2.4. -perfectA copositive matrices exist in all starting dimensions (dimension pointn =1being trivial): For dimensionCOP n 2 the following matrix 2 10... 0 .. .. . 0 12. . . 1 THM: ...... (4) QAn = B 0 . . . 0 C is -perfect B C COP B . . . C B . .. .. 2 1C B C B 0 ... 0 12 C B C @ A is -perfect; 1 Q is a vertex of . COP 2 An R
Proof. Matrix QAn is positive definite since n 1 2 2 2 QAn [x]=x + (xi xi+1) + x for x R. 1 n 2 i=1 X In particular it lies in the interior of the copositive cone. Furthermore,
k min QA =2 with Min QA = ej :1 j k n , COP n COP n 8 9 i=j The matrix QAn is also known as a Gram matrix of the root lattice An,avery important lattice in the theory of sphere packings, see Conway and Sloane [3]. 3. The algorithm In this section we show how one can solve the linear program (2). Our algorithm is similar to the simplex algorithm for linear programming, as it walks along a path of subsequently constructed -perfect matrices, which are vertices of the polyhedral set and which are connectedCOP by edges. R 4M.DutourSikiri´c,A.Sch¨urmann,andF.Vallentin Definition 2.3. A copositive matrix B int( n) is called -perfect if it is uniquely determined by its copositive minimum2 COPmin B andCOP the set Min B attaining it. COP COP In other words, B int( n)is -perfect if and only if it is the unique solution of the system2 of linearCOP equationsCOP B,vvT =min B, for all v Min B. h i COP 2 COP In particular all vertices of are -perfect. R COP Lemma 2.4. -perfectA copositive matrices exist in all starting dimensions (dimension pointn =1being trivial): For dimensionCOP n 2 the following matrix 2 10... 0 .. .. . 0 12. . . 1 THM: ...... (4) QAn = B 0 . . . 0 C is -perfect B C COP B . . . C B . .. .. 2 1C B C B 0 ... 0 12 C B C @ A is -perfect; 1 Q is a vertex of . COP 2 An R Proof. Matrix QAn is positive definite since n 1 2 2 2 QAn [x]=x + (xi xi+1) + x for x R. 1 n 2 i=1 X In particular it lies in the interior of the copositive cone. Furthermore, k min QA =2 with Min QA = ej :1 j k n , COP n COP n 8 9 i=j The matrix QAn is also known as a Gram matrix of the root lattice An,avery important lattice in the theory of sphere packings, see Conway and Sloane [3]. 3. The algorithm In this section we show how one can solve the linear program (2). Our algorithm is similar to the simplex algorithm for linear programming, as it walks along a path of subsequently constructed -perfect matrices, which are vertices of the polyhedral set and which are connectedCOP by edges. R 4M.DutourSikiri´c,A.Sch¨urmann,andF.Vallentin 4M.DutourSikiri´c,A.Sch¨urmann,andF.Vallentin Definition 2.3. A copositive matrix B int( n) is called -perfect if it is Definition 2.3. A copositive2 COP matrix B int(COP ) is called -perfect if it is uniquely determined by its copositive minimum min B2andCOP the setn Min COPB attaining it. uniquely determined by its copositiveCOP minimum min BCOPand the set Min B attaining it. COP COP In other words, B int( n)is -perfect if and only if it is the unique solution of the system2 ofIn linear otherCOP equations words, COPB int( n)is -perfect if and only if it is the unique solution of the system2 of linearCOP equationsCOP T B,vv =min B, forT all v Min B. h i COP B,vv =min2 COPB, for all v Min B. h i COP 2 COP In particular all verticesIn particular of are all vertices-perfect. of are -perfect. R COP R COP Lemma 2.4. -perfectA copositive matrices exist in all starting dimensions (dimension pointn =1being COP Lemma 2.4. -perfect matrices exist in all dimensions (dimension n =1being trivial): For dimensiontrivial):n 2 Forthe dimension followingCOP n matrix2 the following matrix 2 10... 20 10... 0 ...... 0 12. 0. 12. 1 . . . 1 . . . . . . THM:(4) . Q.A = . . . . (4) QAn = B 0 . .n B. 0 0 C. is . -perfect. 0 C B B C COP C B . . . B . C.. .. C B . .. .. B2 . 1C. . 2 1C B B C C B 0 ... 0 B120 C... 0 12C B B C C 1 @ A 1 is -perfect;@ Q is a vertex of A. is -perfect; Q is a vertex of 2 .An COP 2 An COP R R Proof. Matrix QAn is positive definite since Proof. Matrix QAn is positive definite since n 1 n 1 2 2 2 2 QAn [x]=x21 + 2 (xi xi+1) + xn for x R. QAn [x]=x + (xi xi+1) + x for x R. 2 1 in=1 2 i=1 X In particularX it lies in the interior of the copositive cone. Furthermore, In particular it lies in the interior of the copositive cone. Furthermore, k k min QA =2 with Min QA = ej :1 j k n , COP n COP n 8 9 min QA =2 with Min QA = ej :1 j k i=jn , COP n COP n 8 In this section3. The we algorithm show how one can solve the linear program (2). Our algorithm is similar to the simplex algorithm for linear programming, as it walks along a In this section we showpath of how subsequently one can solve constructed the linear program-perfect (2). matrices, Our algorithm which are vertices of the COP is similar to the simplexpolyhedral algorithm set forand linear which programming, are connected by as edges. it walks along a R path of subsequently constructed -perfect matrices, which are vertices of the polyhedral set and which are connectedCOP by edges. R Interior cases (algorithm terminates) 63 EX: A = 32 ✓ ◆ Interior cases (algorithm terminates) 63 EX: A = 32 ✓ ◆ Interior cases (algorithm terminates) 63 EX: A = 32 ✓ ◆ Interior cases (algorithm terminates) 63 EX: A = 32 ✓ ◆ Interior cases (algorithm terminates) 63 EX: A = 32 ✓ ◆ Starting with QA2 one iteration of the algorithm finds 1 3/2 the -perfect matrix BP = and COP 3/23 ✓ ◆ 1 1 > 1 1 > 2 2 > A = + + 0 0 1 1 1 1 ✓ ◆✓ ◆ ✓ ◆✓ ◆ ✓ ◆✓ ◆ 12 M.DutourSikiri´c,A.Sch¨urmann,andF.Vallentin Note that there is always a unique choice in Step 4 in case A is a 2 2 rank-1 matrix. Note also that the vectors represent fractions that converge to ⇥p2. Every second vector corresponds to a convergent of the continued fraction of p2. For instance, 1 99/70 = 1 + . 1 2+ 1 2+ 1 2+ 1 2+ 2 The -perfect matrix after ten iterations of the algorithm is COP 4756 6726 B(10) = . P 6726 9512 ✓ ◆ (i) It can be shown that the matrices BP converge to a multiple of 1 p2 B = satisfying A, B = 0 and X, B 0 for all X 2. p22 h i h i 2 CP ✓ ◆ (i) However, every one of the infinitely many perfect matrices BP satisfies X, B(i) > 0 for all X . h P i 2 CP2 5. Computational Experiments We implemented our algorithm. The source code, written in C++, is available on TODO: github [?]. In this section we report how it performs on several examples, most of them were previously discussed in the literature. TODO: The first example is the matrix A1 = ... Boundary cases from ˜ n which lies in the interior of 5. The second example is from(algorithmCP [11] and terminates lies in the boundarywith a suitable of pivot-rule)CP CP5 85115 58511 0 1 from Groetzner, Dür (2018) EX:A2 = 15851 B11585C not solved by their algorithms B C B51158C B C @ A 12 M.DutourSikiri´c,A.Sch¨urmann,andF.Vallentin Note that there is always a unique choice in Step 4 in case A is a 2 2 rank-1 matrix. Note also that the vectors represent fractions that converge to ⇥p2. Every second vector corresponds to a convergent of the continued fraction of p2. For instance, 1 99/70 = 1 + . 1 2+ 1 2+ 1 2+ 1 2+ 2 The -perfect matrix after ten iterations of the algorithm is COP 4756 6726 B(10) = . P 6726 9512 ✓ ◆ (i) It can be shown that the matrices BP converge to a multiple of 1 p2 B = satisfying A, B = 0 and X, B 0 for all X 2. p22 h i h i 2 CP ✓ ◆ (i) However, every one of the infinitely many perfect matrices BP satisfies X, B(i) > 0 for all X . h P i 2 CP2 5. Computational Experiments We implemented our algorithm. The source code, written in C++, is available on TODO: github [?]. In this section we report how it performs on several examples, most of them were previously discussed in the literature. TODO: The first example is the matrix A1 = ... Boundary cases from ˜ n which lies in the interior of 5. Asimplexalgorithmforrationalcp-factorization 13 The second example is from(algorithmCP [11] and terminates lies in the boundarywith a suitable of pivot-rule)CP CP5 Starting from QA5 our algorithm does five steps to find the factorization A2 = 10 T 85115i=1 vivi with 58511 0P 1 from Groetzner,v1 Dür=(0 (2018), 0, 0, 1, 1) EX:A2 = 15851 B11585C not solved by vtheir2 algorithms=(0, 0 , 1, 1, 0) Asimplexalgorithmforrationalcp-factorizationB C 13 B51158C v =(0, 0, 1, 2, 1) B C 3 @ A Starting from QA5 our algorithm does five steps to find thev4 factorization=(0, 1, 1, 0A, 0)2 = 10 T Starting with QA , our algorithm finds a cp-factorization after 5 iterations i=1 vivi with 5 v5 =(0, 1, 2, 1, 0) P v1 =(0, 0, 0, 1, 1) v6 =(1, 0, 0, 0, 1) v2 =(0, 0, 1, 1, 0) v7 =(1, 0, 0, 1, 2) v3 =(0, 0, 1, 2, 1) v8 =(1, 1, 0, 0, 0) v4 =(0, 1, 1, 0, 0) v9 =(1, 2, 1, 0, 0) v5 =(0, 1, 2, 1, 0) v10 =(2, 1, 0, 0, 1) giving va6 certificate=(1, 0 ,for0, 0 the, 1) matrix to be completely positive Thev7 third=(1 example, 0, 0, 1, 2) is from [17, Example 6.2] and it positive semidefinite, but not completelyv8 =(1 positive, 1, 0, 0, 0) 11001 v9 =(1, 2, 1, 0, 0) 12100 v10 =(2, 1, 0, 0, 1) 0 1 A3 = 01210 B00121C B C B10016C The third example is from [17, Example 6.2] and it positive semidefinite,B but notC @ A completely positive Starting from QA5 our algorithm does 18 steps to find the copositive matrix 11001 363/5 2126/35 2879/70 608/21 4519/210 12100 0 2126/351 1787/35 347/10 1025/42 253/14 A3 = 012100 1 B3 = 2879/70 347/10 829/35 1748/105 371/30 B00121C B B 608/21C 1025/42 1748/105 1237/105 601/70 C B10016B C C B B 4519/210C 253/14 371/30 601/70 671/105 C @ B A C Starting from QA5 our algorithm does 18@ steps to find the copositive matrix A with A3,B3 = 2/5 showing that A3 5. 363/5 2126The/h35 fourth 2879i and/ 70 last example 608/21 is the family451962 CP/ of210 completely positive matrices TODO 2126/35 1787/35 347/10 1025/42 253/14 0 1 B = 2879/70 347/10 829/35 1748/105 371/30 3 B 608/21 1025from/42 [14]. 1748/105 1237/105 601/70 C B C B 4519/210 253/TODO:14 371 Some/30 words601 about/70 the 671 running/105 C time. B C @ A with A3,B3 = 2/5 showing that A3 5. Theh fourthi and last example is the family62 CP of completely positiveAcknowledgement matrices TODO The second author likes to thank Je↵ Lagarias for a helpful discussion about Farey sequences. from [14]. This project has received funding from the European Union’s Horizon 2020 re- TODO: Some words aboutsearch the and running innovation time. programme under the Marie Sklodowska-Curie agreement No 764759. Acknowledgement The second author likes to thank Je↵ Lagarias for a helpfulReferences discussion about Farey sequences. [1] S. Bundfuss and M. D¨ur, Algorithmic copositivity detection by simplicial partition,Linear This project has received fundingAlgebra from Appl. the428 European(2008), 1511–1523. Union’s Horizon 2020 re- search and innovation programme under the Marie Sklodowska-Curie agreement No 764759. References [1] S. Bundfuss and M. D¨ur, Algorithmic copositivity detection by simplicial partition,Linear Algebra Appl. 428 (2008), 1511–1523. Asimplexalgorithmforrationalcp-factorization 13 Starting from QA5 our algorithm does five steps to find the factorization A2 = 10 T i=1 vivi with P v1 =(0, 0, 0, 1, 1) v2 =(0, 0, 1, 1, 0) v3 =(0, 0, 1, 2, 1) v4 =(0, 1, 1, 0, 0) v5 =(0, 1, 2, 1, 0) v6 =(1, 0, 0, 0, 1) v7 =(1, 0, 0, 1, 2) v8 =(1, 1, 0, 0, 0) v9 =(1, 2, 1, 0, 0) Exteriorv10 =(2, 1 , 0cases, 0, 1) The third example is from(algorithm [17, Example conjectured 6.2] and it positiveto terminate) semidefinite, but not completely positive 11001 12100 0 1 EX:A3 = 01210 from Nie (2014) B00121C B C B10016C B C @ A Starting from QA5 our algorithm does 18 steps to find the copositive matrix 363/5 2126/35 2879/70 608/21 4519/210 2126/35 1787/35 347/10 1025/42 253/14 0 1 B = 2879/70 347/10 829/35 1748/105 371/30 3 B 608/21 1025/42 1748/105 1237/105 601/70 C B C B 4519/210 253/14 371/30 601/70 671/105 C B C @ A with A3,B3 = 2/5 showing that A3 5. Theh fourthi and last example is the family62 CP of completely positive matrices TODO from [14]. TODO: Some words about the running time. Acknowledgement The second author likes to thank Je↵ Lagarias for a helpful discussion about Farey sequences. This project has received funding from the European Union’s Horizon 2020 re- search and innovation programme under the Marie Sklodowska-Curie agreement No 764759. References [1] S. Bundfuss and M. D¨ur, Algorithmic copositivity detection by simplicial partition,Linear Algebra Appl. 428 (2008), 1511–1523. Asimplexalgorithmforrationalcp-factorization 13 Starting from QA5 our algorithm does fiveAsimplexalgorithmforrationalcp-factorization steps to find the factorization A2 = 13 10 T i=1 vivi with Starting from QA5 our algorithm does five steps to find the factorization A2 = P 10 Tv1 =(0, 0, 0, 1, 1) i=1 vivi with v2 =(0, 0, 1, 1, 0) P v1 =(0, 0, 0, 1, 1) v3 =(0, 0, 1, 2, 1) v2 =(0, 0, 1, 1, 0) v4 =(0, 1, 1, 0, 0) v3 =(0, 0, 1, 2, 1) v5 =(0, 1, 2, 1, 0) v4 =(0, 1, 1, 0, 0) v6 =(1, 0, 0, 0, 1) v5 =(0, 1, 2, 1, 0) v7 =(1, 0, 0, 1, 2) v6 =(1, 0, 0, 0, 1) v8 =(1, 1, 0, 0, 0) v7 =(1, 0, 0, 1, 2) v9 =(1, 2, 1, 0, 0) v8 =(1, 1, 0, 0, 0) v10 =(2, 1, 0, 0, 1) Exterior casesv9 =(1, 2, 1, 0, 0) v10 =(2, 1, 0, 0, 1) The third example is from(algorithm [17, Example conjectured 6.2] and it positiveto terminate) semidefinite, but not completely positive The third example11001 is from [17, Example 6.2] and it positive semidefinite, but not completely positive 12100 0 1 11001 EX:A3 = 01210 from Nie (2014) 12100 B00121C 0 1 B AC= 01210 B10016C3 B C B00121C Starting from Q our algorithm does@ 18 steps to findA theB copositive matrixC A5 B10016C Starting with Q , after 18 iterationsB our algorithmC finds the -perfect 363/5 Starting2126 from/35Q 2879ourA5/ algorithm70 608 does@/21 18 steps4519 to/ find210A the copositive matrix A5 COP 2126/35 1787/35 347/10 1025/42 253/14 0 363 /5 2126/35 2879/70 6081/21 4519/210 B3 = 2879/70 347/10 829/35 1748/105 371/30 2126/35 1787 /35 347/10 1025/42 253/14 B 608/21 1025/042 1748/105 1237/105 601/70 C 1 B B3 = 2879 /70 347/10 829/35 1748C /105 371/30 B 4519/210 253/14 371/30 601/70 671/105 C B B 608/21 1025 /42 1748/105 1237C/105 601/70 C B C with A ,B@ = 2/5 showing thatB 4519A /210. 253/14 371/30 601A /70 671/105 C 3 3 B 3 5 C Theh fourthi and last example is@ the family62 CP of completely positive matrices TODO A with A3,B3 = 2/5 showing that A3 5. Theh fourthgivingi and a last certificate example for is the the family matrix62 CP of completelynot to be completely positive matrices positiveTODO from [14]. TODO: Some wordsfrom about [14]. the running time. TODO: Some words about the running time. Acknowledgement The second author likes to thank Je↵ LagariasAcknowledgement for a helpful discussion about Farey sequences. The second author likes to thank Je↵ Lagarias for a helpful discussion about This project has receivedFarey sequences. funding from the European Union’s Horizon 2020 re- search and innovation programmeThis project under has received the Marie funding Sklodowska-Curie from the European agreement Union’s Horizon 2020 re- No 764759. search and innovation programme under the Marie Sklodowska-Curie agreement No 764759. References [1] S. Bundfuss and M. D¨ur, Algorithmic copositivity detectionReferences by simplicial partition,Linear Algebra Appl. 428 (2008),[1] S. 1511–1523. Bundfuss and M. D¨ur, Algorithmic copositivity detection by simplicial partition,Linear Algebra Appl. 428 (2008), 1511–1523. Irrational boundary cases (algorithm is known not to terminate) p2 p2 > 2 p2 EX: A = = 1 1 p21 ✓ ◆✓ ◆ ✓ ◆ 12 M.DutourSikiri´c,A.Sch¨urmann,andF.Vallentin Note that there is always a unique choice in Step 4 in case A is a 2 2 rank-1 matrix. Note also that the vectors represent fractions that converge to ⇥p2. Every second vector corresponds to a convergent of the continued fraction of p2. For Irrationalinstance, boundary cases (algorithm is known not to terminate) 1 99/70 = 1 + . 1 2+ 1 p2 p2 > 22+ p2 EX: A = = 1 1 1 p212+ 1 ✓ ◆✓ ◆ ✓ 2+◆ 2 The -perfect matrix after ten iterations of the algorithm is COP 4756 6726 B(10) = . P 6726 9512 ✓ ◆ (i) It can be shown that the matrices BP converge to a multiple of 1 p2 B = satisfying A, B = 0 and X, B 0 for all X 2. p22 h i h i 2 CP ✓ ◆ (i) However, every one of the infinitely many perfect matrices BP satisfies X, B(i) > 0 for all X . h P i 2 CP2 5. Computational Experiments We implemented our algorithm. The source code, written in C++, is available on TODO: github [?]. In this section we report how it performs on several examples, most of them were previously discussed in the literature. TODO: The first example is the matrix A1 = ... which lies in the interior of 5. The second example is fromCP [11] and lies in the boundary of CP5 85115 58511 0 1 A2 = 15851 B11585C B C B51158C B C @ A 12 M.DutourSikiri´c,A.Sch¨urmann,andF.Vallentin Note that there is always a unique choice in Step 4 in case A is a 2 2 rank-1 matrix. Note also that the vectors represent fractions that converge to ⇥p2. Every second vector corresponds to a convergent of the continued fraction of p2. For Irrationalinstance, boundary cases (algorithm is known not to terminate) 1 99/70 = 1 + . 1 2+ 1 p2 p2 > 22+ p2 EX: A = = 1 1 1 p212+ 1 ✓ ◆✓ ◆ ✓ 2+◆ 2 The -perfect matrix after ten iterations of the algorithm is COP 4756 6726 B(10) = . P 6726 9512 ✓ ◆ (i) It can be shown that the matrices BP converge to a multiple of 1 p2 B = satisfying A, B = 0 and X, B 0 for all X 2. p22 h i h i 2 CP ✓ ◆ (i) However, every one of the infinitely many perfect matrices BP satisfies X, B(i) > 0 for all X . h P i 2 CP2 5. Computational Experiments We implemented our algorithm. The source code, written in C++, is available on TODO: github [?]. In this section we report how it performs on several examples, most of them were previously discussed in the literature. TODO: The first example is the matrix A1 = ... which lies in the interior of 5. The second example is fromCP [11] and lies in the boundary of CP5 85115 58511 0 1 A2 = 15851 B11585C B C B51158C B C @ A References • Mathieu Dutour Sikirić, Achill Schürmann and Frank Vallentin, Classification of eight dimensional perfect forms, Electron. Res. Announc. Amer. Math. Soc., 13 (2007). • Achill Schürmann, Enumerating Perfect Forms, AMS Contemporary Mathematics, 437 (2009), 359–378. • Achill Schürmann, Computational Geometry of Positive Definite Quadratic Forms, University Lecture Series, AMS, Providence, RI, 2009. • Achill Schürmann, Exploiting Symmetries in Polyhedral Computations, Fields Institute Communications, 69 (2013), 265–278. • Mathieu Dutour Sikirić, Achill Schürmann and Frank Vallentin, Rational factorizations of completely positive matrices, Linear Algebra and its Applications, 523 (2017), 46–51. • Mathieu Dutour Sikirić, Achill Schürmann and Frank Vallentin, A simplex algorithm for cp-factorization, Preprint, April 2018.