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Variations and Applications of Voronoi’s algorithm
Achill Schürmann(Universität Rostock)
( based on work with Mathieu Dutour Sikiric and Frank Vallentin )
Providence, April 2018
ICERM Conference on Computational Challenges in the Theory of Lattices
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PRELUDE Voronoi’s Algorithm
- classically -
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Lattices and Quadratic Forms
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Lattices and Quadratic Forms
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Lattices and Quadratic Forms
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Reduction Theoryfor positive definite quadratic forms
Task of a reduction theory is to provide a fundamental domain
GLn(Z) acts on Sn>0 by Q 7! UtQU
Classical reductions were obtained by Lagrange, Gauß, Korkin and Zolotareff, Minkowski and others… All the same for :n = 2
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Voronoi’s reduction idea
Observation: The fundamental domain can be obtained frompolyhedral cones that are spanned by rank-1 forms only
Georgy Voronoi (1868 – 1908)
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Voronoi’s reduction idea
Observation: The fundamental domain can be obtained frompolyhedral 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 -equivalenceGLn(Z)
Georgy Voronoi (1868 – 1908)
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Perfect Forms
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Perfect Forms
DEF: min(Q) = minx2Zn\{0}
Q[x] is the arithmetical minimum
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Perfect Forms
DEF: min(Q) = minx2Zn\{0}
Q[x] is the arithmetical minimum
DEF: Q 2 Sn>0 perfect ,
Q is uniquely determined by min(Q) and
MinQ = { x 2 Zn : Q[x] = min(Q) }
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Perfect Forms
DEF: For Q 2 Sn>0, its Voronoi cone is V(Q) = cone{xxt : x 2 MinQ}
DEF: min(Q) = minx2Zn\{0}
Q[x] is the arithmetical minimum
DEF: Q 2 Sn>0 perfect ,
Q is uniquely determined by min(Q) and
MinQ = { x 2 Zn : Q[x] = min(Q) }
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Perfect Forms
DEF: For Q 2 Sn>0, its Voronoi cone is V(Q) = cone{xxt : x 2 MinQ}
THM: Voronoi cones give a polyhedral tessellation of Sn>0
and there are only finitely many up to -equivalence. GLn(Z)
DEF: min(Q) = minx2Zn\{0}
Q[x] is the arithmetical minimum
DEF: Q 2 Sn>0 perfect ,
Q is uniquely determined by min(Q) and
MinQ = { x 2 Zn : Q[x] = min(Q) }
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Perfect Forms
DEF: For Q 2 Sn>0, its Voronoi cone is V(Q) = cone{xxt : x 2 MinQ}
THM: Voronoi cones give a polyhedral tessellation of Sn>0
and there are only finitely many up to -equivalence. GLn(Z)
(Voronoi cones are full dimensional if and only if Q is perfect!)
DEF: min(Q) = minx2Zn\{0}
Q[x] is the arithmetical minimum
DEF: Q 2 Sn>0 perfect ,
Q is uniquely determined by min(Q) and
MinQ = { x 2 Zn : Q[x] = min(Q) }
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Ryshkov Polyhedron
The set of all positive definite quadratic forms / matriceswith arithmetical minimum at least 1 is called
Ryshkov polyhedron
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Ryshkov Polyhedron
The set of all positive definite quadratic forms / matriceswith arithmetical minimum at least 1 is called
Ryshkov polyhedron
R =�Q 2 Sn
>0 : Q[x] � 1 for all x 2 Zn \ {0}
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Ryshkov Polyhedron
The set of all positive definite quadratic forms / matriceswith arithmetical minimum at least 1 is called
Ryshkov polyhedronRyshkov Polyhedra
• R is a locally finite polyhedron
• Vertices of R are perfect forms
• ↵ 7! (det(Q + ↵Q0))1n is strictly concave on Sn
>0
R is a locally finite polyhedron•
R =�Q 2 Sn
>0 : Q[x] � 1 for all x 2 Zn \ {0}
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Ryshkov Polyhedron
The set of all positive definite quadratic forms / matriceswith arithmetical minimum at least 1 is called
Ryshkov polyhedronRyshkov Polyhedra
• R is a locally finite polyhedron
• Vertices of R are perfect forms
• ↵ 7! (det(Q + ↵Q0))1n is strictly concave on Sn
>0
R is a locally finite polyhedron• Vertices of R are perfect•
R =�Q 2 Sn
>0 : Q[x] � 1 for all x 2 Zn \ {0}
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Voronoi’s algorithm
Start with a perfect form Q
1. SVP: Compute Min Q and describing inequalities of the polyhedral cone
P(Q) = { Q02 S
n : Q0[x] � 1 for all x 2 Min Q }
2. PolyRepConv: Enumerate extreme rays R1, . . . , Rk of P(Q)
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
Voronoi’s Algorithm
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Computational Results
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Computational Results
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5000000 Wessel van Woerden, 2018 ?!
Computational Results
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Adjacency Decomposition Method(for vertex enumeration)
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Adjacency Decomposition Method
• Find initial orbit(s) / representing vertice(s)
(for vertex enumeration)
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Adjacency Decomposition Method
• Find initial orbit(s) / representing vertice(s)
• For each new orbit representative
• enumerate neighboring vertices
(for vertex enumeration)
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Adjacency Decomposition Method
• Find initial orbit(s) / representing vertice(s)
• For each new orbit representative
• enumerate neighboring vertices
• add as orbit representative if in a new orbit
(for vertex enumeration)
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Adjacency Decomposition Method
• Find initial orbit(s) / representing vertice(s)
• For each new orbit representative
• enumerate neighboring vertices
• add as orbit representative if in a new orbit
Representation conversion problem
(up to symmetry)
(for vertex enumeration)
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Adjacency Decomposition Method
• Find initial orbit(s) / representing vertice(s)
• For each new orbit representative
• enumerate neighboring vertices
• add as orbit representative if in a new orbit
Representation conversion problem
(up to symmetry)
BOTTLENECK: Stabilizer and In-Orbit computations
=> Need of efficient data structures and algorithms forpermutation groups: BSGS, (partition) backtracking
(for vertex enumeration)
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Representation Conversionin practice
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Representation Conversionin practice
Best known Algorithm:
Mathieu
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Representation Conversionin practice
Best known Algorithm:
Mathieu
• helps to compute linear automorphism groups
• converts polyhedral representations using
Recursive Decomposition Methods (Incidence/Adjacency)
also available through polymake
Thomas Rehn(Phd 2014)
A C++-Tool
http://www.geometrie.uni-rostock.de/software/
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Applicaton: Lattice Sphere Packings
The lattice sphere packing problem can be phrased as:
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Applicaton: Lattice Sphere Packings
The lattice sphere packing problem can be phrased as:
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Part II: Koecher’s generalization
and T-perfect forms
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Koecher’s generalization
Max Koecher, 1924-1990
1960/61 Max Koecher generalized Voronoi’s reduction theory and proofs to a setting with a self-dual cone C
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Koecher’s generalization
Max Koecher, 1924-1990
1960/61 Max Koecher generalized Voronoi’s reduction theory and proofs to a setting with a self-dual cone C
Under certain conditions, he shows that C is covered by a tessellation of polyhedral Voronoi cones and “approximated from inside“ by a Ryshkov polyhedron
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Koecher’s generalization
Max Koecher, 1924-1990
1960/61 Max Koecher generalized Voronoi’s reduction theory and proofs to a setting with a self-dual cone C
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 GLn(OK) on suitable quadratic spaces
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Applications in Math
Ryshkov Polyhedron
���(O�) �������symmetric
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Applications in Math
•
Vertices / Perfect Forms:
Polyhedral complex:
• Reduction theory
Representation Conversion
• Hermite constant
• Hecke operators
• Compactifications of moduli spaces of Abelian varieties
������� ���(O�)• Cohomology of
Ryshkov Polyhedron
���(O�) �������symmetric
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Applications in Math
•
Vertices / Perfect Forms:
Polyhedral complex:
• Reduction theory
Representation Conversion
• Hermite constant
• Hecke operators
• Compactifications of moduli spaces of Abelian varieties
������� ���(O�)• Cohomology of
Ryshkov Polyhedron
���(O�) �������symmetric
See Mathieu’s talk after the coffee break!
AIM Square group 2012: Gangl, Dutour Sikirić, Schürmann, Gunnells, Yasaki, Hanke
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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
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Equivariant theory
For a finite group G ⇢ GLn(Z) the space of invariant forms
TG := { Q 2 Sn : G ⇢ Aut Q }
is a linear subspace of Sn; TG \ Sn>0 is called Bravais space
IDEA (Berge, Martinet, Sigrist, 1992):
Intersect Ryshkov polyhedron R with a linear subspace T ⇢ Sn
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
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Equivariant theory
For a finite group G ⇢ GLn(Z) the space of invariant forms
TG := { Q 2 Sn : G ⇢ Aut Q }
is a linear subspace of Sn; TG \ Sn>0 is called Bravais space
IDEA (Berge, Martinet, Sigrist, 1992):
Intersect Ryshkov polyhedron R with a linear subspace T ⇢ Sn
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
T-perfect and T-extreme formsDEF: Q 2 T \ S
n>0
• is T -extreme if it attains a loc. max. of � within T
• is T -perfect if it is a vertex of R \ T
• is T -eutactic if Q�1| T 2 relint(V(Q) | T )
THM (BMS, 1992): Q T -extreme , Q T -perfect and T -eutactic
• Q,Q02 T \ S
n>0 are called T -equivalent, if 9U 2 GLn(Z) with
Q0 = U tQU and T = U tTU
THM (Jaquet-Chiffelle, 1995): { TG-perfect Q : �(Q) = 1 } / ⇠TGfinite
) Voronoi’s algorithm can be applied to R \ TG
T-perfect and T-extreme formsDEF: Q 2 T \ S
n>0
• is T -extreme if it attains a loc. max. of � within T
• is T -perfect if it is a vertex of R \ T
• is T -eutactic if Q�1| T 2 relint(V(Q) | T )
THM (BMS, 1992): Q T -extreme , Q T -perfect and T -eutactic
• Q,Q02 T \ S
n>0 are called T -equivalent, if 9U 2 GLn(Z) with
Q0 = U tQU and T = U tTU
THM (Jaquet-Chiffelle, 1995): { TG-perfect Q : �(Q) = 1 } / ⇠TGfinite
) Voronoi’s algorithm can be applied to R \ TG
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T-Algorithm
SVPs: Obtain a T -perfect form Q
1. SVP: Compute Min Q and describing inequalities of the polyhedral cone
P(Q) = { Q02 T : Q0[x] � 1 for all x 2 Min Q }
2. PolyRepConv: Enumerate extreme rays R1, . . . , Rk of P(Q)
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
Voronoi’s Algorithmfor a linear subspace T
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T-Algorithm
SVPs: Obtain a T -perfect form Q
1. SVP: Compute Min Q and describing inequalities of the polyhedral cone
P(Q) = { Q02 T : Q0[x] � 1 for all x 2 Min Q }
2. PolyRepConv: Enumerate extreme rays R1, . . . , Rk of P(Q)
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
Voronoi’s Algorithmfor a linear subspace T
Possibleexistence of“Dead-Ends“(for PQFs R)
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Equivariant theory
For a finite group G ⇢ GLn(Z) the space of invariant forms
TG := { Q 2 Sn : G ⇢ Aut Q }
is a linear subspace of Sn; TG \ Sn>0 is called Bravais space
IDEA (Berge, Martinet, Sigrist, 1992):
Intersect Ryshkov polyhedron R with a linear subspace T ⇢ Sn
G-invariant theory
T-perfect and T-extreme formsDEF: Q 2 T \ S
n>0
• is T -extreme if it attains a loc. max. of � within T
• is T -perfect if it is a vertex of R \ T
• is T -eutactic if Q�1| T 2 relint(V(Q) | T )
THM (BMS, 1992): Q T -extreme , Q T -perfect and T -eutactic
• Q,Q02 T \ S
n>0 are called T -equivalent, if 9U 2 GLn(Z) with
Q0 = U tQU and T = U tTU
THM (Jaquet-Chiffelle, 1995): { TG-perfect Q : �(Q) = 1 } / ⇠TGfinite
T-perfect and T-extreme formsDEF: Q 2 T \ S
n>0
• is T -extreme if it attains a loc. max. of � within T
• is T -perfect if it is a vertex of R \ T
• is T -eutactic if Q�1| T 2 relint(V(Q) | T )
THM (BMS, 1992): Q T -extreme , Q T -perfect and T -eutactic
• Q,Q02 T \ S
n>0 are called T -equivalent, if 9U 2 GLn(Z) with
Q0 = U tQU and T = U tTU
THM (Jaquet-Chiffelle, 1995): { TG-perfect Q : �(Q) = 1 } / ⇠TGfinite
T-perfect and T-extreme formsDEF: Q 2 T \ S
n>0
• is T -extreme if it attains a loc. max. of � within T
• is T -perfect if it is a vertex of R \ T
• is T -eutactic if Q�1| T 2 relint(V(Q) | T )
THM (BMS, 1992): Q T -extreme , Q T -perfect and T -eutactic
• Q,Q02 T \ S
n>0 are called T -equivalent, if 9U 2 GLn(Z) with
Q0 = U tQU and T = U tTU
THM (Jaquet-Chiffelle, 1995): { TG-perfect Q : �(Q) = 1 } / ⇠TGfinite
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Examples/Applications
n 2 4 6 8 10 12# E-perfect 1 1 2 5 1628 ?maximum � 0.9069 . . . 0.6168 . . . 0.3729 . . . 0.2536 . . . 0.0360 . . .
Perfect Eisenstein forms
n 2 4 6 8 10 12# G-perfect 1 1 1 2 � 8192 ?maximum � 0.7853 . . . 0.6168 . . . 0.3229 . . . 0.2536 . . .
Perfect Gaussian forms
n 4 8 12 16# Q-perfect 1 1 8 ?maximum � 0.6168 . . . 0.2536 . . . 0.03125 . . .
Perfect Quaternion forms
Applicaton: Lattice Sphere Packingswith prescribed symmetry
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PART III: A new Generalization
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Further Generalization? … and application!
IDEA: Generalize Voronoi’s theory to other convex cones and their duals
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Further Generalization? … and application!
IDEA: Generalize Voronoi’s theory to other convex cones and their duals
In particular to the completely positive cone
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CPn ⇢ Sn>0 ⇢ COPn
Further Generalization? … and application!
IDEA: Generalize Voronoi’s theory to other convex cones and their duals
A SIMPLEX ALGORITHM FOR RATIONAL
CP-FACTORIZATION
MATHIEU DUTOUR SIKIRIC, ACHILL SCHURMANN, AND FRANK VALLENTIN
Abstract. We describe and analyze an algorithmic procedure, similar to thesimplex algorithm for linear programming, which tests whether or not a givensymmetric matrix is completely positive. For matrices in the rational closureof the cone of completely positive matrices our procedure computes a rationalcp-factorization. For matrices which are not completely positive a certificatein form of a separating hyperplane is computed, whenever our procedure ter-minates. We conjecture that there exists a pivot rule so that our procedurealways terminates for rational input matrices.
1. Introduction
Copositive programming gives a common framework to formulate many di�cultoptimization problems as convex conic ones. In fact, many NP-hard problemsare known to have such reformulations (see for example the surveys [2, 8]). Allthe di�culty of these problems appears to be “converted” into the di�culty ofunderstanding the cone of copositive matrices COPn which consists of all symmetricn⇥ n matrices B 2 S
n with xTBx � 0 for all x 2 Rn�0. Its dual cone is the cone
CPn = cone{xxT : x 2 Rn�0}
of completely positive n ⇥ n matrices. Therefore, it seems no surprise that manybasic questions about this cone are still open and appear to be di�cult.
One important problem is to find an algorithmic test deciding whether or nota given symmetric matrix A is completely positive. If possible one would like toobtain a certificate for either A 2 CPn or A 62 CPn. In terms of the definitions themost natural certificate for A 2 CPn is giving a cp-factorization
A =mX
i=1
xixTi with x1, . . . , xm 2 Rn
�0
and for A 62 CPn giving a separating hyperplanes defined by B 2 COPn so thathB,Ai < 0. Dickinson and Gijben [5] showed that this (strong) membership prob-lem is NP-hard.
Date: April 15, 2018.2010 Mathematics Subject Classification. 90C20.Key words and phrases. copositive programming, complete positivity, matrix factorization,
copositive minimum.The first author was supported by the Humboldt foundation. The third author is partially
supported by the SFB/TRR 191 “Symplectic Structures in Geometry, Algebra and Dynamics”,funded by the DFG.
1
and its dual, the copositive cone
2 M. Dutour Sikiric, A. Schurmann, and F. Vallentin
From the algorithmic side some successful ideas have been proposed by Jarreand Schmallowsky [14], Nie [17], Sponsel and Dur [20], Groetzner and Dur [11],and Anstreicher, Burer, and Dickinson [4, Section 3.3].
In this paper we describe a new procedure that works for all matrices in therational closure
˜CPn = cone{xxT : x 2 Qn�0}
of the cone of completely positive matrices. Moreover, it also computes certificatesin case A 62 CPn.
In contrast to most of the other approaches mentioned above (the exception beingthe approach by Anstreicher, Burer and Dickinson whose factorization method isbased on the ellipsoid method), our algorithm is exact in the sense that it usesrational numbers only if the input matrix is rational and so does not have to copewith numerical instabilities. In particular it finds a rational cp-factorization if itexists. However, questions of convergence remain, in particular for all A which arenot contained in the “irrational boundary part” (bd CPn) \ ˜CPn.
2. The copositive minimum and copositive perfect matrices
By Sn we denote the Euclidean vector space of symmetric n ⇥ n matrices with
inner product hA,Bi = Trace(AB) =Pn
i,j=1 AijBij . With respect to this innerproduct we have the following duality relations between the cone of copositivematrices and and the cone of completely positive matrices
COPn = (CPn)⇤ = {B 2 S
n : hA,Bi � 0 for all A 2 CPn},
andCPn = (COPn)
⇤.
So, in order to show that a given symmetric matrix A is not completely positive, itsu�ces to find a copositive matrix B 2 COPn with hB,Ai < 0. We call B a sepa-rating witness for A 62 CPn in this case, because the linear hyperplane orthogonalto B separates A and CPn.
Using the notation B[x] for xTBx = hB, xxTi, we obtain
COPn = {B 2 Sn : B[x] � 0 for all x 2 Rn
�0}.
Definition 2.1. For a symmetric matrix B 2 Sn we define the copositive minimum
asminCOP(B) = inf
�B[v] : v 2 Zn
�0 \ {0} ,
and we denote the set of vectors attaining it by
MinCOP(B) =�v 2 Zn
�0 : B[v] = minCOP(B) .
The following proposition shows that matrices in the interior of the cone ofcopositive matrices attain their copositive minimum.
Lemma 2.2. Let B be a matrix in the interior of the cone of copositive matrices.Then, the copositive minimum of B is strictly positive and it is attained by onlyfinitely many vectors.
Proof. Since B is copositive, we have the inequality minCOP(B) � 0. Suppose thatminCOP(B) = 0. Then there is a sequence vi 2 Zn
�0 \{0} of pairwise distinct latticepoints such that B[vi] tends to zero when i tends to infinity. From the sequencevi we construct a new sequence ui of points on the unit sphere Sn�1 by setting
2 M. Dutour Sikiric, A. Schurmann, and F. Vallentin
From the algorithmic side some successful ideas have been proposed by Jarreand Schmallowsky [14], Nie [17], Sponsel and Dur [20], Groetzner and Dur [11],and Anstreicher, Burer, and Dickinson [4, Section 3.3].
In this paper we describe a new procedure that works for all matrices in therational closure
˜CPn = cone{xxT : x 2 Qn�0}
of the cone of completely positive matrices. Moreover, it also computes certificatesin case A 62 CPn.
In contrast to most of the other approaches mentioned above (the exception beingthe approach by Anstreicher, Burer and Dickinson whose factorization method isbased on the ellipsoid method), our algorithm is exact in the sense that it usesrational numbers only if the input matrix is rational and so does not have to copewith numerical instabilities. In particular it finds a rational cp-factorization if itexists. However, questions of convergence remain, in particular for all A which arenot contained in the “irrational boundary part” (bd CPn) \ ˜CPn.
2. The copositive minimum and copositive perfect matrices
By Sn we denote the Euclidean vector space of symmetric n ⇥ n matrices with
inner product hA,Bi = Trace(AB) =Pn
i,j=1 AijBij . With respect to this innerproduct we have the following duality relations between the cone of copositivematrices and and the cone of completely positive matrices
COPn = (CPn)⇤ = {B 2 S
n : hA,Bi � 0 for all A 2 CPn},
andCPn = (COPn)
⇤.
So, in order to show that a given symmetric matrix A is not completely positive, itsu�ces to find a copositive matrix B 2 COPn with hB,Ai < 0. We call B a sepa-rating witness for A 62 CPn in this case, because the linear hyperplane orthogonalto B separates A and CPn.
Using the notation B[x] for xTBx = hB, xxTi, we obtain
COPn = {B 2 Sn : B[x] � 0 for all x 2 Rn
�0}.
Definition 2.1. For a symmetric matrix B 2 Sn we define the copositive minimum
asminCOP(B) = inf
�B[v] : v 2 Zn
�0 \ {0} ,
and we denote the set of vectors attaining it by
MinCOP(B) =�v 2 Zn
�0 : B[v] = minCOP(B) .
The following proposition shows that matrices in the interior of the cone ofcopositive matrices attain their copositive minimum.
Lemma 2.2. Let B be a matrix in the interior of the cone of copositive matrices.Then, the copositive minimum of B is strictly positive and it is attained by onlyfinitely many vectors.
Proof. Since B is copositive, we have the inequality minCOP(B) � 0. Suppose thatminCOP(B) = 0. Then there is a sequence vi 2 Zn
�0 \{0} of pairwise distinct latticepoints such that B[vi] tends to zero when i tends to infinity. From the sequencevi we construct a new sequence ui of points on the unit sphere Sn�1 by setting
In particular to the completely positive cone
hA,Bi = h`�+2(A ·B) /2MQi2b i?2 bi�M/�`/ BMM2` T`Q/m+i QM Sn
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Application: Copositive Optimization
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Application: Copositive Optimization
Copositive optimization problems are convex conic problems • min hC,Qi such that hQ, Aii = bi, i = 1, . . . ,m
and Q 2 CONE
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Application: Copositive Optimization
Copositive optimization problems are convex conic problems • min hC,Qi such that hQ, Aii = bi, i = 1, . . . ,m
and Q 2 CONE
Linear Programming (LP)
CONE = Rn�0
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Application: Copositive Optimization
Copositive optimization problems are convex conic problems • min hC,Qi such that hQ, Aii = bi, i = 1, . . . ,m
and Q 2 CONE
Linear Programming (LP)
CONE = Rn�0
Semidefinite Programming (SDP)CONE = Sn
�0
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Application: Copositive Optimization
Copositive optimization problems are convex conic problems • min hC,Qi such that hQ, Aii = bi, i = 1, . . . ,m
and Q 2 CONE
Linear Programming (LP)
CONE = Rn�0
Copositive Programming (CP)CONE = CPn or COPn
Semidefinite Programming (SDP)CONE = Sn
�0
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Application: Copositive Optimization
Copositive optimization problems are convex conic problems • min hC,Qi such that hQ, Aii = bi, i = 1, . . . ,m
and Q 2 CONE
Linear Programming (LP)
CONE = Rn�0
Copositive Programming (CP)CONE = CPn or COPn
Semidefinite Programming (SDP)CONE = Sn
�0NP-hard (2000)
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Application: Copositive Optimization
Copositive optimization problems are convex conic problems • min hC,Qi such that hQ, Aii = bi, i = 1, . . . ,m
and Q 2 CONE
Linear Programming (LP)
CONE = Rn�0
Copositive Programming (CP)CONE = CPn or COPn
Semidefinite Programming (SDP)CONE = Sn
�0NP-hard (2000)
Such problems have a duality theory and allow certificates for solutions!
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cp-factorizations and certificates
DEF: A finite set X ⇢ Rn�0 is called a certificate
if it gives a cp-factorization Q =X
x2Xxx>
for Q 2 Sn being completely positive,
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cp-factorizations and certificates
DEF:
PROBLEM: How to find a cp-factorization for a given Q ?
A finite set X ⇢ Rn�0 is called a certificate
if it gives a cp-factorization Q =X
x2Xxx>
for Q 2 Sn being completely positive,
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cp-factorizations and certificates
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)
DEF:
PROBLEM: How to find a cp-factorization for a given Q ?
A finite set X ⇢ Rn�0 is called a certificate
if it gives a cp-factorization Q =X
x2Xxx>
for Q 2 Sn being completely positive,
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cp-factorizations and certificates
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)
DEF:
PROBLEM: How to find a cp-factorization for a given Q ?
Non of these approaches is exact and latter do not even guarantee to find solutions!
A finite set X ⇢ Rn�0 is called a certificate
if it gives a cp-factorization Q =X
x2Xxx>
for Q 2 Sn being completely positive,
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Copositive minimum
DEF: minCOP Q = minx2Zn
�0\{0}Q[x] is the copositive minimum
(COP-SVP)
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Copositive minimum
DEF: minCOP Q = minx2Zn
�0\{0}Q[x] is the copositive minimum
Difficult to compute!
(COP-SVP)
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Copositive minimum
DEF: minCOP Q = minx2Zn
�0\{0}Q[x] is the copositive minimum
Difficult to compute!
in the standard simplex � =�x 2 Rn
�0 : x1 + . . . xn = 1
THM: (Bundfuss and Dür, 2008)
such that each �k has vertices v1, . . . vn with v>i Qvj > 0
For Q 2 int COPn we can construct a family of simplices �k
(COP-SVP)
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Copositive minimum
DEF: minCOP Q = minx2Zn
�0\{0}Q[x] is the copositive minimum
Difficult to compute!
Computation in practice:
”Fincke-Pohst strategy” to compute minCOP Q in each cone�k
in the standard simplex � =�x 2 Rn
�0 : x1 + . . . xn = 1
THM: (Bundfuss and Dür, 2008)
such that each �k has vertices v1, . . . vn with v>i Qvj > 0
For Q 2 int COPn we can construct a family of simplices �k
(COP-SVP)
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Generalized Ryshkov polyhedron
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Generalized Ryshkov polyhedron The set of all copositive quadratic forms / matrices
with copositive minimum at least 1 is called Ryshkov polyhedron
R =�Q 2 COPn : Q[x] � 1 for all x 2 Zn
�0 \ {0}
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Generalized Ryshkov polyhedron The set of all copositive quadratic forms / matrices
with copositive minimum at least 1 is called Ryshkov polyhedron
R =�Q 2 COPn : Q[x] � 1 for all x 2 Zn
�0 \ {0}
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Generalized Ryshkov polyhedron
DEF: Q 2 int COPn is called COP-perfect if and only if
Q is uniquely determined by minCOP Q and
MinCOPQ =�x 2 Zn
�0 : Q[x] = minCOPQ
The set of all copositive quadratic forms / matriceswith copositive minimum at least 1 is called
Ryshkov polyhedron
R =�Q 2 COPn : Q[x] � 1 for all x 2 Zn
�0 \ {0}
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Generalized Ryshkov polyhedron
DEF: Q 2 int COPn is called COP-perfect if and only if
Q is uniquely determined by minCOP Q and
MinCOPQ =�x 2 Zn
�0 : Q[x] = minCOPQ
The set of all copositive quadratic forms / matriceswith copositive minimum at least 1 is called
Ryshkov polyhedron
R =�Q 2 COPn : Q[x] � 1 for all x 2 Zn
�0 \ {0}
R is a locally finite polyhedron• (with dead-ends / rays)
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Generalized Ryshkov polyhedron
DEF: Q 2 int COPn is called COP-perfect if and only if
Q is uniquely determined by minCOP Q and
MinCOPQ =�x 2 Zn
�0 : Q[x] = minCOPQ
The set of all copositive quadratic forms / matriceswith copositive minimum at least 1 is called
Ryshkov polyhedron
R =�Q 2 COPn : Q[x] � 1 for all x 2 Zn
�0 \ {0}
Vertices of R are COP-perfect• R is a locally finite polyhedron• (with dead-ends / rays)
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Voronoi-type simplex algorithm Input: A 2 Sn
>0
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Voronoi-type simplex algorithm Input: A 2 Sn
>0
COP-SVPs: Obtain an initial COP-perfect matrix BP
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Voronoi-type simplex algorithm Input: A 2 Sn
>0
COP-SVPs: Obtain an initial COP-perfect matrix BP
1. B7 hBP, Ai < 0 i?2M QmiTmi A 62 CPn (with witness BP)
2. LP: B7 A 2 cone�xx> : x 2 MinCOPBP
i?2M QmiTmi A 2 ˜CPn
3. COP-SVP: Compute MinCOPBP and the polyhedral cone
P(BP) = { B 2 Sn : B[x] � 1 for all x 2 MinCOPBP }
4. PolyRepConv: Determine a generator R of an extreme ray of P(BP)with hA, Ri < 0.
5. LPs: B7 R 2 COPn i?2M QmiTmi A 62 CPn (with witness R)
6. COP-SVPs: Determine the contiguous COP-perfect matrix
BN := BP + �R with � > 0 and minCOPBN = 1
7. Set BP := BN and ;QiQ 1.
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Voronoi-type simplex algorithm Input: A 2 Sn
>0
COP-SVPs: Obtain an initial COP-perfect matrix BP
˜CPn = cone�xx> : x 2 Qn
1. B7 hBP, Ai < 0 i?2M QmiTmi A 62 CPn (with witness BP)
2. LP: B7 A 2 cone�xx> : x 2 MinCOPBP
i?2M QmiTmi A 2 ˜CPn
3. COP-SVP: Compute MinCOPBP and the polyhedral cone
P(BP) = { B 2 Sn : B[x] � 1 for all x 2 MinCOPBP }
4. PolyRepConv: Determine a generator R of an extreme ray of P(BP)with hA, Ri < 0.
5. LPs: B7 R 2 COPn i?2M QmiTmi A 62 CPn (with witness R)
6. COP-SVPs: Determine the contiguous COP-perfect matrix
BN := BP + �R with � > 0 and minCOPBN = 1
7. Set BP := BN and ;QiQ 1.
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Voronoi-type simplex algorithm Input: A 2 Sn
>0
COP-SVPs: Obtain an initial COP-perfect matrix BP
˜CPn = cone�xx> : x 2 Qn
1. B7 hBP, Ai < 0 i?2M QmiTmi A 62 CPn (with witness BP)
2. LP: B7 A 2 cone�xx> : x 2 MinCOPBP
i?2M QmiTmi A 2 ˜CPn
3. COP-SVP: Compute MinCOPBP and the polyhedral cone
P(BP) = { B 2 Sn : B[x] � 1 for all x 2 MinCOPBP }
4. PolyRepConv: Determine a generator R of an extreme ray of P(BP)with hA, Ri < 0.
5. LPs: B7 R 2 COPn i?2M QmiTmi A 62 CPn (with witness R)
6. COP-SVPs: Determine the contiguous COP-perfect matrix
BN := BP + �R with � > 0 and minCOPBN = 1
7. Set BP := BN and ;QiQ 1.
( flexible ”pivot-rule” )
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A copositive starting point
4 M. Dutour Sikiric, A. Schurmann, and F. Vallentin
Definition 2.3. A copositive matrix B 2 int(COPn) is called COP-perfect if it isuniquely determined by its copositive minimum minCOP B and the set MinCOP Battaining it.
In other words, B 2 int(COPn) is COP-perfect if and only if it is the uniquesolution of the system of linear equations
hB, vvTi = minCOP B, for all v 2 MinCOP B.
In particular all vertices of R are COP-perfect.
Lemma 2.4. COP-perfect matrices exist in all dimensions (dimension n = 1 beingtrivial): For dimension n � 2 the following matrix
(4) QAn =
0
BBBBBBB@
2 �1 0 . . . 0
�1 2. . .
. . ....
0. . .
. . .. . . 0
.... . .
. . . 2 �10 . . . 0 �1 2
1
CCCCCCCA
is COP-perfect; 12QAn is a vertex of R.
Proof. Matrix QAn is positive definite since
QAn [x] = x21 +
n�1X
i=1
(xi � xi+1)2 + x2
n for x 2 R.
In particular it lies in the interior of the copositive cone. Furthermore,
minCOP QAn = 2 with MinCOP QAn =
8<
:
kX
i=j
ej : 1 j k n
9=
; ,
where ej is the j-th standard unit basis vector of Rn. Thus, the�n+1
2
�vectors at-
taining the copositive minimum have a continued sequence of 1s in their coordinatesand otherwise 0s. Now it is easy to see that the rank-1-matrices
0
@kX
i=j
ej
1
A
0
@kX
i=j
ej
1
AT
, where 1 j k n,
are linearly independent and span the space of symmetric matrices which showsthat QAn is COP-perfect. ⇤
The matrix QAn is also known as a Gram matrix of the root lattice An, a veryimportant 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 algorithmis similar to the simplex algorithm for linear programming, as it walks along apath of subsequently constructed COP-perfect matrices, which are vertices of thepolyhedral set R and which are connected by edges.
is COP-perfectTHM:
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A copositive starting point
4 M. Dutour Sikiric, A. Schurmann, and F. Vallentin
Definition 2.3. A copositive matrix B 2 int(COPn) is called COP-perfect if it isuniquely determined by its copositive minimum minCOP B and the set MinCOP Battaining it.
In other words, B 2 int(COPn) is COP-perfect if and only if it is the uniquesolution of the system of linear equations
hB, vvTi = minCOP B, for all v 2 MinCOP B.
In particular all vertices of R are COP-perfect.
Lemma 2.4. COP-perfect matrices exist in all dimensions (dimension n = 1 beingtrivial): For dimension n � 2 the following matrix
(4) QAn =
0
BBBBBBB@
2 �1 0 . . . 0
�1 2. . .
. . ....
0. . .
. . .. . . 0
.... . .
. . . 2 �10 . . . 0 �1 2
1
CCCCCCCA
is COP-perfect; 12QAn is a vertex of R.
Proof. Matrix QAn is positive definite since
QAn [x] = x21 +
n�1X
i=1
(xi � xi+1)2 + x2
n for x 2 R.
In particular it lies in the interior of the copositive cone. Furthermore,
minCOP QAn = 2 with MinCOP QAn =
8<
:
kX
i=j
ej : 1 j k n
9=
; ,
where ej is the j-th standard unit basis vector of Rn. Thus, the�n+1
2
�vectors at-
taining the copositive minimum have a continued sequence of 1s in their coordinatesand otherwise 0s. Now it is easy to see that the rank-1-matrices
0
@kX
i=j
ej
1
A
0
@kX
i=j
ej
1
AT
, where 1 j k n,
are linearly independent and span the space of symmetric matrices which showsthat QAn is COP-perfect. ⇤
The matrix QAn is also known as a Gram matrix of the root lattice An, a veryimportant 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 algorithmis similar to the simplex algorithm for linear programming, as it walks along apath of subsequently constructed COP-perfect matrices, which are vertices of thepolyhedral set R and which are connected by edges.
is COP-perfectTHM:
![Page 80: Variations and Applications of Voronoi’s algorithm](https://reader031.vdocument.in/reader031/viewer/2022012916/61c6e01e205dbf35826b28e9/html5/thumbnails/80.jpg)
A copositive starting point
4 M. Dutour Sikiric, A. Schurmann, and F. Vallentin
Definition 2.3. A copositive matrix B 2 int(COPn) is called COP-perfect if it isuniquely determined by its copositive minimum minCOP B and the set MinCOP Battaining it.
In other words, B 2 int(COPn) is COP-perfect if and only if it is the uniquesolution of the system of linear equations
hB, vvTi = minCOP B, for all v 2 MinCOP B.
In particular all vertices of R are COP-perfect.
Lemma 2.4. COP-perfect matrices exist in all dimensions (dimension n = 1 beingtrivial): For dimension n � 2 the following matrix
(4) QAn =
0
BBBBBBB@
2 �1 0 . . . 0
�1 2. . .
. . ....
0. . .
. . .. . . 0
.... . .
. . . 2 �10 . . . 0 �1 2
1
CCCCCCCA
is COP-perfect; 12QAn is a vertex of R.
Proof. Matrix QAn is positive definite since
QAn [x] = x21 +
n�1X
i=1
(xi � xi+1)2 + x2
n for x 2 R.
In particular it lies in the interior of the copositive cone. Furthermore,
minCOP QAn = 2 with MinCOP QAn =
8<
:
kX
i=j
ej : 1 j k n
9=
; ,
where ej is the j-th standard unit basis vector of Rn. Thus, the�n+1
2
�vectors at-
taining the copositive minimum have a continued sequence of 1s in their coordinatesand otherwise 0s. Now it is easy to see that the rank-1-matrices
0
@kX
i=j
ej
1
A
0
@kX
i=j
ej
1
AT
, where 1 j k n,
are linearly independent and span the space of symmetric matrices which showsthat QAn is COP-perfect. ⇤
The matrix QAn is also known as a Gram matrix of the root lattice An, a veryimportant 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 algorithmis similar to the simplex algorithm for linear programming, as it walks along apath of subsequently constructed COP-perfect matrices, which are vertices of thepolyhedral set R and which are connected by edges.
is COP-perfectTHM:
4 M. Dutour Sikiric, A. Schurmann, and F. Vallentin
Definition 2.3. A copositive matrix B 2 int(COPn) is called COP-perfect if it isuniquely determined by its copositive minimum minCOP B and the set MinCOP Battaining it.
In other words, B 2 int(COPn) is COP-perfect if and only if it is the uniquesolution of the system of linear equations
hB, vvTi = minCOP B, for all v 2 MinCOP B.
In particular all vertices of R are COP-perfect.
Lemma 2.4. COP-perfect matrices exist in all dimensions (dimension n = 1 beingtrivial): For dimension n � 2 the following matrix
(4) QAn =
0
BBBBBBB@
2 �1 0 . . . 0
�1 2. . .
. . ....
0. . .
. . .. . . 0
.... . .
. . . 2 �10 . . . 0 �1 2
1
CCCCCCCA
is COP-perfect; 12QAn is a vertex of R.
Proof. Matrix QAn is positive definite since
QAn [x] = x21 +
n�1X
i=1
(xi � xi+1)2 + x2
n for x 2 R.
In particular it lies in the interior of the copositive cone. Furthermore,
minCOP QAn = 2 with MinCOP QAn =
8<
:
kX
i=j
ej : 1 j k n
9=
; ,
where ej is the j-th standard unit basis vector of Rn. Thus, the�n+1
2
�vectors at-
taining the copositive minimum have a continued sequence of 1s in their coordinatesand otherwise 0s. Now it is easy to see that the rank-1-matrices
0
@kX
i=j
ej
1
A
0
@kX
i=j
ej
1
AT
, where 1 j k n,
are linearly independent and span the space of symmetric matrices which showsthat QAn is COP-perfect. ⇤
The matrix QAn is also known as a Gram matrix of the root lattice An, a veryimportant 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 algorithmis similar to the simplex algorithm for linear programming, as it walks along apath of subsequently constructed COP-perfect matrices, which are vertices of thepolyhedral set R and which are connected by edges.
![Page 81: Variations and Applications of Voronoi’s algorithm](https://reader031.vdocument.in/reader031/viewer/2022012916/61c6e01e205dbf35826b28e9/html5/thumbnails/81.jpg)
Interior cases
EX:
(algorithm terminates)
A =
✓6 33 2
◆
![Page 82: Variations and Applications of Voronoi’s algorithm](https://reader031.vdocument.in/reader031/viewer/2022012916/61c6e01e205dbf35826b28e9/html5/thumbnails/82.jpg)
Interior cases
EX:
(algorithm terminates)
A =
✓6 33 2
◆
![Page 83: Variations and Applications of Voronoi’s algorithm](https://reader031.vdocument.in/reader031/viewer/2022012916/61c6e01e205dbf35826b28e9/html5/thumbnails/83.jpg)
Interior cases
EX:
(algorithm terminates)
A =
✓6 33 2
◆
![Page 84: Variations and Applications of Voronoi’s algorithm](https://reader031.vdocument.in/reader031/viewer/2022012916/61c6e01e205dbf35826b28e9/html5/thumbnails/84.jpg)
Interior cases
EX:
(algorithm terminates)
A =
✓6 33 2
◆
![Page 85: Variations and Applications of Voronoi’s algorithm](https://reader031.vdocument.in/reader031/viewer/2022012916/61c6e01e205dbf35826b28e9/html5/thumbnails/85.jpg)
Interior cases
EX:
(algorithm terminates)
A =
✓6 33 2
◆
A =
✓10
◆✓10
◆>+
✓11
◆✓11
◆>+
✓21
◆✓21
◆>
Starting with QA2 one iteration of the algorithm finds
the COP-perfect matrix BP =✓
1 �3/2�3/2 3
◆and
![Page 86: Variations and Applications of Voronoi’s algorithm](https://reader031.vdocument.in/reader031/viewer/2022012916/61c6e01e205dbf35826b28e9/html5/thumbnails/86.jpg)
12 M. Dutour Sikiric, A. Schurmann, and F. Vallentin
Note that there is always a unique choice in Step 4 in case A is a 2 ⇥ 2 rank-1matrix. Note also that the vectors represent fractions that converge to
p2. Every
second vector corresponds to a convergent of the continued fraction ofp2. For
instance,
99/70 = 1 +1
2 +1
2 +1
2 +1
2 +1
2
.
The COP-perfect matrix after ten iterations of the algorithm is
B(10)P =
✓4756 �6726�6726 9512
◆.
It can be shown that the matrices B(i)P converge to a multiple of
B =
✓1 �
p2
�p2 2
◆satisfying hA,Bi = 0 and hX,Bi � 0 for all X 2 CP2.
However, every one of the infinitely many perfect matrices B(i)P satisfies
hX,B(i)P i > 0 for all X 2 CP2.
5. Computational Experiments
We implemented our algorithm. The source code, written in C++, is available onTODO: 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 CP5.The second example is from [11] and lies in the boundary of CP5
A2 =
0
BBBB@
8 5 1 1 55 8 5 1 11 5 8 5 11 1 5 8 55 1 1 5 8
1
CCCCAEX:
(algorithm terminates with a suitable pivot-rule)
from Groetzner, Dür (2018)
not solved by their algorithms
Boundary cases from ˜CPn
![Page 87: Variations and Applications of Voronoi’s algorithm](https://reader031.vdocument.in/reader031/viewer/2022012916/61c6e01e205dbf35826b28e9/html5/thumbnails/87.jpg)
12 M. Dutour Sikiric, A. Schurmann, and F. Vallentin
Note that there is always a unique choice in Step 4 in case A is a 2 ⇥ 2 rank-1matrix. Note also that the vectors represent fractions that converge to
p2. Every
second vector corresponds to a convergent of the continued fraction ofp2. For
instance,
99/70 = 1 +1
2 +1
2 +1
2 +1
2 +1
2
.
The COP-perfect matrix after ten iterations of the algorithm is
B(10)P =
✓4756 �6726�6726 9512
◆.
It can be shown that the matrices B(i)P converge to a multiple of
B =
✓1 �
p2
�p2 2
◆satisfying hA,Bi = 0 and hX,Bi � 0 for all X 2 CP2.
However, every one of the infinitely many perfect matrices B(i)P satisfies
hX,B(i)P i > 0 for all X 2 CP2.
5. Computational Experiments
We implemented our algorithm. The source code, written in C++, is available onTODO: 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 CP5.The second example is from [11] and lies in the boundary of CP5
A2 =
0
BBBB@
8 5 1 1 55 8 5 1 11 5 8 5 11 1 5 8 55 1 1 5 8
1
CCCCAEX:
(algorithm terminates with a suitable pivot-rule)
from Groetzner, Dür (2018)
not solved by their algorithms A simplex algorithm for rational cp-factorization 13
Starting from QA5 our algorithm does five steps to find the factorization A2 =P10i=1 viv
Ti with
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)
v10 = (2, 1, 0, 0, 1)
The third example is from [17, Example 6.2] and it positive semidefinite, but notcompletely positive
A3 =
0
BBBB@
1 1 0 0 11 2 1 0 00 1 2 1 00 0 1 2 11 0 0 1 6
1
CCCCA
Starting from QA5 our algorithm does 18 steps to find the copositive matrix
B3 =
0
BBBB@
363/5 �2126/35 2879/70 608/21 �4519/210�2126/35 1787/35 �347/10 1025/42 253/142879/70 �347/10 829/35 �1748/105 371/30608/21 1025/42 �1748/105 1237/105 �601/70
�4519/210 253/14 371/30 �601/70 671/105
1
CCCCA
with hA3, B3i = �2/5 showing that A3 62 CP5.The fourth and last example is the family 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 aboutFarey sequences.
This project has received funding from the European Union’s Horizon 2020 re-search and innovation programme under the Marie Sk lodowska-Curie agreementNo 764759.
References
[1] S. Bundfuss and M. Dur, Algorithmic copositivity detection by simplicial partition, LinearAlgebra Appl. 428 (2008), 1511–1523.
A simplex algorithm for rational cp-factorization 13
Starting from QA5 our algorithm does five steps to find the factorization A2 =P10i=1 viv
Ti with
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)
v10 = (2, 1, 0, 0, 1)
The third example is from [17, Example 6.2] and it positive semidefinite, but notcompletely positive
A3 =
0
BBBB@
1 1 0 0 11 2 1 0 00 1 2 1 00 0 1 2 11 0 0 1 6
1
CCCCA
Starting from QA5 our algorithm does 18 steps to find the copositive matrix
B3 =
0
BBBB@
363/5 �2126/35 2879/70 608/21 �4519/210�2126/35 1787/35 �347/10 1025/42 253/142879/70 �347/10 829/35 �1748/105 371/30608/21 1025/42 �1748/105 1237/105 �601/70
�4519/210 253/14 371/30 �601/70 671/105
1
CCCCA
with hA3, B3i = �2/5 showing that A3 62 CP5.The fourth and last example is the family 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 aboutFarey sequences.
This project has received funding from the European Union’s Horizon 2020 re-search and innovation programme under the Marie Sk lodowska-Curie agreementNo 764759.
References
[1] S. Bundfuss and M. Dur, Algorithmic copositivity detection by simplicial partition, LinearAlgebra Appl. 428 (2008), 1511–1523.
giving a certificate for the matrix to be completely positive
Starting with QA5 , our algorithm finds a cp-factorization after 5 iterations
Boundary cases from ˜CPn
![Page 88: Variations and Applications of Voronoi’s algorithm](https://reader031.vdocument.in/reader031/viewer/2022012916/61c6e01e205dbf35826b28e9/html5/thumbnails/88.jpg)
Exterior cases
A simplex algorithm for rational cp-factorization 13
Starting from QA5 our algorithm does five steps to find the factorization A2 =P10i=1 viv
Ti with
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)
v10 = (2, 1, 0, 0, 1)
The third example is from [17, Example 6.2] and it positive semidefinite, but notcompletely positive
A3 =
0
BBBB@
1 1 0 0 11 2 1 0 00 1 2 1 00 0 1 2 11 0 0 1 6
1
CCCCA
Starting from QA5 our algorithm does 18 steps to find the copositive matrix
B3 =
0
BBBB@
363/5 �2126/35 2879/70 608/21 �4519/210�2126/35 1787/35 �347/10 1025/42 253/142879/70 �347/10 829/35 �1748/105 371/30608/21 1025/42 �1748/105 1237/105 �601/70
�4519/210 253/14 371/30 �601/70 671/105
1
CCCCA
with hA3, B3i = �2/5 showing that A3 62 CP5.The fourth and last example is the family 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 aboutFarey sequences.
This project has received funding from the European Union’s Horizon 2020 re-search and innovation programme under the Marie Sk lodowska-Curie agreementNo 764759.
References
[1] S. Bundfuss and M. Dur, Algorithmic copositivity detection by simplicial partition, LinearAlgebra Appl. 428 (2008), 1511–1523.
EX:
(algorithm conjectured to terminate)
from Nie (2014)
![Page 89: Variations and Applications of Voronoi’s algorithm](https://reader031.vdocument.in/reader031/viewer/2022012916/61c6e01e205dbf35826b28e9/html5/thumbnails/89.jpg)
Exterior cases
A simplex algorithm for rational cp-factorization 13
Starting from QA5 our algorithm does five steps to find the factorization A2 =P10i=1 viv
Ti with
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)
v10 = (2, 1, 0, 0, 1)
The third example is from [17, Example 6.2] and it positive semidefinite, but notcompletely positive
A3 =
0
BBBB@
1 1 0 0 11 2 1 0 00 1 2 1 00 0 1 2 11 0 0 1 6
1
CCCCA
Starting from QA5 our algorithm does 18 steps to find the copositive matrix
B3 =
0
BBBB@
363/5 �2126/35 2879/70 608/21 �4519/210�2126/35 1787/35 �347/10 1025/42 253/142879/70 �347/10 829/35 �1748/105 371/30608/21 1025/42 �1748/105 1237/105 �601/70
�4519/210 253/14 371/30 �601/70 671/105
1
CCCCA
with hA3, B3i = �2/5 showing that A3 62 CP5.The fourth and last example is the family 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 aboutFarey sequences.
This project has received funding from the European Union’s Horizon 2020 re-search and innovation programme under the Marie Sk lodowska-Curie agreementNo 764759.
References
[1] S. Bundfuss and M. Dur, Algorithmic copositivity detection by simplicial partition, LinearAlgebra Appl. 428 (2008), 1511–1523.
EX:
(algorithm conjectured to terminate)
giving a certificate for the matrix not to be completely positive
A simplex algorithm for rational cp-factorization 13
Starting from QA5 our algorithm does five steps to find the factorization A2 =P10i=1 viv
Ti with
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)
v10 = (2, 1, 0, 0, 1)
The third example is from [17, Example 6.2] and it positive semidefinite, but notcompletely positive
A3 =
0
BBBB@
1 1 0 0 11 2 1 0 00 1 2 1 00 0 1 2 11 0 0 1 6
1
CCCCA
Starting from QA5 our algorithm does 18 steps to find the copositive matrix
B3 =
0
BBBB@
363/5 �2126/35 2879/70 608/21 �4519/210�2126/35 1787/35 �347/10 1025/42 253/142879/70 �347/10 829/35 �1748/105 371/30608/21 1025/42 �1748/105 1237/105 �601/70
�4519/210 253/14 371/30 �601/70 671/105
1
CCCCA
with hA3, B3i = �2/5 showing that A3 62 CP5.The fourth and last example is the family 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 aboutFarey sequences.
This project has received funding from the European Union’s Horizon 2020 re-search and innovation programme under the Marie Sk lodowska-Curie agreementNo 764759.
References
[1] S. Bundfuss and M. Dur, Algorithmic copositivity detection by simplicial partition, LinearAlgebra Appl. 428 (2008), 1511–1523.
Starting with QA5 , after 18 iterations our algorithm finds the COP-perfect
from Nie (2014)
![Page 90: Variations and Applications of Voronoi’s algorithm](https://reader031.vdocument.in/reader031/viewer/2022012916/61c6e01e205dbf35826b28e9/html5/thumbnails/90.jpg)
Irrational boundary cases(algorithm is known not to terminate)
EX: A =
✓p21
◆✓p21
◆>
=
✓2
p2p
2 1
◆
![Page 91: Variations and Applications of Voronoi’s algorithm](https://reader031.vdocument.in/reader031/viewer/2022012916/61c6e01e205dbf35826b28e9/html5/thumbnails/91.jpg)
Irrational boundary cases
12 M. Dutour Sikiric, A. Schurmann, and F. Vallentin
Note that there is always a unique choice in Step 4 in case A is a 2 ⇥ 2 rank-1matrix. Note also that the vectors represent fractions that converge to
p2. Every
second vector corresponds to a convergent of the continued fraction ofp2. For
instance,
99/70 = 1 +1
2 +1
2 +1
2 +1
2 +1
2
.
The COP-perfect matrix after ten iterations of the algorithm is
B(10)P =
✓4756 �6726�6726 9512
◆.
It can be shown that the matrices B(i)P converge to a multiple of
B =
✓1 �
p2
�p2 2
◆satisfying hA,Bi = 0 and hX,Bi � 0 for all X 2 CP2.
However, every one of the infinitely many perfect matrices B(i)P satisfies
hX,B(i)P i > 0 for all X 2 CP2.
5. Computational Experiments
We implemented our algorithm. The source code, written in C++, is available onTODO: 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 CP5.The second example is from [11] and lies in the boundary of CP5
A2 =
0
BBBB@
8 5 1 1 55 8 5 1 11 5 8 5 11 1 5 8 55 1 1 5 8
1
CCCCA
(algorithm is known not to terminate)
EX: A =
✓p21
◆✓p21
◆>
=
✓2
p2p
2 1
◆
![Page 92: Variations and Applications of Voronoi’s algorithm](https://reader031.vdocument.in/reader031/viewer/2022012916/61c6e01e205dbf35826b28e9/html5/thumbnails/92.jpg)
Irrational boundary cases
12 M. Dutour Sikiric, A. Schurmann, and F. Vallentin
Note that there is always a unique choice in Step 4 in case A is a 2 ⇥ 2 rank-1matrix. Note also that the vectors represent fractions that converge to
p2. Every
second vector corresponds to a convergent of the continued fraction ofp2. For
instance,
99/70 = 1 +1
2 +1
2 +1
2 +1
2 +1
2
.
The COP-perfect matrix after ten iterations of the algorithm is
B(10)P =
✓4756 �6726�6726 9512
◆.
It can be shown that the matrices B(i)P converge to a multiple of
B =
✓1 �
p2
�p2 2
◆satisfying hA,Bi = 0 and hX,Bi � 0 for all X 2 CP2.
However, every one of the infinitely many perfect matrices B(i)P satisfies
hX,B(i)P i > 0 for all X 2 CP2.
5. Computational Experiments
We implemented our algorithm. The source code, written in C++, is available onTODO: 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 CP5.The second example is from [11] and lies in the boundary of CP5
A2 =
0
BBBB@
8 5 1 1 55 8 5 1 11 5 8 5 11 1 5 8 55 1 1 5 8
1
CCCCA
(algorithm is known not to terminate)
EX: A =
✓p21
◆✓p21
◆>
=
✓2
p2p
2 1
◆
![Page 93: Variations and Applications of Voronoi’s algorithm](https://reader031.vdocument.in/reader031/viewer/2022012916/61c6e01e205dbf35826b28e9/html5/thumbnails/93.jpg)
References
• Achill Schürmann, Exploiting Symmetries in Polyhedral Computations, Fields Institute Communications, 69 (2013), 265–278.
• Achill Schürmann, Computational Geometry of Positive Definite Quadratic Forms, University Lecture Series, AMS, Providence, RI, 2009.
• 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.
• 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.