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A Simplex-Type Voronoi Algorithm Based on Short Vector Computations of Copositive Quadratic Forms

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A Simplex-Type Voronoi Algorithm Based on Short Vector Computations of Copositive Quadratic Forms Simons workshop Lattices Geometry, Algorithms and Hardness Berkeley, February 21st 2020 A simplex-type Voronoi algorithm based on short vector computations of copositive quadratic forms Achill Schürmann (Universität Rostock) based on work with Mathieu Dutour Sikirić and Frank Vallentin ( for Q n positive definite ) Perfect Forms 2 S>0 DEF: min(Q)= min Q[x] is the arithmetical minimum • x n 0 2Z \{ } Q is uniquely determined by min(Q) and Q perfect • , n MinQ = x Z : Q[x]=min(Q) { 2 } V(Q)=cone xxt : x MinQ is Voronoi cone of Q • { 2 } (Voronoi cones are full dimensional if and only if Q is perfect!) THM: Voronoi cones give a polyhedral tessellation of n S>0 and there are only finitely many up to GL n ( Z ) -equivalence. Voronoi’s Reduction Theory n t GLn(Z) acts on >0 by Q U QU S 7! Georgy Voronoi (1868 – 1908) Task of a reduction theory is to provide a fundamental domain Voronoi’s algorithm gives a recipe for the construction of a complete list of such polyhedral cones up to GL n ( Z ) -equivalence Ryshkov Polyhedron The set of all positive definite quadratic forms / matrices with arithmeticalRyshkov minimum Polyhedra at least 1 is called Ryshkov polyhedron = Q n : Q[x] 1 for all x Zn 0 R 2 S>0 ≥ 2 \{ } is a locally finite polyhedron • R is a locally finite polyhedron • R Vertices of are perfect forms Vertices of are• perfect R • R 1 n n ↵ (det(Q + ↵Q0)) is strictly concave on • 7! S>0 Voronoi’s algorithm Voronoi’s 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 ( graph traversal search on edge graph of Ryshkov polyhedron ) 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 MATHIEU DUTOUR SIKIRIC,´ ACHILL SCHURMANN,¨ ˜ AND FRANKT VALLENTINn n = cone xx : x Q 0 2M.DutourSikiri´c,A.Sch¨urmann,andF.VallentinCP { 2 ≥ } of the cone of completely positive matrices. Moreover, it also computes certificates Abstract. Wein case describeA and analyze. an algorithmic procedure, similar to the From the algorithmic62 CPn side some successful ideas have been proposed by Jarre simplex algorithmand SchmallowskyIn for contrast linear programming, to [14], most Nie of the [17], which other Sponsel tests approaches whether and D¨ur mentioned or [20], not a Groetzner given above (the and exception D¨ur [11], being symmetric matrix is completely positive. For matrices in the rational closure andthe Anstreicher, approach Burer, by Anstreicher, and Dickinson Burer [4, and Section Dickinson 3.3]. whose factorization method is 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 cp-factorization.In this For paper matrices we which describe are not a new completely procedure positive that a works certificate for all matrices in the rational numbers only if the input matrix is rational 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 CP { 2 ≥ } always terminatesof theexists. cone for However, of rational completely input questions matrices. positive of convergence matrices. Moreover, remain, in it also particular computes for all certificatesA which are not contained in the “irrational boundary part” (bd n) ˜ n. in case A n. CP \ CP In contrast62 CP to most of the other approaches mentioned above (the exception being 2. The copositive minimum and copositive perfect matrices the approach by Anstreicher, Burer and Dickinson whose factorization method is basedBy on then we ellipsoid denote method), the Euclidean our algorithm vector space is exact of symmetric in the sensen thatn matrices it uses with S n ⇥ rationalinner numbers product1. onlyA,Introduction B if the= Trace( input matrixAB)= is rationalA andB .Withrespecttothisinner so does not have to cope h i i,j=1 ij ij Copositive programmingwithproduct numericalGeneralization we gives instabilities. have a common the following In framework particular duality to it relations formulatefinds a rational… between manyand cp-factorizationapplication! di thefficult cone of copositive if it exists. However, questions of convergence remain,P in particular for all A which are optimization problemsmatrices as convex and and conic the cone ones. of completely In fact, many positive NP-hard matrices problems not containedIDEA: in Generalize the “irrational Voronoi’s boundary theory part”n to (bd n) ˜ n. are known to have such reformulationsn =( (seen)⇤ for= exampleB : theA, B surveys0 for [2, all 8]).A Alln , otherCOP convexCP cones and{ 2theirS dualsh iCP \ CP 2 CP } the difficulty of theseand2. problemsThe copositive appears(Opgenorth, minimum to be 2001) “converted” and copositive into the perfect difficulty 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 productsuffices 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 fficult. 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 matrixDefinitionA is 2.1. completelyFor a symmetric positive.CP matrix IfCOP possibleB n onewe define would the likecopositive to minimum So, in order to show that a given symmetric matrix2 S A is not completely positive, it obtain a certificateas for either A n or A n n. In terms of the definitions the suffices 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 minin this(B)=inf case, becauseB[v]: thev linearZ 0 hyperplane0 , orthogonal 2 CP62 CPn 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 Definitionn givingcopositive a 2.1. separating matricesFor a symmetric attain hyperplanes their matrix copositive definedB minimum. byn weB define thencopositiveso that 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 andfinitely we denote many the vectors. set of vectors attaining it by 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. copositive programming, complete positivity, matrixn factorization,COP ≥ Themin following(B) = proposition 0. Then there shows is a sequence that matricesvi Z in0 the0 interiorof pairwise of the distinct cone oflattice copositive minimum. COP 2 ≥ \{ } copositivepoints matrices such that attainB[vi]tendstozerowhen their copositive minimum.i tends to infinity. From the sequence The first author wasv supportedwe construct by the a newHumboldt sequence foundation.u of points The third on the author unit is sphere partiallySn 1 by setting supported by the SFB/TRRLemmai 2.2. 191 “SymplecticLet B be a Structures matrix in in the Geometry,i interior of Algebra the cone and of Dynamics”, copositive− matrices. 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.
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