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NEW APPROXIMATIONS FOR THE CONE OF COPOSITIVE MATRICES AND ITS DUAL 2 1 JEANB.LASSERRE 0 2 Abstract. WeprovideconvergenthierarchiesfortheconvexconeCofcopos- n itive matrices and its dual C∗, the cone of completely positive matrices. In a both cases the corresponding hierarchy consists of nested spectrahedra and J provide outer (resp. inner) approximations forC (resp. for its dual C∗), thus 9 complementing previousinner(resp. outer)approximationsforC (forC∗). In 1 particular, both inner and outer approximations have avery simpleinterpre- tation. Finally, extension to K-copositivity and K-complete positivity for a ] closedconvexconeK,isstraightforward. C O . h t 1. Introduction a m In recent years the convex cone of copositive matrices and its dual cone ∗ C C [ of completely positive matrices have attracted a lot of attention, in part because severalinterestingNP-hardproblemscanbemodelledasconvexconicoptimization 2 problemsoverthosecones. Forasurveyofresultsanddiscussionon andits dual, v C 2 theinterestedreaderisreferredtoe.g. AnstreicherandBurer[1],Bomze[2],Bomze 5 et al. [3], Burer [5], Du¨r [8] and Hiriart-Urruty and Seeger [11].. 5 As optimizing over (or its dual) is in generaldifficult, a typical approachis to 2 C optimizeoversimplerandmoretractablecones. Inparticular,nestedhierarchiesof . 2 tractableconvexcones ,k N,thatprovideinnerapproximationsof havebeen k 1 proposed, notably by PCarrilo∈[13], deKlerk and Pasechnik [7], Bomze aCnd deKlerk 0 [4], as well as Penã et al. [14]. For example, denoting by (resp. ) the convex 1 N S+ cone of nonnegative (resp. positive semidefinite) matrices, the first cone in the : v hierarchyof[7] is , and + inthat of[13], whereasthe hierarchyof[14]is in + i N N S X sandwich between that of [7] and [13]. Of course, to each such hierarchy of inner approximations ( ), k N, of , one may associate the hierarchy ( ∗), k N, of ar dual cones whichCpkrovid∈es outerCapproximations of ∗. However, quCokting∈Du¨r in C [8]: “Wearenotawareofcomparableapproximationschemesthatapproximatethe completely positive cone (i.e. ∗) from the interior.” C Let us also mention Gaddum’s characterizationof copositive matrices described inGaddum[9]forwhichthe setmembershipproblem“A ”(withAknown)re- ∈C ducestosolvingalinearprogramming(LP)problem(whosesizeisnotpolynomially bounded in the input size of the matrix A). However, Gaddum’s characterization is not convex in the coefficients of the matrix A, and so cannot be used to provide ahierarchyofouterapproximationsof . Onthe otherhand,deKlerkandPasech- C nik [6] have used Gaddum’s characterization to help solve quadratic optimization 1991 Mathematics Subject Classification. 15B4890C22. Key words and phrases. copositive matrices; completely positive matrices; semidefinite relaxations. 1 2 JEANB.LASSERRE problemsonthe simplexviasolvingahierarchyofLP-relaxationsofincreasingsize and with finite convergence of the process. Contribution. The contributionofthis note is precisely to describeanexplicit hierarchy of tractable convex cones that provide outer approximations of which C converge monotonically and asymptotically to . And so, by duality, the corre- C sponding hierarchy of dual cones provides inner approximations of ∗ converging C to ∗,answeringDu¨r’squestionandalsoshowingthat and ∗ canbe sandwiched C C C between two converging hierarchies of tractable convex cones. The present outer approximations are not polyhedral and hence are different from the outer polyhedral approximations n of defined in Yildirim [20]. n are based on a certain discretization ∆(n,r)Oorf theCsimplex ∆ := x Rn O: reTx = 1 whose size is n { ∈ + } parametrized by r. All matrices associated with quadratic forms nonnegative on ∆(n,r) belong to r for all r, and = ∞ r; Combining with the inner ap- On C ∩r=1On proximations of de Klerk and Pasechnik [7], the author provides tight bounds on the gap between upper and lower bounds for quadratic optimization problems; for more details the interested reader is referred to Yildirim [20]. Infact,ourresultisaspecializationofamoregeneralresultof[12]aboutnonneg- ativity of polynomials on a closed set K Rn, in the specific context of quadratic forms and K = Rn. In such a context,⊂and identifying copositive matrices with + quadratic forms nonnegative on K = Rn, this specialization yields inner approxi- + mationsfor ∗ withaverysimpleinterpretationdirectlyrelatedtothedefinitionof C ∗, which might be of interest for the community interestedin and ∗ but might C C C be “lost” in the general case treated in [12], whence the present note. Finally, fol- lowingBurer[5],and Rn beingaclosedconvexcone,onemayalsoconsiderthe K⊂ convexcone of -copositive matrices,i.e., realsymmetric matrices A suchthat K C K xTAx 0on , andits dualcone ∗ of -completely positive matrices. Thenthe ≥ K CK K outer approximations previously defined have an immediate and straightforward analogue (as well as the inner approximations of the dual). 2. Main result 2.1. Notation and definitions. Let R[x] be the ring of polynomials in the variables x=(x ,...,x ). Denote by R[x] R[x] the vector space of polynomials of 1 n d ⊂ degree at most d, which forms a vector space of dimension s(d)= n+d , with e.g., d theusualcanonicalbasis(xα)ofmonomials. Also,letNn := α Nn : α d and denote by Σ[x] R[x] (resp. Σ[x] R[x] ) thed spa{ce ∈of (cid:0)sums(cid:1)ofi sqiu≤are}s d 2d (s.o.s.) polynomials (⊂resp. s.o.s. polynomi⊂als of degree at most 2d). IfPf R[x] , d write f(x) = f xα in the canonical basis and denote by f = (f )∈ Rs(d) α∈Nnd α α ∈ its vector of coefficients. Finally, let n denote the space of n n real symmetric P S × matrices, with inner product A,B = traceAB, and where the notation A 0 h i (cid:23) (resp. A 0) stands for A is positive semidefinite. Given≻K Rn, denote by clK (resp. convK) the closure (resp. the convex hull) of K. R⊆ecall that given a convex cone K Rn, the convex cone K∗ = y Rn : y,x 0 x K is called the dual cone⊆of K, and satisfies (K∗)∗ = c{lK∈. Morehover,ig≥iven∀tw∈o con}vex cones K ,K Rn, 1 2 ⊆ K∗ K∗ = (K +K )∗ = (K K )∗ 1 ∩ 2 1 2 1∪ 2 (K K )∗ = cl(K∗+K∗) = cl(conv(K∗ K∗)). 1∩ 2 1 2 1∪ 2 See for instance Rockafellar [15, Theorem 3.8; Corollary 16.4.2]. NONNEGATIVITY 3 Moment matrix. With a sequence y = (y ), α Nn, let L : R[x] R be the α y ∈ → linear functional h (= h xα) L (h) = h y , h R[x]. α y α α 7→ ∈ α α X X With d N, let M (y) be the symmetric matrix with rows and columns indexed d in Nn, a∈nd defined by: d (2.1) M (y)(α,β) := L (xα+β) = y , (α,β) Nn Nn. d y α+β ∈ d × d The matrix M (y) is called the moment matrix associated with y, and it is d straightforwardto check that L (g2) 0 g R[x] M (y) 0, d=0,1,.... y d ≥ ∀ ∈ ⇔ (cid:23) Localizing(cid:2) matrix. Similarly, wit(cid:3)h y=(yα), α Nn, and f R[x] written ∈ ∈ x f(x) = f xγ, γ 7→ γ∈Nn X let M (fy) be the symmetric matrix with rows and columns indexed in Nn, and d d defined by: (2.2) M (fy)(α,β) := L f(x)xα+β = f y , (α,β) Nn Nn. d y γ α+β+γ ∈ d × d γ (cid:0) (cid:1) X ThematrixM (fy)iscalledthelocalizingmatrixassociatedwithy andf R[x]. d ∈ Observe that (2.3) g,M (fy)g = L (g2f), g R[x] , d y d h i ∀ ∈ and so if y has a representing finite Borel measure µ, i.e., if y = xαdµ, α Nn, α ZRn ∀ ∈ then (2.3) reads (2.4) g,M (fy)g = L (g2f) = g(x)2f(x)dµ(x), g R[x] . d y d h i ZRn ∀ ∈ Actually, the localizing matrix M (fy) is nothing less than the moment matrix d associated with the sequence z = fy = (z ), α Nn, with z = f y . α ∈ α γ γ α+γ In particular, if f is nonnegative on the support suppµ of µ then the localizing P matrixM (fy)isjustthemomentmatrixassociatedwiththefiniteBorelmeasure d dµ = fdµ, absolutely continuous with respect to µ (denoted µ µ), and with f f ≪ density f. For instance, with d=1 one may write 1 xT 1 xT | | M (fy) = f(x)dµ(x) = dµ (x), 1 f ZRn  −x x−xT  ZRn  −x x−xT  | | or, equivalently,     1 E (x)T | µ˜f (2.5) M (fy) = mass(µ ) , 1 f ×  − −  E (x) E (xxT) µ˜f | µ˜f   where E () denotes the expectation operator associated with the normalization µ˜f · µ˜ ofµ (asaprobabilitymeasure),andE (xxT)denotesthematrixofnoncentral f f µ˜f second-order moments of µ˜ (and the covariance matrix of µ˜ if E (x)=0). f f µ˜f 4 JEANB.LASSERRE 2.2. Main result. In[12,Theorem3.2]the authorhasshownina generalcontext that a polynomial f R[x] is nonnegative on a closed set K Rn if and only if ∈ ⊆ (2.6) g2fdµ 0, g R[x], ≥ ∀ ∈ ZK where µ is a given finite Borel measure with support suppµ being exactly K; if K is compact then µ is arbitrary whereas if K is not compact then µ has to satisfy a certain growth condition on its moments. If y = (y ), α Nn, is the sequence of α ∈ moments of µ then (2.6) is in turn equivalent to M (fy) 0, d=0,1,..., d (cid:23) ∀ where M (fy) is the localizing matrix associated with f and y, defined in (2.2). d Inthis sectionwe particularizethis resultto the caseof copositivematrices viewed as homogeneous forms of degree 2, nonnegative on the closed set K=Rn. + SowithA=(a ) n,letdenotebyf R[x] thequadraticformx xTAx, ij A 2 ∈S ∈ 7→ and let µ be the joint probability measure associated with n i.i.d. exponential variates (with mean 1), with support suppµ = Rn, and with moments y = (y ), + α α Nn, given by: ∈ n n (2.7) y = xαdµ(x) = xα exp( x )dx = α !, α Nn. α i i Rn Rn − ∀ ∈ Z + Z + Xi=1 iY=1 Recall that a matrix A n is copositive if f (x) 0 for all x Rn, and denote ∈S A ≥ ∈ + by n the cone of copositive matrices, i.e., C ⊂S (2.8) := A n : f (x) 0 x Rn . C { ∈S A ≥ ∀ ∈ +} Itsdualconeistheclosedconvexconeofcompletelypositive,i.e.,matricesof nthat can be written as the sum of finitely many rank-one matrices xxT, with xS Rn, ∈ + i.e., (2.9) ∗ = conv xxT : x Rn . C { ∈ +} Next, introduce the following sets n, d=0,1,..., defined by: d C ⊂S (2.10) := A n : M (f y) 0 , d=0,1,... d d A C { ∈S (cid:23) } where M (f y) is the localizing matrix defined in (2.2), associated with the qua- d A dratic form f and the sequence y in (2.7). A Observethatinviewofthedefinition(2.2)ofthelocalizingmatrix,theentriesof the matrix M (f y) are homogeneous and linear in A. Therefore, the condition d A M (f y) 0 is a homogeneous Linear Matrix Inequality (LMI) and defines a d A closed conv(cid:23)ex cone (in fact a spectrahedron) of n (or equivalently, of Rn(n+1)/2). S Each n isaconvexconedefinedsolelyintermsoftheentries(a )ofA n, d ij and thCe h⊂ieSrarchy of spectrahedra ( ), d N, provides a nested sequence of∈ouSter d C ∈ approximations of . C Theorem 2.1. Let y be as in (2.7) and let n, d=0,1,..., be the hierarchy d C ⊂S ∞ of convex cones defined in (2.10). Then and = . 0 1 d d C ⊃C ···⊃C ···⊃C C C d=0 \ NONNEGATIVITY 5 Theproofisadirectconsequenceof[12,Theorem3.3]withK=Rn andf =f . + A Since f is homogeneous, alternatively one may use the probability measure ν A uniformly supported on the n-dimensional simplex ∆= x Rn : x 1 and { ∈ + i i ≤ } invoke [12, Theorem 3.2]. The moments y˜ = (y˜ ) of ν are also quite simple to α P obtain and read: α ! α ! (2.11) y˜ = xαdx = 1 ··· n , α Nn. α (n+ α )! ∀ ∈ Z∆ i i P (See e.g. Grundmann [10].) Observethatthesetmembershipproblem“A ”,i.e.,testingwhetheragiven d ∈C matrixA n belongsto ,isaneigenvalueproblemasonehastocheckwhether d ∈S C the smallest eigenvalue of M (f y) is nonnegative. Therefore, instead of using d A standardpackagesfor LinearMatrix Inequalities,onemayuse powerfulspecialized softwares for computing eigenvalues of real symmetric matrices. Wenextdescribeaninnerapproximationoftheconvexcone ∗ viathehierarchy of convex cones ( ∗), d N, where each ∗ is the dual cone ofC in Theorem 2.1. Cd ∈ Cd Cd Recall that Σ[x] is the space of polynomials that are sums of squares of poly- d nomials of degree at most d. A matrix A n is also identified with a vector a Rn(n+1)/2,andconversely,withanyvecto∈raS Rn(n+1)/2 isassociatedamatrix ∈ ∈ A n. For instance, with n=2, ∈S a a b (2.12) A = a = 2b . b c ↔   (cid:20) (cid:21) c   SowewillnotdistinguishbetweenaconvexconeinRn(n+1)/2andthecorresponding cone in n. S Theorem 2.2. Let n be the convex cone defined in (2.10). Then d C ⊂S (2.13) ∗ =cl ( X,M (x x y) ) : X s(d) . Cd h d i j i 1≤i≤j≤n ∈S+ n o Equivalently: ∗ = cl xxT σ(x)dµ(x) : σ Σ[x] Cd ZRn+ dµσ(x) ∈ d (2.14) = cl mass(µσ)E|µ˜σ({xzxT)}: σ ∈Σ[x]d , (cid:8) (cid:9) with µ˜ and E (xxT) as in (2.5). σ µ˜σ Proof. Let ∆ := ( X,M (x x y) ) : X s(d) , d h d i j i 1≤i≤j≤n ∈S+ n o 6 JEANB.LASSERRE so that ∆∗ = a Rn(n+1)/2 : a X,M (x x y 0 X s(d) , d  ∈ ijh d i j i≥ ∀ ∈S+   1≤Xi≤j≤n    = a Rn(n+1)/2 : X,M ( a x x )y 0 X s(d) ,  ∈ * d ij i j +≥ ∀ ∈S+   1≤Xi≤j≤n  = A n : X,M (f y) 0 X s(d) [with A,a as in (2.12)] { ∈S h d A i≥ ∀ ∈S+ }  = A n : M (f y) 0 = . d A d { ∈S (cid:23) } C And so we obtain the desired result ∗ = (∆∗)∗ = cl(∆ ). Next, writing the singular decomposition of X as s Cdq qT fodr some s d N and some vectors k=0 k k ∈ (q ) Rs(d), one obtains that for every 1 i j n, k ⊂ P ≤ ≤ ≤ s s X,M (x x y) = q qT,M (x x y) = q ,M (x x y)q h d i j i h k k d i j i h k d i j ki k=0 k=0 X X s = x x q (x)2dµ(x) [by (2.4)] i j k Rn kX=0Z + = x x σ(x)dµ(x), i j Rn Z + dµσ(x) where σ(x)= s q (x)2 Σ[x] , a|nd µ{z(B)}= σdµ for all Borel sets B. (cid:3) k=0 k ∈ d σ B So TheoremP2.2 states that ∗ is the closure oRf the convex cone generated by Cd second-order moments of measures dµ =σdµ, absolutely continuous with respect σ to µ (hence with support on Rn) and with density being a s.o.s. polynomial σ of + degree at most 2d. Of course we immediately have: Corollary 2.3. Let ∗, d N, be as in (2.14). Then ∗ ∗ for all d N, and Cd ∈ Cd ⊂Cd+1 ∈ ∞ ∗ = cl ∗. C Cd d=0 [ Proof. As ∗ ∗ for all d N, the result follows from Cd ⊂Cd+1 ∈ ∞ ∗ ∞ ∞ ∗ = = cl conv ∗ = cl ∗. C Cd! Cd! Cd d=0 d=0 d=0 \ [ [ (cid:3) Inotherwords, ∗ approximates ∗ =conv xxT :x Rn (i.e.,theconvexhull of second-order moCmd ents of Dirac Cmeasures w{ith supp∈ort+in} Rn) from inside by + second-order moments of measures µ µ whose density is a s.o.s. polynomial σ σ ≪ ofdegreeatmost2d,andbetter andbetter approximationsareobtainedbyletting d increase. a b Example 1. For instance, with n = 2, and A = , it is known that A b c (cid:20) (cid:21) is copositive if and only if a,c 0 and b + √ac 0; see e.g. [11]. Let µ be ≥ ≥ NONNEGATIVITY 7 1 0.8 0.6 0.4 0.2 b 0 −0.2 −0.4 −0.6 −0.8 −0.2 0 0.2 0.4 0.6 0.8 a Figure 1. n = 2: Projection on the (a,b)-plane of versus , 1 C C both intersected with the unit ball the exponential measure on R2 with moments defined in (2.7). With f (x) := + A ax2+2bx x +cx2 and d=1, the condition M (f y) 0 which reads 1 1 2 2 d A (cid:23) a+b+c 3a+2b+c a+2b+3c (2.15) 2 3a+2b+c 12a+6b+2c 3a+4b+3c 0,   (cid:23) a+2b+3c 3a+4b+3c 2a+6b+12c defines the convex cone R3. It is a connected component of the basic semi- 1 C ⊂ algebraic set (a,b,c):det(M (f y)) 0 , that is, elements (a,b,c) such that: 1 A { ≥ } (2.16) 3a3+15a2b+29a2c+16ab2+50abc+29ac2+4b3+16b2c+15bc2+3c3 0. ≥ Alternatively,by homogeneity,insteadof µ we maytake the probability measureν uniformlysupportedonthe simplex ∆andwith momentsy˜ =(y˜ )givenin(2.11), α in which case the corresponding matrix M (f y˜) now reads: 1 A 30(a+b+c) 6(3a+2b+c) 6(a+2b+3c) 1 (2.17) 6(3a+2b+c) 12a+6b+2c 3a+4b+3c 0. 360  (cid:23) 6(a+2b+3c) 3a+4b+3c 2a+6b+12c   Itturns outthatup to a multiplicative factorboth matrices(2.15) and(2.17) have same determinant and so define the same convex cone (the same connected 1 C component of (2.16)). Figure 1 below displays the projectionon the (a,b)-plane of the sets and intersected with the unit ball. 1 C C 2.3. An alternative representation of the cone ∗. From its definition (2.13) Cd inTheorem2.2, the cone ∗ n is definedthroughthe matrixvariableX s(d) Cd ⊂S ∈S which lives in a (lifted) space of dimension s(d)(s(d) +1)/2 and with the linear 8 JEANB.LASSERRE matrixinequality(LMI)constraintX 0ofsizes(d). Incontrast,the convexcone (cid:23) n defined in (2.10) is defined solely in terms of the entries of the matrix d C ⊂ S A n, that is, no projection from a higher dimensional space (or no lifting) is ∈ S needed. We next provide another explicit description on how ∗ can be generated with Cd no LMI constraintand with only s(d) variables,but of course this characterization is not well suited for optimization purposes. Since everyg Σ[x] canbe written g2 for finitely many polynomials(g ) R[x] , the convex∈conedΣ[x] of s.o.s. polyℓnoℓmials can be written ℓ ⊂ d d P Σ[x] = conv g2 : g R[x] . d d { ∈ } Next, for g R[x] , with vector of coefficients g =(g ) Rs(d), let g(2) =(g(2)) d α α Rs(2d) be th∈e vector of coefficients of g2, that is, ∈ ∈ g(x)= g xα g(x)2 = g(2)xα. α → α α∈Nn α∈Nn Xd X2d Notice that for each α Nn , g(2) is quadratic in g. For instance, with n= 2 and ∈ 2d α d=1, g(x)=g +g x +g x with g=(g , g ,g )T Rs(1), and so 00 10 1 01 2 00 10 01 ∈ g(2) = (g2 , 2g g , 2g g , g2 , 2g g , g2 )T Rs(2). 00 00 10 00 01 10 10 01 02 ∈ Next, forg R[x] andevery1 i j n,letG =(G ) n be defined by: d d ij ∈ ≤ ≤ ≤ ∈S (2.18) G := g(x)2x x dµ(x) = g(2)(α +1)!(α +1)! α !, ij i j α i j k ZRn α∈Nn k6=i,j X2d Y for all 1 i j n. We can now describe ∗. ≤ ≤ ≤ Cd Corollary 2.4. Let G n be as in (2.18). Then: d ∈S (2.19) ∗ = cl conv G : g Rs(d) . Cd { d ∈ } (cid:16) (cid:17) Proof. From Theorem 2.2 ∗ =cl ( X,M (x x y) ) : X s(d) . Cd h d i j i 1≤i≤j≤n ∈S+ n o As in the proof of Theorem 2.2, writing X 0 as s g gT for some vectors (cid:23) k=1 k k (g ) Rs(d) (and associated polynomials (g ) R[x] ), k k d ⊂ ⊂ P s s X,M (x x y) = x x g (x)2dµ, = Gk, h d i j i Rn i j k ij kX=1Z + kX=1 for all 1 i j n, where Gk is as in (2.18) (but now associatedwith g instead ≤ ≤ ≤ ij k of g). Hence s ( X,M (x x y) ) = Gk conv G : g Rs(d) . h d i j i 1≤i≤j≤n d ∈ { d ∈ } k=1 X Hence ∗ cl conv G : g Rs(d) . d C ⊆ { ∈ } (cid:0) (cid:1) NONNEGATIVITY 9 Conversely,leth conv G :g Rs(d) ,i.e.,forsomeintegersandpolynomials d (g ) R[x] , and fo∈r all 1{ i j ∈n, } k d ⊂ ≤ ≤ ≤ s s h = λ Gk = λ g2(x) x x dµ(x) ij k ij k k i j Rn ! Xk=1 >0 Z + Xk=1 = X,M (x x y) |{z} h d i j i where X= s λ g gT s(d). Hence cl conv G : g Rs(d) ∗. (cid:3) k=1 k k k ∈S+ { d ∈ } ⊆C The charPacterization(2.19) of ∗ should b(cid:0)e compared with the ch(cid:1)aracterization conv xxT : x Rn of ∗. Cd { ∈ +} C Example 2. With n=2 and d=1, G reads: 1 2g020+12g10(g00+g01)+4g01(g00+g01)+24g120 g020+4g00(g10+g01)+6(g120+g021)+8g10g01 (cid:20) g020+4g00(g10+g01)+6(g120+g021)+8g10g01 2g020+4g10(g00+g10)+12g01(g00+g10)+24g021 (cid:21) 2.4. -copositive and -completely positive matrices. Let Rn be a K K K ⊂ closed convex cone and let be the convex cone of -copositive matrices, i.e., K C K matrices A n such that xTAx 0 on . Its dual cone ∗ n is the cone of ∈S ≥ K CK ⊂S -completely positive matrices. K Then one may define a hierarchy of convex cones ( ) and ( ∗) , d N, CK d CK d ∈ formally exactly as in Theorem 2.1 and Theorem 2.2, but now y is the moment sequence of a finite Borel measure µ with suppµ = (instead of suppµ = Rn in (2.7)), i.e., y=(y ), α Nn, with: K + α ∈ (2.20) y := xαdµ, α Nn (and where suppµ= ). α ∀ ∈ K ZK And so with y as in (2.20) Theorem 2.1 and 2.2, as well as Corollary 2.3, are still valid. (In (2.14 replace with ). Therefore, Rn K + ∞R R ∞ = (C ) and ∗ = cl ( )∗. CK K d CK CK d d=0 d=0 \ [ But of course, for practical implementation, one need to know the sequence y = (y ),α Nn,whichwaseasywhen =Rn. Forinstance,if isapolyhedralcone, α ∈ K + K by homogeneity of f , one may equivalently consider a compact base of (which A K is a polytope ′), and take for µ the Lebesgue measure on ′. Then all moments K K of µ can be calculated exactly. The same argument works for every convex cone K forwhichone maycompute allmoments ofa finite Borelmeasureµ whose support is a compact base of . K Acknowledgement Theauthorwishestothanktwoanonymousrefereesfortheirveryhelpfulremarks and suggestions to improve the initial version of this note. References [1] K.M.Anstreicher,S.Burer.Computablerepresentationsforconvexhullsoflow-dimensional quadraticforms,Math.Program.Sér.B124(2010), pp.33–43. [2] I.M. Bomze. Copositive optimization - recent developments and applications, European J. 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