New approximations for the cone of copositive matrices and its dual

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New approximations for the cone of copositive matrices and its dual

Jean-Bernard Lasserre

To cite this version:

Jean-Bernard Lasserre. New approximations for the cone of copositive matrices and its dual. Math-

ematical Programming, Series A, Springer, 2014, 144, pp.265-276. �10.1007/s10107-013-0632-5�. �hal-

00545755v2�

NEW APPROXIMATIONS FOR THE CONE OF COPOSITIVE MATRICES AND ITS DUAL

JEAN B. LASSERRE

Abstract. We provide convergent hierarchies for the convex coneCof copos- itive matrices and its dualC^∗, the cone of completely positive matrices. In both cases the corresponding hierarchy consists of nested spectrahedra and provide outer (resp. inner) approximations forC(resp. for its dualC^∗), thus complementing previous inner (resp. outer) approximations forC(forC^∗). In particular, both inner and outer approximations have a very simple interpre- tation. Finally, extension toK-copositivity and K-complete positivity for a closed convex coneK, is straightforward.

1. Introduction

In recent years the convex cone C of copositive matrices and its dual cone C^∗ of completely positive matrices have attracted a lot of attention, in part because several interesting NP-hard problems can be modelled as convex conic optimization problems over those cones. For a survey of results and discussion onCand its dual, the interested reader is referred to e.g. Anstreicher and Burer [1], Bomze [2], Bomze et al. [3], Burer [5], D¨ur [8] and Hiriart-Urruty and Seeger [11]..

As optimizing overC (or its dual) is in general difficult, a typical approach is to optimize over simpler and more tractable cones. In particular, nested hierarchies of tractable convex conesC^k,k∈N, that provideinnerapproximations ofChave been proposed, notably by Parrilo [13], deKlerk and Pasechnik [7], Bomze and deKlerk [4], as well as Pe˜na et al. [14]. For example, denoting byN (resp. S⁺) the convex cone of nonnegative (resp. positive semidefinite) matrices, the first cone in the hierarchy of [7] isN, andN+S⁺ in that of [13], whereas the hierarchy of [14] is in sandwich between that of [7] and [13]. Of course, to each such hierarchy of inner approximations (C^k),k∈N, ofC, one may associate the hierarchy (Ck^∗),k∈N, of dual cones which provides outer approximations of C^∗. However, quoting D¨ur in [8]: “We are not aware of comparable approximation schemes that approximate the completely positive cone (i.e. C^∗) from the interior.”

Let us also mention Gaddum’s characterization of copositive matrices described in Gaddum [9] for which the set membership problem “A∈ C” (withAknown) re- duces to solving a linear programming (LP) problem (whose size is not polynomially bounded in the input size of the matrixA). However, Gaddum’s characterization isnotconvex in the coefficients of the matrixA, and so cannot be used to provide a hierarchy of outer approximations ofC. On the other hand, de Klerk and Pasech- nik [6] have used Gaddum’s characterization to help solve quadratic optimization

1991Mathematics Subject Classification. 15B48 90C22.

Key words and phrases. copositive matrices; completely positive matrices; semidefinite relaxations.

1

problems on the simplex via solving a hierarchy of LP-relaxations of increasing size and withfiniteconvergence of the process.

Contribution. The contribution of this note is precisely to describe an explicit hierarchy of tractable convex cones that provideouterapproximations of C which converge monotonically and asymptotically to C. And so, by duality, the corre- sponding hierarchy of dual cones provides innerapproximations of C^∗ converging toC^∗, answering D¨ur’s question and also showing thatCandC^∗can be sandwiched between two converging hierarchies of tractable convex cones. The present outer approximations arenot polyhedral and hence are different from the outer polyhe- dral approximations Oⁿr of C defined in Yildirim [20]. Oⁿr are based on a certain discretization ∆(n, r) of the simplex ∆n := {x ∈ Rⁿ₊ : e^Tx = 1} whose size is parametrized by r. All matrices associated with quadratic forms nonnegative on

∆(n, r) belong to O^rn for all r, and C = ∩^∞r=1O^rn; Combining with the inner ap- 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].

In fact, our result is a specialization of a more general result of [12] about nonneg- ativity of polynomials on a closed setK⊂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- mations forC^∗with a very simple interpretation directly related to the definition of C^∗, which might be of interest for the community interested inC andC^∗but might be “lost” in the general case treated in [12], whence the present note. Finally, fol- lowing Burer [5], andK ⊂Rⁿbeing a closed convex cone, one may also consider the convex coneC^K ofK-copositive matrices, i.e., real symmetric matricesAsuch that x^TAx≥0 onK, and its dual coneCK^∗ ofK-completely positive matrices. Then the 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. LetR[x] be the ring of polynomials in the vari- ablesx= (x1, . . . , xn). Denote byR[x]d⊂R[x] the vector space of polynomials of degree at mostd, which forms a vector space of dimensions(d) = ^n+d_d

, with e.g., the usual canonical basis (x^α) of monomials. Also, letNⁿ_d :={α∈Nⁿ : P

iαi≤d} and denote by Σ[x] ⊂ R[x] (resp. Σ[x]d ⊂ R[x]2d) the space of sums of squares (s.o.s.) polynomials (resp. s.o.s. polynomials of degree at most 2d). Iff ∈R[x]d, write f(x) =P

α∈Nn

dfαx^α in the canonical basis and denote byf = (fα)∈R^s(d) its vector of coefficients. Finally, let Sⁿ denote the space of n×n real symmetric matrices, with inner product hA,Bi= traceAB, and where the notation A 0 (resp. A≻0) stands forAis positive semidefinite.

Given K ⊆ Rⁿ, denote by clK (resp. convK) the closure (resp. the convex hull) ofK. Recall that given a convex coneK⊆Rⁿ, the convex coneK^∗={y∈ Rⁿ :hy,xi ≥0∀x∈K} is called thedual cone of K, and satisfies (K^∗)^∗ = clK . Moreover, given two convex conesK₁,K₂⊆Rⁿ,

K^∗₁ ∩ K^∗₂ = (K₁+K₂)^∗ = (K₁∪K₂)^∗ (K1∩K₂)^∗ = cl (K^∗₁+K^∗₂) = cl (conv (K^∗₁∪K^∗₂) ).

See for instance Rockafellar [15, Theorem 3.8; Corollary 16.4.2].

NONNEGATIVITY 3

Moment matrix. With a sequencey = (yα),α∈Nⁿ, let Ly :R[x]→Rbe the linear functional

h (=X

α

hαx^α) 7→ Ly(h) = X

α

hαyα, h∈R[x].

With d∈N, let Md(y) be the symmetric matrix with rows and columns indexed inNⁿ

d, and defined by:

(2.1) M_d(y)(α, β) := Ly(x^α+β) = yα+β, (α, β)∈Nⁿ_d×Nⁿ_d.

The matrix M_d(y) is called the moment matrix associated with y, and it is straightforward to check that

Ly(g²)≥0 ∀g∈R[x]

⇔ Md(y) 0, d= 0,1, . . . . Localizing matrix. Similarly, withy= (yα),α∈Nⁿ, andf ∈R[x] written

x7→f(x) = X

γ∈Nn

fγx^γ,

letM_d(fy) be the symmetric matrix with rows and columns indexed inNⁿ_d, and defined by:

(2.2) M_d(fy)(α, β) := Ly f(x)x^α+β

= X

γ

fγyα+β+γ, (α, β)∈Nⁿ_d×Nⁿ_d. The matrixMd(fy) is called the localizing matrix associated withyandf∈R[x].

Observe that

(2.3) hg,M_d(fy)gi = Ly(g²f), ∀g∈R[x]d, and so ifyhas a representing finite Borel measureµ, i.e., if

yα = Z

Rn

x^αdµ, ∀α∈Nⁿ, then (2.3) reads

(2.4) hg,M_d(fy)gi = Ly(g²f) = Z

Rn

g(x)²f(x)dµ(x), ∀g∈R[x]d. Actually, the localizing matrix Md(fy) is nothing less than the moment matrix associated with the sequence z = fy = (zα), α ∈ Nⁿ, with zα = P

γfγyα+γ. In particular, if f is nonnegative on the support suppµ of µ then the localizing matrixM_d(fy) is just the moment matrix associated with the finite Borel measure dµf =f dµ, absolutely continuous with respect to µ(denoted µf ≪µ), and with densityf. For instance, withd= 1 one may write

M₁(fy) = Z

Rn





1 | x^T

− −

x | xx^T



f(x)dµ(x) = Z

Rn





1 | x^T

− −

x | xx^T



dµf(x), or, equivalently,

(2.5) M₁(fy) = mass(µf)×





1 | Eµ˜f(x)^T

− −

Eµ˜_f(x) | Eµ˜_f(xx^T)



,

where Eµ˜f(·) denotes the expectation operator associated with the normalization

˜

µfofµf (as a probability measure), and Eµ˜_f(xx^T) denotes the matrix of noncentral second-order moments of ˜µf (and the covariance matrix of ˜µf if Eµ˜_f(x) = 0).

2.2. Main result. In [12, Theorem 3.2] the author has shown in a general context that a polynomialf ∈R[x] is nonnegative on a closed setK⊆Rⁿ if and only if (2.6)

Z

K

g²f dµ ≥ 0, ∀g∈R[x],

whereµis a given finite Borel measure with support suppµbeing exactlyK; ifK is compact thenµis arbitrary whereas ifKis not compact thenµhas to satisfy a certain growth condition on its moments. If y= (yα), α∈Nⁿ, is the sequence of moments ofµthen (2.6) is in turn equivalent to

M_d(fy) 0, ∀d= 0,1, . . . ,

where M_d(fy) is the localizing matrix associated with f and y, defined in (2.2).

In this section we particularize this result to the case of copositive matrices viewed as homogeneous forms of degree 2, nonnegative on the closed setK=Rⁿ₊.

So withA= (aij)∈ Sⁿ, let denote byfA∈R[x]2the quadratic formx7→x^TAx, 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= (yα), α∈Nⁿ, given by:

(2.7) yα= Z

Rⁿ

+

x^αdµ(x) = Z

Rⁿ

+

x^α exp(− Xn

i=1

xi)dx = Yn

i=1

αi!, ∀α∈Nⁿ. Recall that a matrixA∈ Sⁿ is copositive iffA(x)≥0 for allx∈Rⁿ₊, and denote byC ⊂ Sⁿ the cone of copositive matrices, i.e.,

(2.8) C := {A∈ Sⁿ : fA(x) ≥ 0 ∀x∈Rⁿ₊}.

Its dual cone is the closed convex cone ofcompletely positive, i.e., matrices ofSⁿthat can be written as the sum of finitely many rank-one matrices xx^T, with x∈Rⁿ₊, i.e.,

(2.9) C^∗ = conv{xx^T : x∈Rⁿ₊}.

Next, introduce the following setsC^d⊂ Sⁿ,d= 0,1, . . ., defined by:

(2.10) C^d := {A∈ Sⁿ : M_d(fAy) 0}, d= 0,1, . . .

whereM_d(fAy) is the localizing matrix defined in (2.2), associated with the qua- dratic formfA and the sequencey in (2.7).

Observe that in view of the definition (2.2) of the localizing matrix, the entries of the matrixM_d(fAy) are homogeneous and linear in A. Therefore, the condition M_d(fAy) 0 is a homogeneous Linear Matrix Inequality (LMI) and defines a closed convex cone (in fact a spectrahedron) ofSⁿ (or equivalently, ofR^n(n+1)/2).

EachC^d⊂ Sⁿis a convex cone defined solely in terms of the entries (aij) ofA∈ Sⁿ, and the hierarchy of spectrahedra (C^d),d∈N, provides a nested sequence of outer approximations ofC.

Theorem 2.1. Let ybe as in (2.7) and let C^d⊂ Sⁿ,d= 0,1, . . ., be the hierarchy of convex cones defined in (2.10). Then C⁰⊃ C¹· · · ⊃ C^d· · · ⊃ C andC=

\∞

d=0

C^d.

NONNEGATIVITY 5

The proof is a direct consequence of [12, Theorem 3.3] withK=Rⁿ₊andf =fA. Since fA is homogeneous, alternatively one may use the probability measure ν uniformly supported on then-dimensional simplex ∆ ={x∈Rⁿ₊ :P

ixi≤1}and invoke [12, Theorem 3.2]. The moments ˜y = (˜yα) of ν are also quite simple to obtain and read:

(2.11) y˜α =

Z

∆

x^αdx = α1!· · ·αn! (n+P

iαi)!, ∀α∈Nⁿ. (See e.g. Grundmann [10].)

Observe that the set membership problem “A∈ C^d”, i.e., testing whether a given matrixA∈ Sⁿ belongs toC^d, is an eigenvalue problem as one has to check whether the smallest eigenvalue of M_d(fAy) is nonnegative. Therefore, instead of using standard packages for Linear Matrix Inequalities, one may use powerful specialized softwares for computing eigenvalues of real symmetric matrices.

We next describe an inner approximation of the convex coneC^∗via the hierarchy of convex cones (C^∗d),d∈N, where eachCd^∗ is the dual cone ofC^d in Theorem 2.1.

Recall that Σ[x]d is the space of polynomials that are sums of squares of poly- nomials of degree at most d. A matrix A ∈ Sⁿ is also identified with a vector a∈R^n(n+1)/2, and conversely, with any vectora∈R^n(n+1)/2is associated a matrix A∈ Sⁿ. For instance, withn= 2,

(2.12) A =

a b b c

↔ a =



 a 2b c



.

So we will not distinguish between a convex cone inR^n(n+1)/2and the corresponding cone in Sⁿ.

Theorem 2.2. Let C^d⊂ Sⁿ be the convex cone defined in (2.10). Then (2.13) Cd^∗= cln

(hX,M_d(xixjy)i)_{1≤i≤j≤n} : X∈ S+^s(d)

o.

Equivalently:

C^∗d = cl





 Z

Rn +

xx^T σ(x)dµ(x)

| {z }

dµσ(x)

: σ∈Σ[x]d







= cl

mass(µσ)Eµ˜σ(xx^T) : σ∈Σ[x]d , (2.14)

withµ˜σ andEµ˜σ(xx^T) as in (2.5).

Proof. Let

∆d := n

(hX,M_d(xixjy)i)_{1≤i≤j≤n} : X∈ S+^s(d)

o,

so that

∆^∗_d =







a∈R^n(n+1)/2 : X

1≤i≤j≤n

aijhX,Md(xixjyi ≥0 ∀X∈ S+^s(d)





 ,

=







a∈R^n(n+1)/2 :

* X,M_d



( X

1≤i≤j≤n

aijxixj)y



 +

≥0 ∀X∈ S+^s(d)





,

= {A∈ Sⁿ : hX,M_d(fAy)i ≥0 ∀X∈ S+^s(d)} [with A,a as in (2.12)]

= {A∈ Sⁿ : M_d(fAy) 0} = C^d.

And so we obtain the desired result Cd^∗ = (∆^∗_d)^∗ = cl (∆d). Next, writing the singular decomposition of X as Ps

k=0q_kq^T_k for some s ∈ N and some vectors (qk)⊂R^s(d), one obtains that for every 1≤i≤j≤n,

hX,M_d(xixjy)i = Xs

k=0

hq_kq^T_k,M_d(xixjy)i = Xs

k=0

hq_k,M_d(xixjy)qki

= Xs

k=0

Z

Rn +

xixjqk(x)²dµ(x) [by (2.4)]

= Z

Rⁿ

+

xixj σ(x)dµ(x)

| {z }

dµσ(x)

,

whereσ(x) =Ps

k=0qk(x)²∈Σ[x]d, and µσ(B) =R

Bσ dµ for all Borel setsB.

So Theorem 2.2 states that Cd^∗ is the closure of the convex cone generated by second-order moments of measuresdµσ=σdµ, absolutely continuous with respect to µ(hence with support on Rⁿ₊) and with density being a s.o.s. polynomialσ of degree at most 2d. Of course we immediately have:

Corollary 2.3. Let Cd^∗,d∈N, be as in (2.14). Then Cd^∗⊂ Cd+1^∗ for all d∈N, and C^∗ = cl

[∞

d=0

Cd^∗.

Proof. AsCd^∗⊂ Cd+1^∗ for alld∈N, the result follows from C^∗ =

\∞

d=0

C^d

!∗

= cl conv [∞

d=0

Cd^∗

!

= cl [∞

d=0

Cd^∗.

In other words,Cd^∗approximatesC^∗= conv{xx^T :x∈Rⁿ₊}(i.e., the convex hull of second-order moments of Dirac measures with support in Rⁿ₊) from inside by second-order moments of measuresµσ ≪µwhose density is a s.o.s. polynomialσ of degree at most 2d, and better and better approximations are obtained by letting dincrease.

Example 1. For instance, with n = 2, and A = a b

b c

, it is known that A is copositive if and only if a, c ≥ 0 and b+√ac ≥ 0; see e.g. [11]. Let µ be

NONNEGATIVITY 7

−0.2 0 0.2 0.4 0.6 0.8

−0.8

−0.6

−0.4

−0.2 0 0.2 0.4 0.6 0.8 1

a

b

Figure 1. n = 2: Projection on the (a, b)-plane of C versus C¹, both intersected with the unit ball

the exponential measure on R²₊ with moments defined in (2.7). With fA(x) :=

ax²₁+ 2bx1x2+cx²₂ andd= 1, the conditionM_d(fAy)0 which reads

(2.15) 2





a+b+c 3a+ 2b+c a+ 2b+ 3c 3a+ 2b+c 12a+ 6b+ 2c 3a+ 4b+ 3c a+ 2b+ 3c 3a+ 4b+ 3c 2a+ 6b+ 12c



 0,

defines the convex cone C¹ ⊂R³. It is a connected component of the basic semi- algebraic set{(a, b, c) : det (M1(fAy))≥0}, that is, elements (a, b, c) such that:

(2.16) 3a³+ 15a²b+ 29a²c+ 16ab²+ 50abc+ 29ac²+ 4b³+ 16b²c+ 15bc²+ 3c³ ≥ 0.

Alternatively, by homogeneity, instead ofµwe may take the probability measureν uniformly supported on the simplex ∆ and with moments ˜y= (˜yα) given in (2.11), in which case the corresponding matrixM₁(fAy) now reads:˜

(2.17) 1

360





30(a+b+c) 6(3a+ 2b+c) 6(a+ 2b+ 3c) 6(3a+ 2b+c) 12a+ 6b+ 2c 3a+ 4b+ 3c 6(a+ 2b+ 3c) 3a+ 4b+ 3c 2a+ 6b+ 12c



 0.

It turns out that up to a multiplicative factor both matrices (2.15) and (2.17) have same determinant and so define the same convex cone C¹ (the same connected component of (2.16)). Figure 1 below displays the projection on the (a, b)-plane of the setsC¹ andCintersected with the unit ball.

2.3. An alternative representation of the coneCd^∗. From its definition (2.13) in Theorem 2.2, the coneCd^∗⊂ Sⁿis defined through the matrix variableX∈ S^s(d) which lives in a (lifted) space of dimension s(d)(s(d) + 1)/2 and with the linear

matrix inequality (LMI) constraintX0 of sizes(d). In contrast, the convex cone C^d ⊂ Sⁿ defined in (2.10) is defined solely in terms of the entries of the matrix A ∈ Sⁿ, that is, no projection from a higher dimensional space (or no lifting) is needed.

We next provide another explicit description on how Cd^∗ can be generated with no LMI constraint and with onlys(d) variables, but of course this characterization is not well suited for optimization purposes.

Since everyg∈Σ[x]d can be writtenP

ℓg_ℓ²for finitely many polynomials (gℓ)⊂ R[x]d, the convex cone Σ[x]d of s.o.s. polynomials can be written

Σ[x]d = conv{g² : g∈R[x]d}.

Next, forg∈R[x]d, with vector of coefficientsg= (gα)∈R^s(d), letg⁽²⁾ = (gα⁽²⁾)∈ R^s(2d) be the vector of coefficients ofg², that is,

g(x) = X

α∈Nn d

gαx^α → g(x)² = X

α∈Nn 2d

g_α⁽²⁾x^α.

Notice that for eachα∈Nⁿ_2d,g⁽²⁾α is quadratic ing. For instance, with n= 2 and d= 1,g(x) =g00+g10x1+g01x2 withg= (g00, g10, g01)^T ∈R^s(1), and so

g⁽²⁾ = (g²₀₀, 2g00g10,2g00g01, g₁₀² ,2g10g01, g₀₂² )^T ∈R^s(2).

Next, forg∈R[x]dand every 1≤i≤j≤n, letG_d= (Gij)∈ Sⁿbe defined by:

(2.18) Gij :=

Z

Rn

g(x)²xixjdµ(x) = X

α∈Nn 2d

g⁽²⁾_α (αi+ 1)!(αj+ 1)! Y

k6=i,j

αk!, for all 1≤i≤j≤n. We can now describeCd^∗.

Corollary 2.4. Let G_d∈ Sⁿ be as in (2.18). Then:

(2.19) Cd^∗ = cl

conv{G_d : g∈R^s(d)} . Proof. From Theorem 2.2

Cd^∗= cln

(hX,Md(xixjy)i)_{1≤i≤j≤n} : X∈ S+^s(d)

o . As in the proof of Theorem 2.2, writing X 0 as Ps

k=1g_kg^T_k for some vectors (gk)⊂R^s(d)(and associated polynomials (gk)⊂R[x]d),

hX,M_d(xixjy)i = Xs

k=1

Z

Rⁿ

+

xixjgk(x)²dµ,= Xs

k=1

G^k_ij,

for all 1≤i≤j≤n, whereG^k_ij is as in (2.18) (but now associated withgk instead ofg). Hence

(hX,M_d(xixjy)i)_{1≤i≤j≤n}= Xs

k=1

G^k_d ∈ conv{G_d : g∈R^s(d)}. HenceC^∗⊆cl conv{Gd : g∈R^s(d)}

.

NONNEGATIVITY 9

Conversely, leth∈conv{Gd:g∈R^s(d)}, i.e., for some integersand polynomials (gk)⊂R[x]d, and for all 1≤i≤j≤n,

hij = Xs

k=1

λk

|{z}>0

G^k_ij = Z

Rn +

Xs

k=1

λkg_k²(x)

!

xixjdµ(x)

= hX,M_d(xixjy)i whereX=Ps

k=1λkg_kg^T_k ∈ S+^s(d). Hence cl conv{G_d : g∈R^s(d)}

⊆ C^∗. The characterization (2.19) ofCd^∗should be compared with the characterization conv{xx^T : x∈Rⁿ₊}ofC^∗.

Example 2. Withn= 2 andd= 1,G1 reads:

2g²₀₀+ 12g10(g00+g01) + 4g01(g00+g01) + 24g₁₀² g²₀₀+ 4g00(g10+g01) + 6(g²₁₀+g²₀₁) + 8g10g01

g²₀₀+ 4g00(g10+g01) + 6(g₁₀² +g²₀₁) + 8g10g01 2g²₀₀+ 4g10(g00+g10) + 12g01(g00+g10) + 24g²₀₁

2.4. K-copositive and K-completely positive matrices. Let K ⊂ Rⁿ be a closed convex cone and let CK be the convex cone of K-copositive matrices, i.e., matricesA∈ Sⁿ such that x^TAx≥0 onK. Its dual coneCK^∗ ⊂ Sⁿ is the cone of K-completely positive matrices.

Then one may define a hierarchy of convex cones (C^K)d and (CK^∗)d, d ∈ N, formally exactly as in Theorem 2.1 and Theorem 2.2, but now y is the moment sequence of a finite Borel measureµ with suppµ=K (instead of suppµ=Rⁿ₊ in (2.7)), i.e.,y= (yα),α∈Nⁿ, with:

(2.20) yα :=

Z

K

x^αdµ, ∀α∈Nⁿ (and where suppµ=K).

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 replaceR

Rn + with R

K). Therefore, CK =

\∞

d=0

(CK)d and CK^∗ = cl [∞

d=0

(CK)^∗_d.

But of course, for practical implementation, one need to know the sequence y = (yα),α∈Nⁿ, which was easy whenK=Rⁿ₊. For instance, ifKis a polyhedral cone, by homogeneity offA, one may equivalently consider a compact base ofK(which is a polytopeK^′), and take for µthe Lebesgue measure onK^′. Then all moments ofµcan be calculated exactly. The same argument works for every convex coneK for which one may compute all moments of a finite Borel measureµwhose support is a compact base ofK.

Acknowledgement

The author wishes to thank two anonymous referees for their very helpful remarks and suggestions to improve the initial version of this note.

References

[1] K.M. Anstreicher, S. Burer. Computable representations for convex hulls of low-dimensional quadratic forms, Math. Program. S´er. B124(2010), pp. 33–43.

[2] I.M. Bomze. Copositive optimization - recent developments and applications, European J.

Oper. Res., 2011. To appear.

[3] I.M. Bomze, W. Schachinger and G. Uchida. Think co(mpletely) positive! - matrix properties, examples and a clustered bibliography on copositive optimization, J. Global Optim. 2011.

To appear.

[4] I.M. Bomze, E. de Klerk. Solving standard quadratic optimization problems via linear, semi- denite and copositive programming, J. Global Optim.24(2002), 163–185.

[5] S. Burer. Copositive programming, in Handbook of Semidefinite, Conic and Polynomial Optimization, M. Anjos and J.B. Lasserre, Eds., Springer, New York, 2012, pp. 201–218.

[6] E. de Klerk, D.V. Pasechnik. A linear programming formulation of the standard quadratic optimization problem, J. Global Optim.37(2007), pp. 75–84.

[7] E. de Klerk, D.V. Pasechnik. Approximation of the stability number of a graph via copositive programming, SIAM J. Optim.12(2002), 875–892.

[8] M. D¨ur. Copositive Programming - a survey, In Recent Advances in Optimization and its Applications in Engineering, M. Diehl, F. Glineur, E. Jarlebring and W. Michiels Eds., Springer, New York, 2010, pp. 3 – 20.

[9] J.W. Gaddum. Linear inequalities and quadratic forms, Pacific J. Math.8(1958), pp. 411–

414.

[10] A. Grundmann and H.M. Moeller. Invariant integration formulas for the n-simplex by com- binatorial methods, SIAM J. Numer. Anal.15(1978), pp. 282–290.

[11] J.-B. Hiriart-Urruty, A. Seeger. A variational approach to copositive matrices, SIAM Rev.

52(2010), 593–629.

[12] J.B. Lasserre. A new look at nonnegativity and polynomial optimization, SIAM J. Optim.

21(2011), pp. 864–885.

[13] P. Parrilo. Structured semidefinite programs and semialgebraic geometry methods in robust- ness and optimization, Ph.D. Dissertation, California Institute of Technology, 2000.

[14] J. Pe˜na, J. Vera, L. Zuluaga. Computing the stability number of a graph via linear and semidenite programming, SIAM J. Optim.18(2007), 87–105.

[15] R.T. Rockafellar.Convex Analysis, Princeton University Preess, Priceton, New Jersey, 1970.

[16] K. Schm¨udgen. TheK-moment problem for compact semi-algebraic sets, Math. Ann.289 (1991), 203–206.

[17] M. Schweighofer. Global optimization of polynomials using gradient tentacles and sums of squares, SIAM J. Optim.17(2006), pp. 920–942.

[18] G. Stengel. A Nullstellensatz and a Positivstellensatz in semialgebraic geometry, Math. Ann.

207, pp. 87–97.

[19] L. Vandenberghe and S. Boyd. Semidefinite programming, SIAM Rev.38(1996), pp. 49–95.

[20] E. A. Yildirim. On the accuracy of uniform polyhedral approximations of the copositive cone, Optim. methods Softw., 2011. To appear.

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E-mail address: lasserre@laas.fr