J B Lasserre
email: lasserre@laas.fr

LEVEL SETS AND NON GAUSSIAN INTEGRALS OF POSITIVELY HOMOGENEOUS FUNCTIONS

Keywords: 1991 Mathematics Subject Classification. 26B15 65K10 90C22 90C25 Positively homogeneous functions, sublevel sets, non gaussian integrals, volume, convex optimization

We investigate various properties of the sublevel set {x : g(x) ≤ 1} and the integration of h on this sublevel set when g and h are positively homogeneous functions. For instance, the latter integral reduces to integrating h exp(-g) on the whole space R n (a non Gaussian integral) and when g is a polynomial, then the volume of the sublevel set is a convex function of the coefficients of g. In fact, whenever h is nonnegative, the functional φ(g(x))h(x)dx is a convex function of g for a large class of functions φ : R+ → R. We also provide a numerical approximation scheme to compute the volume or integrate h (or, equivalently to approximate the associated non Gaussian integral). We also show that finding the sublevel set {x : g(x) ≤ 1} of minimum volume that contains some given subset K is a (hard) convex optimization problem for which we also propose two convergent numerical schemes. Finally, we provide a Gaussian-like property of non Gaussian integrals for homogeneous polynomials that are sums of squares and critical points of a specific function.

Introduction

Positively homogeneous functions (PHF) form a wide class of functions that one encounters in many applications. As a consequence of homogeneity, they enjoy very particular properties, and among them the celebrated and very useful Euler's identity which allows to deduce additional properties of PHFs in various contexts. One goal of this paper is to show that another not well-known property of PHFs yields surprising and unexpected results, some of them already known in particular cases. Namely, we are concerned with PHFs, their sublevel sets and in particular, the integral (1.1) y → I g,h (y) := {x : g(x)≤y} h(x) dx, as a function I g,h : R + → R when g, h are PHFs. With y fixed, we are also interested in I g,h (y) now as a function of g, especially when g is a nonnegative homogeneous polynomial. Nonnegative homogeneous polynomials are particularly interesting as they can be used to approximate norms; see e.g. Barvinok [4,Theorem 3.4]. Interestingly, the integral (1.1) is related in a simple and remarkable manner to the non-Gaussian integral R n h exp(-g)dx and therefore, any information on either one can help understanding and evaluating the other.

Functional integrals appear frequently in quantum Physics. In the important particular case when h = 1, Morosov et Shakirov [START_REF] Morosov | New and old results in resultant theory[END_REF] proved that for all forms g of degree k,

(1.2) I g,1 (1) = cte(k) • R n exp(-g)dx,
where the constant depends only on k. In fact, a formula of exactly the same flavor was already known for convex sets, and was the initial motivation and starting point of this paper. Namely, if C ⊂ R n is convex, its support function σ C is a PHF of degree 1, and the polar

C • ⊂ R n of C is the convex set {x : σ C (x) ≤ 1}. Then vol (C • ) = 1 n! R n exp(-σ C (x)) dx, ∀C.
(See e.g. Hiriart-Urruty [3, Exercise 65, p. 243-244] and Barvinok [4,Problem 4,p. 207].) In fact, as we will see, the intriguing result (1.2) is a particular case of a more general and striking result about PHFs. The connection between I g1 and exp(-g) is a consequence of what the authors in [START_REF] Morosov | New and old results in resultant theory[END_REF] call action-independence of integral discriminants. That is, for a very general class of univariate functions φ, and for every form g of degree d, (1.3) φ(g(x))dx = C(φ, d) • θ(g),

for some function θ, where the constant C(φ, d) depends only on φ and d; see [17, (73)]. And so for instance, with the choices t → φ 1 (t) = 1 [0,1] (t) and φ 2 (t) = exp(-t), and with R n as integration contour,

{x : g(x)≤1} dx = C(φ 1 , d) C(φ 2 , d) R n exp(-g) dx, ∀g,
whenever any of the above integrals finite. In Morosov and Shakirov [START_REF] Morosov | New and old results in resultant theory[END_REF] the motivation of the authors was not to connect I g1 with exp(-g) but rather to study a specific class of non-Gaussian integrals, the so-called integral discriminants. In fact, the theory of integral discriminants which lies in the field of non linear algebra, extends to integrals on more general contours of integration, and Ward identities permits to study properties of such integrals that are invariant under a change of contour of integration. Hence the goal in [START_REF] Morosov | New and old results in resultant theory[END_REF] is to provide exact formulas for integral discriminants in terms of algebraic invariants of the form g involved in the exponent; several highly non trivial examples are provided in [START_REF] Morosov | Introduction to integral discriminants[END_REF][START_REF] Shakirov | Nonperturbative approach to finite-dimensional non-Gaussian integrals[END_REF] and some of their results are further analyzed in Fuji [START_REF] Fujii | Beyond the Gaussian[END_REF] and Stoyanovsky [START_REF] Stoyanovsky | On integral of exponent of a homogeneous polynomial[END_REF].

Contribution. In the present paper we do not study non Gaussian integrals from the point of view of algebraic invariants as in e.g. [START_REF] Morosov | New and old results in resultant theory[END_REF][START_REF] Morosov | Introduction to integral discriminants[END_REF][START_REF] Shakirov | Nonperturbative approach to finite-dimensional non-Gaussian integrals[END_REF]. We are interested in properties of the integral (1.1) (and variants as well) and its associated non-Gaussian integral R n h exp(-g)dx for PHFs that are not necessarily polynomials. Among other things, we are also interested in methods for their evaluation (or approximation). More precisely: (a) We first extend the action-independence property (1.3) to PHFs of degree 0 = d ∈ R. Namely, given a measurable mapping φ : R + → R, and g, h PHFs of respective degree 0 = d, p ∈ R and such that |h| exp(-g)dx is finite,

(1.4) R n φ(g(x))h(x) dx = C(φ, d, p) • R n h exp(-g) dx,
where the constant C(φ, d, p) depends only on φ, d, p. In particular, if the sublevel set {x : g(x) ≤ 1} is bounded, then for every nonnegative y ∈ R, (1.5) {x : g(x)≤y}

h dx = y (n+p)/d Γ(1 + (n + p)/d) R n h exp(-g) dx,
with Γ being the standard Gamma function 1 . And so the (Lebesgue) volume of the sublevel set {x : g(x) ≤ y} is given by:

(1.6) vol ({x :

g(x) ≤ y}) = y n/d Γ(1 + n/d) R n exp(-g) dx.
As already mentioned, when h(x) = 1 for all x, and g is a positive definite form, (1.6) was already proved in Morosov et Shakirov [17, (75) p. 39]. In fact, the same arguments as in [START_REF] Morosov | New and old results in resultant theory[END_REF] can be adapted to PHFs g and with h = 1. The same technique applies for the integral I g 1 ,...,gm,h := {x : g k (x)≤1, k=1,...,m} hdx, for a finite family (g k ), h of PHFs with same degree 0 < d ∈ R.

In addition, we obtain the identity

{x : g(x)≤y} exp(-g) h dx R n exp(-g) h dx = y 0 exp(-z)z (n+p)/d-1 dz Γ((n + p)/d) , which expresses how fast µ({x : g(x) ≤ y}) converges to µ(R n )(= exp(-g)hdx)
when y → ∞ and µ has the non Gaussian density h exp(-g) with respect to the Lebesgue measure. It is the same speed with which the truncated (Gamma) integral

y 0 exp(-z)z (n+p)/d-1 dz converges to Γ((n + p)/d). (b)
We also provide an alternative and simple proof based on Laplace transform (in the spirit of Lasserre and Zeron [START_REF] Lasserre | Solving a class of multivariable integration problems via Laplace techniques[END_REF] to provide 1 The Gamma function Γ : R → R is defined by a → Γ(a) := ∞ 0 t a-1 exp(-t)dt, for every a > -1.

explicit expressions for certain integrals on specific domains like e.g., a simplex or an ellipsoid). This proof permits to interpret (1.2) as duality result in the usual (•, +)-algebra which parallels duality in convex optimization, i.e., in the (max, +)-algebra. This complements the list of the many remarkable properties of homogeneous functions; for instance in [START_REF] Lasserre | Integration and homogeneous functions[END_REF] for integration, in [START_REF] Lasserre | Mathematical properties of optimization problems defined by positively homogeneous functions[END_REF] for optimization, and in [START_REF] Lasserre | Homogeneous functions and conjugacy[END_REF] for several properties of some conjugates of homogeneous functions.

(c) Convexity: (1.5) also helps analyze properties of the mapping f h :

C d → R, g → f h (g) := {x : g(x)≤1} h(x) dx, ∀ g ∈ C d ,
where C d is the space of nonnegative PHFs of degree 0 = d ∈ R with bounded sublevel set {x : g(x) ≤ 1}, and h is a given PHF. Whenever h is nonnegative, f h is strictly convex and when g is a homogeneous polynomial, we provide an explicit expression of its gradient and Hessian on the interior of its domain. This is the analogue in our context of a formula already provided in Müller et al. [START_REF] Müller | Derivative of an integral over a convex polytope[END_REF] when the integration domain is a polytope. This convexity property is important. For instance, given K ⊂ R n , it shows that the problem of computing a homogeneous polynomial g such that {x : g(x) ≤ 1} contains K and has minimum volume, is a finite-dimensional convex optimization problem (but hard to solve even though it is convex)! This problem has received a lot of attention with an elegant solution in the case d = 2 (that is, finding the ellipsoid of minimum volume). Notice that here with d > 2 the problem is still convex even though the sublevel set {x : g(x) ≤ 1} is not required to be convex. We present two numerical approximation schemes based on semidefinite programming, that provide sequences of upper and lower bounds that converge to minimum volume.

(d) Polarity: When g is a convex PHF of degree d ∈ R with compact sublevel set G = {x : g(x) ≤ 1}, the polar G • is a certain sublevel set of the Legendre-Fenchel conjugate g * of g, which is itself a PHF of degree q with 1/d + 1/q = 1 and therefore we also obtain the volume of G • in the form of non Gaussian integral, the dual analogue of the one obtained for G.

(e) Numerical approximation: Evaluating a non Gaussian integral exp(-g) is a very hard problem even if in some cases exact results can be obtained from some algebraic invariants of g, as in e.g. [START_REF] Morosov | New and old results in resultant theory[END_REF][START_REF] Morosov | Introduction to integral discriminants[END_REF][START_REF] Shakirov | Nonperturbative approach to finite-dimensional non-Gaussian integrals[END_REF]. However, it turns out that (1.5) is indeed very useful to approximate non Gaussian integrals from polynomials. Indeed, when h is a polynomial and g is a homogeneous polynomial, we show how to approximate the non Gaussian integral h exp(-g)dx, as closely as desired, by solving a hierarchy of so-called semidefinite programs. This numerical approximation scheme complements the exact results mentioned above.

(f) A Gaussian-like property: Finally we also prove a (again intriguing) Gaussian-like property of homogeneous polynomials that are sums of squares (SOS). Recall that every homogeneous and SOS polynomial g of degree 2d can be written as ṽd (x) T Σṽ d (x) for some positive semidefinite matrix Σ (not unique in general) and where ṽd (x) is the vector of all monomials of degree d. If we now consider the function

Σ → θ(Σ) = (det Σ) k exp(-kṽ d (x) T Σṽ d (x)),
where

k = n/(2dℓ(d)) (with ℓ(d) = n+d-1 d
), then its critical points are the positive semidefinite matrices Σ such that Σ -1 is the matrix of moments of order 2d associated with the non Gaussian density exp(-kṽ d (x) T Σṽ d (x)) (normalized to be a probability density) ... exactly like in the Gaussian case (i.e. when d = 1) where Σ -1 is a covariance matrix! (In the latter case, the function θ is constant, hence all points Σ are critical). A polynomial f ∈ R[x] d is written

sublevel sets and non Gaussian integrals

x → f (x) = α∈N n f α x α ,
for some vector of coefficients f = (f α ) ∈ R s (d) .

A real-valued continuous function f : R n → R is homogeneous (resp. positively homogeneous) of degree k (k ∈ R) if f (λx) = λ k f (x) for all λ = 0 (resp. for all λ > 0) and all x ∈ R. For instance if x → f (x) is homogeneous of degree k, then |f | is positively homogeneous of degree k (but not homogeneous). Let us denote by:

• C d the convex cone of PHFs g of degree d ∈ R with bounded sub- level set G := {x : g(x) ≤ 1}. Notice that necessarily d = 0 and any g ∈ C d is nonnegative. Indeed, if d = 0 then x ∈ G implies λx ∈ G for every λ > 0, in contradiction with boundedness of G.
Similarly, assume that g(x 0 ) < 0 for some some x 0 (hence x 0 ∈ G).

As g(λx 0 ) = λ d g(x 0 ) < 0 for every λ > 0, λx 0 ∈ G for every λ > 0, again in contradiction with boundedness of G. Remember that a convex function is proper if f (x) > -∞ for all x and f (x) < +∞ for some x. A proper convex function is closed if it is lower semicontinuous; see Rockafellar [START_REF] Rockafellar | Convex Analysis[END_REF].

• P[x] d ⊂ R[x] d , d ∈ N,
In the sequel we will need the following result which extends to PHFs of degree d ∈ R, one already used in [START_REF] Morosov | New and old results in resultant theory[END_REF] for forms of degree d ∈ N (and also used in Hiriart-Urruty [START_REF] Azé | Analyse Variationnelle et Optimisation[END_REF] for the volume of the polar of a convex set).

Lemma 2.1. Let d, p ∈ R with d = 0, and let φ : R + → R + be a measurable function such that

∞ 0 t n+p-1 φ(t d ) dt < ∞,
and let g, h be nonnegative PHFs of degree 0 = d ∈ R and p ∈ R respectively.

The functional

R n φ(g(x))h(x) dx is finite if and only if R n h exp(-g)dx <
+∞, in which case:

(2.1) R n φ(g(x))h(x) dx = d ∞ 0 t n+p-1 φ(t d ) dt Γ((n + p)/d) cte(n,p,d,φ) • R n h exp(-g) dx.
Proof. With z = (z 1 , . . . , z n-1 ), do the change of variable x 1 = t, x 2 = tz 1 , . . . , x n = tz n-1 ) so that one may decompose R n φ(g(x))h(x)dx into the sum

R + ×R n-1 t n+p-1 φ(t d g(1, z))h(1, z) dt dz + R + ×R n-1 t n+p-1 φ(t d g(-1, -z))h(-1, -z) dt dz, = R n-1 ∞ 0 t n+p-1 φ(t d g(1, z)) dt h(1, z) dz + R n-1 ∞ 0 t n+p-1 φ(t d g(-1, -z)) dt h(-1, -z) dz,
where the last two integrals are obtained from the sum of the previous two by using Tonelli's Theorem (since φ and h are nonnegative functions) and whether or not the integrals are finite; see e.g. Dunford and Schwartz [START_REF] Dunford | Linear Operators: Part I: General Theory[END_REF]Theorem 14,p. 194].

Next, with the change of variable u = t g(1, z) 1/d and u = t g(-1, -z)

1/d (2.2) R n φ(g(x)) h(x)dx = R + u n+p-1 φ(u d ) du • A(g, h), with (2.3) A(g, h) = R n-1 h(1, z) g(1, z) (n+p)/d + h(-1, -z) g(-1, -z) (n+p)/d dz.
In particular, the choice t → φ(t) = exp(-t) yields:

(2.4) A(g, h) = d Γ((n + p)/d) R n h exp(-
Proof. Write h = h + -h -with h + (x) := max[0, h(x)], h -(x) := max[0, -h(x)]
for all x. Observe that |h|, h + and h -are nonnegative and positively homogeneous of degree p, and

|h| = h + + h -, h = h + -h -.
With A as in (2.2), using Lemma 2.1 with φ(t) = exp(-t), yields

R n |h| exp(-g) dx = Γ(1 + (n + p)/d) n + p • A(g, |h|)
where all integrals are finite since the right-hand-side is finite. Similarly, using Lemma 2.1 with φ(t) = 1 [0,1] (t), yields

{x:g(x)≤1} |h| dx = 1 n + p • A(g, |h|),
and therefore,

{x : g(x)≤1} |h| dx = 1 Γ(1 + (n + p)/d) R n |h| exp(-g) dx.
Next, by homogeneity,

{x : g(x)≤y} |h| dx = y (n+p)/d {x : g(x)≤1}
|h| dx, and so

(2.7)

{x : g(x)≤y} |h| dx = y (n+p)/d Γ(1 + (n + p)/d) R n |h| exp(-g) dx.
Obviously When y = 1, g is a positive definite form with d ∈ N, and h is identical to the constant function 1, (2.5) is already proved in Morosov and Shakirov [17, (75)

p. 39].
In particular, when g is the quadratic form x → g 2 (x) := 1 2 x T Qx for some real symmetric positive definite matrix Q, one retrieves that the volume of the ellipsoid ξ(y) := {x : g 2 (x) ≤ y} is simply related to the determinant of Q by the formula

(2.8) vol (ξ(y)) = y n/2 Γ(1 + n/2) R n exp(- 1 2 x T Qx) dx =(2π) n/2 / √ det Q So Theorem 2.
2 states that the volume of the sublevel set is simply related to the integral of exp(-g(x)) over the entire domain R n which happens to be simply related to the determinant of Q when g is the quadratic form x T Qx. One goal of the theory of integral discriminants is precisely to express exp(-g) in terms of invariants of g when g is a form. See e.g. Morosov and Shakirov [START_REF] Morosov | New and old results in resultant theory[END_REF][START_REF] Morosov | Introduction to integral discriminants[END_REF] and Shakirov [START_REF] Shakirov | Nonperturbative approach to finite-dimensional non-Gaussian integrals[END_REF].

Intersection of sublevel sets. Suppose that with h being positively homogeneous of degree p, one wishes to compute the integral Ω h(x) dx where

Ω := {x ∈ R n : g k (x) ≤ z k , k = 1, . . . , m},
for some m PHFs g 1 , . . . , g m of degree 0 = d ∈ R, and some (strictly) positive vector z ∈ R m . Equivalently,

Ω := {x ∈ R n : gk (x; z) ≤ 1, k = 1, . . . , m}, for the functions x → gk (x; z) := g k (z -1/d k x), k = 1, . . . , m,
which are also PHFs of degree d ∈ R. Hence with no loss of generality, one may restrict to sets of the form (2.9) Ω(y) := {x :

g k (x) ≤ y, k = 1, . . . , m}
for some positive scalar y ∈ R, and PHFs g 1 , . . . , g m of same degree

d ∈ R. Notice that x → ψ(x) := max[g 1 (x), . . . , g m (x)] is a PHF of degree d.
Corollary 2.3. Let h be a PHF of degree p ∈ R, let g 1 , . . . , g m be PHFs of degree 0 = d ∈ R, and assume that the set {x : g k (x) ≤ 1, k = 1, . . . , m} is bounded and for every y > 0, let Ω(y) be as in (2.9). Then:

(2.10) vol (Ω(y)) = y n/d Γ(1 + n/d) R n exp(-ψ) dx,

and

(2. 11)

Ω(y) h(x) dx = y (n+p)/d Γ(1 + (n + p)/d) R n h exp(-ψ) dx, whenever R n |h| exp(-ψ) dx. Proof. Notice that x → ψ(x) := max[g 1 (x), . . . , g m (x)] is also a PHF of degree 0 = d ∈ R, and Ω(y) = {x : ψ(x) ≤ y}. And so if Ω(y) is bounded then ψ ∈ C d . Hence (2.11)-(2.10) is just (2.5)-(2.6
) with h and ψ in lieu of h and g.

2.

3. An alternative proof with a duality interpretation. Next, we present an alternative proof of Theorem 2.2 that uses Laplace transform techniques and provides an interpretation of the result in an appropriate duality framework. Suppose that g, h ∈ C d . Since g is nonnegative, the function

I g,h vanishes on (-∞, 0]. Its Laplace transform L I g,h : C → R is the function λ → L I g,h (λ) := ∞ 0 exp(-λy)I g,h (y) dy,
and observe that

L I g,h (λ) = ∞ 0 exp(-λy) {x:g(x)≤y} hdx dy = R n h(x) ∞ g(x) exp(-λy)dy dx [by Fubini's Theorem] = 1 λ R n h(x) exp(-λg(x)) dx = λ -p/d λ R n h(λ 1/d x) exp(-g(λ 1/d x)) dx [by homogeneity] = 1 λ 1+(n+p)/d R n h(z) exp(-g(z)) dz [by λ -1/d x → z] = R n h(z) exp(-g(z)) dz Γ(1 + (n + p)/d) L y (n+p)/d (λ).
And so, by uniqueness of the Laplace transform,

I g,h (y) = y (n+p)/d Γ(1 + (n + p)/d) R n h(x) exp(-g(x)) dx,
which is the desired result. So the above expression of I g,h (y) is obtained by "inverting" the Laplace transform L I g,h at the point y, which in fact, as we next see, is solving a "dual" problem.

For analogy purposes, consider, the optimization problem

ρ g,h (y) = sup x {h(x) : g(x) ≤ y},
where y is fixed, h and -g are concave and h is nonnegative. Equivalently, ρ g,h (y) = exp(θ g,h (y)) where y → θ g,h (y) is the optimal value function of the optimization problem

P y : θ g,h (y) = sup x {ln h(x) : g(x) ≤ y }.
Associated with P y is the dual problem P * y : inf λ {G(λ) : λ ≥ 0}, where

G : R + → R is the function λ → G(λ) := sup x { ln h(x) + λ(y -g(x))}. Observe that: λ → G(λ) = λy + sup x { ln h(x) -λg(x)} = λy + sup z { ln(λ -p/d h(z)) -g(z)} [via z = λ 1/d x] = λy + - p d ln λ + sup z {ln h(z) -g(z)}.
And so the dual problem P * y reads

P * y : γ = sup z {ln h(z) -g(z)} + inf λ≥0 {λy - p d ln λ } = ln y p/d + ln sup z {h(z) exp(-g(z))} + p d (1 -ln p d )
In particular, if h is log-concave and g is convex, by a standard argument of convex optimization, γ = θ g,h (y) (= ln ρ g,h (y)). And therefore,

ρ g,h (y) = exp(θ g,h (y)) = y p/d exp(p/d) (p/d) p/d sup x { h(x) exp(-g(x)) }
to compare with

I g,h (y) = y (n+p)/d 1 Γ(1 + (n + p)/d) R n h(x) exp(-g(x)) dx.
Alternatively, the Legendre-Fenchel transform of θ g,h (y) (for the concave version) is the function

θ * g,h (λ) = inf y {λy -θ g,h (y)} = inf λ {λy -sup x {ln h(x) : g(x) ≤ y} } = inf x {λg(x) -ln h(x)} = -ln sup x {h(x) exp(-λg(x))}
and so when h is log-concave and g is convex, θ g,h (y) = (θ * g,h ) * (y), so that

θ g,h (y) = inf λ {λy -θ * g,h (λ)} = inf λ {λy + ln sup x {h(x) exp(-λg(x))} } = ln y p/d + ln sup z {h(z) exp(-g(z))} + p d (1 -ln p d ).
In summary, to the Laplace transform step (2.12)

L I g,h (λ) = R n
exp(-λy)I g,h (y) dy in the usual (•, +)-algebra, corresponds to the Legendre Fenchel transform

θ * g,h (λ) = inf y {λy -θ g,h (y)} = -sup y {-λy + θ g,h (y)} = -ln sup y {exp(-λy)ρ g,h (y)},
where to emphasize the analogy, the latter term can be written ln " " exp(-λy) ρ g,h (y) ,

i.e., the " sup " operator is integration " " in the (max, +)-algebra. And with this convention exp(θ * g,h (λ)) ≃ " " exp(-λy) ρ g,h (y).

(Compare with (2.12).) Similarly, the Laplace inverse transform step (2.13)

I g,h (y) = ω+i∞ ω-i∞ exp(λy) L I g,h (λ) dλ
(where ω ∈ C is on the right of all singularities of L I g,h ) and which yields

I g,h (y) = y (n+p)/d Γ(1 + (n + p)/d) R n h exp(-g)dx,
is solving the "dual" and corresponds to the Legendre-Fenchel transform (which is involutive

) applied to θ * g,h exp(θ g,h (y)) = exp(inf λ {λy -θ * g,h (λ)}) ≃ -" " exp(-λy) exp(θ * g,h (λ)) dλ
(compare with (2.13)), and which yields

exp(θ g,h (y)) = y p/d exp(p/d) (p/d) p/d " " h(x) exp(-g(x)).
A rigorous analysis of the links between integration and linear optimization on a polytope has been already investigated in [START_REF] Lasserre | Linear and Integer Programming vs Linear Integration and Counting[END_REF].

2.4. Approximating non Gaussian integrals. As we have already mentioned, in some cases the non Gaussian integral can be computed explicitly in terms of some on algebraic invariants of g; see e.g. Morosov and Shakirov [START_REF] Morosov | New and old results in resultant theory[END_REF] and Shakirov [START_REF] Shakirov | Nonperturbative approach to finite-dimensional non-Gaussian integrals[END_REF]. But so far there is no general formula and therefore an alternative is to seak for a numerical scheme for its evaluation, or at least, its approximation. For this purpose, we next show that Theorem 2.2 is helpful as it provides a means to compute any moment of the measure dµ = exp(-g)dx on R n by computing the same moment but now of the Lebesgue measure on the sublevel set {x : g(x) ≤ 1}. Indeed, for every

α ∈ N n , letting x → h(x) := x α in Theorem 2.2, yields (2.14) R n x α exp(-g(x)) dx = Γ(1 + (n + |α|)/d) {x : g(x)≤1}
x α dx, where |α| = i α i . Have we made any progress with this equivalence?

The answer is yes. If g is a (non necessarily homogeneous) polynomial, it turns out that every moment of the Lebesgue measure on the sublevel set {x : g(x) ≤ 1} can be approximated as closely as desired by solving a hierarchy of semidefinite programs 2 as described in Henrion et al. [START_REF] Henrion | Approximate volume and integration of basic semi-algebraic sets[END_REF]. In fact, for every α ∈ N n fixed, the moment

z α := {x : g(x)≤1}
x α dx can be approximated to arbitrary precision ǫ > 0 fixed in advance, by solving two sequences of semidefinite programs, one which provides a monotone non decreasing sequence of upper bounds u k , k ∈ N, while the other provides a monotone non increasing sequence of lower bounds ℓ k , k ∈ N. The procedure stops whenever u k -ℓ k < ǫ, in which case one may set

z α ≈ zα := (u k + ℓ k )/2.
This requires to solve two sequences of semidefinite programs for each α ∈ N n . In fact, if one is ready to relax the monotonicity property of the upper and lower bounds {u k , ℓ k }, it is enough to solve a single sequence of semidefinite programs, e.g., the one defined to approximate the mass z 0 . Then if d ∈ N and ǫ > 0 are fixed, and k is large enough, not only |u k -z 0 | < ǫ but also from the solution of the semidefinite program at step k one obtains scalars zα such that |z α -z α | < ǫ, for all α ∈ N n such that |α| < d. However, in contrast to the case of upper and lower bounds, there is no simple stopping criterion to guarantee the ǫ-approximation. For more details, the interested reader is referred to Henrion et al. [START_REF] Henrion | Approximate volume and integration of basic semi-algebraic sets[END_REF]. And therefore, once the 2 A semidefinite program is a finite-dimensional convex conic optimization problem, that up to arbitrary (fixed) precision, can be solved efficiently, i.e., in time polynomial in the input size of the problem. For more details the interested reader is referred to e.g. [START_REF] Wolkowicz | Handbook of Semidefinite Programming[END_REF].

zα have been computed, using Theorem 2.2, we obtain:

R n x α exp(-g(x)) dx -zα Γ(1 + (n + |α|)/d) < ǫ, ∀α ∈ N n d ,
which provides an ǫ-approximation guarantee for the non Gaussian integral R n exp(-g)dx (and more generally for the integral R n h exp(-g)dx whenever h is any polynomial).

2.5. Sensitivity analysis and convexity. Recall that when d ∈ N, P[x] d ⊂ C d is the convex cone of nonnegative and homogeneous polynomials of degree d, with compact sublevel set {x : g(x) ≤ 1}. Formula (2.6) of Theorem 2.2 allows us to provide insights into the function f : C d → R, defined by:

(2.15) g → f h (g) := {x : g(x)≤1} h(x) dx, g ∈ C d ,
where h is a PHF. In particular when one wishes to see how f h changes when some coefficient of g ∈ P

∂f (g) ∂g α = x α exp(-g) dµ, ∀α, |α| ≤ d. (2.16) ∂ 2 f (g) ∂g α ∂g β = x α+β exp(-g) dµ, ∀α, β, |α|, |β| ≤ d. (2.17) Proof. (a) Observe that f is nonnegative. Let α ∈ [0, 1] and let g, q ∈ C d . To prove f (αg + (1 -α)q) ≤ αf (g) + (1 -α)f (q),
we only need consider the case where f (g), f (q) < +∞, for which we have

f (αg + (1 -α)q) = exp(-αg -(1 -α)q) dµ.
By convexity of u → exp(-u),

f (αg + (1 -α)q) ≤ [ α exp(-g) + (1 -α) exp(-q) ] dµ = αf (g) + (1 -α)f (q),
and so f is convex. Now, in view of the strict convexity of u → exp(-u), equality may occur only if g(x) = q(x), µ-almost everywhere. If g, q ∈ C 0 d , the set ∆ := {x : g(x) -q(x) = 0} is closed and so if ∆ = R n then µ(∆) < µ(R n ) because supp µ = R n . Therefore, equality occurs only if g = q so that f is strictly convex on C 0 d .

(b) Next, if d ∈ N and g ∈ int(Θ µ ), write g in the canonical basis as g(x) = α g α x α . For every α ∈ N n d , let e α = (e α (β)) ∈ R s(d) be such that e α (β) = δ β=α (with δ being the Kronecker symbol). Then for every t ≥ 0,

f (g + te α ) -f (g) t = exp(-g)     exp(-tx α ) -1 t ψ(t,x)     dµ(x)
Notice that for every x, by convexity of the function t → exp(-tx α ),

lim t↓0 ψ(t, x) = inf t≥0 ψ(t, x) = exp(-tx α ) ′ |t=0 = -x α ,
because for every x, the function t → ψ(t, x) is nondecreasing; see e.g. Rockafellar [START_REF] Rockafellar | Convex Analysis[END_REF]Theorem 23.1]. Hence, the one-sided directional derivative f ′ (g; e α ) in the direction e α satisfies

f ′ (g; e α ) = lim t↓0 f (g + te α ) -f (g) t = lim t↓0 exp(-g) ψ(t, x) dµ(x) = exp(-g) lim t↓0 ψ(t, x) dµ(x) = -x α exp(-g) dµ(x),
where the third equality follows from the Extended Monotone Convergence Theorem [2, 1.6.7]. Indeed for all t < t 0 with t 0 sufficiently small, the function ψ(t, •) is bounded above by ψ(t 0 , •) and exp(-g)ψ(t 0 , x)dµ < ∞.

Similarly, for every t > 0

f (g -te α ) -f (g) t = exp(-g) exp(tx α ) -1 t ξ(t,x) dµ(x),
and by convexity of the function t → exp(tx α )

lim t↓0 ξ(t, x) = inf t≥0 ξ(t, x) = exp(tx α ) ′ |t=0 = x α .
Therefore, with exactly same arguments as before,

f ′ (g; -e α ) = lim t↓0 f (g -te α ) -f (g) t = x α exp(-g) dµ(x) = -f ′ (g; e α ),
and so ∂f (g)

∂g α = - R n x α exp(-g) dµ(x),
for every α with |α| ≤ d, which yields (2.16). Similar arguments can used for the Hessian ∇ 2 f (g) which yields (2.17). So the Hessian ∇ 2 f (g) is the matrix M d ∈ S ℓ(d) whose rows and columns are indexed in the set Γ d := {α ∈ N n : |α| = d} and with entries

M d (α, β) = R n x α+β exp(-g) dµ dν , α, β ∈ Γ d ,
i.e., M d is the matrix of 2d-moments of the finite Borel measure ν. Let h ∈ R ℓ(d) be the coefficient vector of a non trivial and arbitrary homogeneous polynomial

h ∈ R[x] d . Then h, M d h = R n h(x) 2 dν(x) > 0 because µ(O) > 0 (hence ν(O) > 0) on some open set O ⊂ R n . Therefore ∇ 2 f (g) ≻ 0
, which in turn implies that f is strictly convex on int(Θ µ ).

Corollary 2.5. Let h be a PHF of degree p ∈ R and with 0 = d ∈ R, consider the function f h : C d → R defined by:

(2.18) g → f h (g) := {x:g(x)≤1} h(x) dx, ∀g ∈ C d .
The function f h is a PHF of degree -(n + p)/d and convex whenever h is nonnegative (and strictly convex if h > 0 on R n \ {0}). In addition, if d ∈ N and h is continuous with |h| exp(-g)dx < ∞, then:

(a) f h is twice differentiable on int(P[x] d ), and for every α, β ∈ N n d : ∂f h (g)

∂g α = -1 Γ(1 + (n + p)/d) R n x α h(x) exp(-g(x)) dx (2.19) = -Γ(2 + (n + p)/d) Γ(1 + (n + p)/d) {x : g(x)≤1} x α h(x) dx. (2.20) ∂ 2 f h (g) ∂g α ∂g β = 1 Γ(1 + (n + p)/d) R n x α+β h(x) exp(-g(x)) dx. (2.21) (b) If h is
non trivial and nonnegative, then f h is strictly convex on int(P[x] d ), and its Hessian ∇ 2 f h (g) is the matrix of 2d-moments of the measure

ν(B) = 1 Γ(1 + (n + p)/d) B h exp(-g) dx, B ∈ B.
Proof. With λ > 0 and g ∈ C d ,

f h (λg) = {x:λg(x)≤1} h dx = {x:g(λ 1/d x)≤1}
h dx, and so, doing the change of variable z = λ

f h (g) = 1 Γ(1 + (n + p)/d) R n exp(-g) dµ.
Hence by Lemma 2.4, f h is convex (and strictly convex if h > 0 on R n \ {0}).

In addition, if d ∈ N and h is continuous,

f h (g) < ∞ if g ∈ int(P[x] d ) (

21). This proves (b).

To prove (a) when h is not nonnegative, write h = h + -h -with h + := max[0, h] and h -:= max[0, -h]. Both h + and h -are continuous PHFs of degree p, and nonnegative. Moreover, f h = f h + -f h -and so applying (b) to f h + and f h -, yields (a) by additivity.

Remark 2.6. (a) Notice that proving convexity of f h directly from its definition (2.18) is not obvious at all whereas it becomes much easier when using Theorem 2.2.

(b) In Lemma 2.4, differentiability of f on the convex cone C d should be now in the sense of Gâteaux-differentiablity, not explored here.

We end up with the following relatively surprising results which even though are again particular cases of Lemma 2.1, deserve special mention.

Lemma 2.7. Let y ≥ 0 be fixed, let h be a PHF of degree p ∈ R and let ξ, ψ : R + → R be measurable functions such that {t:ψ(t d )≤y}

t n+p-1 ξ(t d ) dt < +∞. Let f h : C d → R, 0 = d ∈ R, be the function: g → f h (g) := {x : ψ(g(x))≤y} ξ(g(x)) h(x) dx, g ∈ C d .
Then whenever |h| exp(-g)dx < +∞, f h (g) is finite, and

(2.22) f h (g) = d {t:ψ(t d )≤y} t n+p-1 ξ(t d ) dt Γ((n + p)/d) cte(y,p,d,ξ,ψ) • R n h exp(-g)dx,
where the constant depends only on ξ, ψ, d, p, y and neither on g nor h.

Therefore, f h is convex whenever h is nonnegative (and strictly convex on C 0 d whenever h > 0 on R n \ {0}). In particular,

R n gh exp(-g) dx = n + p d R n h exp(-g) dx (2.23) {x : g(x)≤1} gh dx = 1 Γ((n + p)/d) R n h exp(-g) dx (2.24) {x : g(x)≤y} exp(-g) dx R n exp(-g) dx = y 0 exp(-z)z n/d-1 dz Γ(n/d) (2.25) {x : g(x)≤y} exp(g) dx = R n exp(-g) dx Γ(n/d) y 0 exp(z)z n/d-1 dz. (2.26)
Proof. We first assume that h is nonnegative and h exp(-g) < +∞, so that (2.22) follows from Lemma 2.1 with t → φ(t

) := ξ(t)I [0,y] (ψ(t)), which yields f h (g) = cte(y, p, d, ξ, ψ) • A(g, h), with A(•, •) as in (2.3) and cte(y, p, d, ξ, ψ) = {t:ψ(t d )≤y} t n+p-1 ξ(t d ) dt,
and the result follows by recalling that with φ(t) = exp(-t) one had already obtained in (2.4)

A(g, h) = d Γ((n + p)/d) R n h exp(-g) dx ( < +∞).
When h is not nonnegative, writing h = h + -h -where both h + and h -are also PHFs of degree p, the result follows by additivity since |h| exp(-g)dx < +∞ only if both h + exp(-g)dx and h -exp(-g)dx are finite. Finally, (2.23)-(2.26) are special cases of (2.22) with respective choices t → ψ(t) := 0, ξ(t) := exp(-t), then t → ψ(t) := t, ξ(t) = t and finally, t → ψ(t) = t, ξ(t) := exp(-t) and t → ξ(t) := exp(t).

At last, when h is nonnegative, convexity and strict convexity follow from Lemma 2.4(a) with dµ = hdx (and so supp µ = R n if h > 0). So Lemma 2.7 shows that f is convex provided that h is nonnegative and no matter how the functions ξ and ψ behave! Next, if µ is the non Gaussian measure dµ = exp(-g)dx on R n , then (2.25) shows how fast µ({x : g(x) ≤ y}) converges to the non Gaussian integral R n exp(-g)dx as y → ∞. It converges as fast as the one-dimensional integral y 0 t n/d-1 exp(-t)dt converges to the Gamma function Γ(n/d).

2.6. Polarity. We here investigate the polar G • set of the sublevel set G := {x : g(x) ≤ 1} assumed to be compact and when g is a proper closed convex PHF. In this case G is a convex body, and in fact g is a gauge.

The polar C • of a set C ⊂ R n is the convex set defined by

C • = {x ∈ R n : σ C (x) ≤ 1} with σ C (x) := sup y { x, y : y ∈ C}. and (C • ) • is the smallest convex balanced set that contains C. Recall that the Legendre-Fenchel conjugate f * : R n → R ∪ {+∞, -∞} of f : R n → R is defined by f * (u) := sup x { u, x -f (x) } u ∈ R n .
The conjugate g * of a PHF g of degree d ∈ R is tself a PHF of degree q with 1 d + 1 q = 1 (where d does not need to be positive); see e.g. Lasserre [START_REF] Lasserre | Homogeneous functions and conjugacy[END_REF]. Proposition 2.8. Let g be closed proper convex PHF of degree 1 < d ∈ R, and let G := {x : g(x) ≤ 1/d}. Then with

1 d + 1 q = 1, G • = {x ∈ R n : g * (x) ≤ 1/q}, (2.27)
where g * is the Legendre-Fenchel conjugate of g. In other words, G • is a sublevel set the PHF g * of degree q. Moreover, if G is bounded then

(2.28) vol (G • ) = 1 q n/q Γ(1 + n/q) R n exp(-g * ) dx.
Proof.

x 4 1 + x 4 2 ≤ 1/4}. Then g * (x) = 3(x 4/3 1 + x 4/3
2 )/4 4/3 (a PHF of degree 4/3) and G • = {x :

x 4/3 1 + x 4/3 2 ≤ 1/4 1/3 }.

2.7.

A variational property of homogeneous polynomials. We end up this section with an intriguing variational property of homogeneous polynomials that are sums of squares.

Let ṽd (x) be the vector of all monomials (x α ) of degree d and let g ∈ R[x] 2d be homogeneous and a sum of squares, that is,

(2.29) g(x) = - 1 2 ṽd (x) T Σṽ d (x), x ∈ R n , for some real symmetric ℓ(d) × ℓ(d) matrix Σ which is positive definite (denoted Σ ≻ 0). If d = 1 it is well-known that R n exp(-g)dx = (2π) n/2 √ det Σ ,
and

R n ṽd (x)ṽ T d (x) exp(-g)dx = (2π) n/2 √ det Σ Σ -1 ,
that is, Σ -1 is the covariance matrix associated with the Gaussian probability density (2π) -n/2 √ det Σ exp(-g). When d > 1 the non Gaussian integral can be still expressed as a (possibly complicated) combination of several algebraic invariants of g, but in general not in terms of the single algebraic invariant det Σ.

It turns out that det Σ and exp(- (2.30)

θ d (Σ) := (det Σ) k R n exp(-kṽ d (x) T Σ ṽd (x)) dx, Σ ∈ S ++ ℓ(d) ,
where k = n/(2dℓ(d)) and let

M d (Σ) := R n ṽd (x)ṽ d (x) T exp(-kṽ d (x) T Σ ṽd (x)) dx R n
exp(-kṽ d (x) T Σ ṽd (x)) dx , be the matrix of moments of order d, associated with the non Gaussian probability measure

(2.31) µ(B) := B exp(-kṽ d (x) T Σṽ d (x)) dx R n exp(-kṽ d (x) T Σṽ d (x)) dx , B ∈ B.
Observe that θ d is nonnegative and positively homogeneous of degree 0; therefore θ d is constant in any fixed direction Σ. In particular, if d = 1 then k = 1/2, µ is a Gaussian probability measure, M 1 (Σ) is the associated covariance matrix Σ -1 and θ d (Σ) is constant. In fact, Lemma 2.9. Let θ d be the function defined in (2.30). Then M d (Σ), Σ = ℓ(d) for all Σ in the domain of θ d and ∇θ d (Σ) = 0 if

(2.32) M d (Σ) = Σ -1 .
Proof. Let g(x) = kṽ d (x)Σṽ d (x) so that

M d (Σ), Σ R n exp(-g) dx = R n ṽd (x)ṽ d (x) T exp(-g) dx, Σ = k -1 R n kṽ d (x) T Σṽ d (x) exp(-g) dx = k -1 R n g exp(-g) dx = k -1 n 2d R n exp(-g) dx [by (2.23)],
which yields the desired result M d (Σ), Σ = ℓ(d). Next, write the gradient ∇θ(Σ) in the form A 1 + A 2 with

A 1 = ∇((det Σ) k ) R n exp(-kṽ d (x) T Σṽ d (x)) dx = k(det Σ) k-1 Σ A R n exp(-kṽ d (x) T Σṽ d (x)) dx = k Σ A det Σ (det Σ) k R n exp(-kṽ d (x) T Σṽ d (x)) dx = kΣ -1 θ d (Σ),
where Σ A is the adjugate of Σ (see e.g. [5, p. 411]), and

A 2 = (det Σ) k ∇ R n exp(-kṽ d (x) T Σṽ d (x)) dx = -k(det Σ) k R n ṽd (x)ṽ d (x) T exp(-kṽ d (x) T Σṽ d (x)) dx = -kM d (Σ) θ d (Σ).
This yields

A 1 +A 2 = kθ d (Σ)(Σ -1 -M d (Σ)) and so ∇θ d (Σ) = 0 if M d (Σ) = Σ -1 .
Lemma 2.9 states that for all critical points Σ, or equivalently for all critical SOS homogeneous polynomials g of the function θ d (assuming that at least one such critical point exists), their associated non Gaussian measure dµ = exp(-g)dµ (rescaled to a probability measure) has the Gaussian-like property that Σ -1 is the "d-covariance" matrix of µ!

sublevel set of minimum volume containing a compact set

If K ⊂ R n is a convex body, computing the ellipsoid of minimum volume that contains K is a classical problem which has an optimal solution called the Löwner-John ellipsoid; see e.g. Barvinok [4,p. 209]. In this section we consider the following generalization: P: Find a homogeneous polynomial g of degree 2d such that its sublevel set G := {x : g(x) ≤ 1} contains K and has minimum volume among all such sublevel sets with this inclusion property.

With K ⊂ R n , let C 2d (K) ⊂ R[x]
2d be the convex cone of polynomials of degree at most 2d that are nonnegative on K. Recall that P

[x] 2d ⊂ R[x] 2d
is the convex cone of homogeneous polynomials of degree 2d with compact sublevel set {x : g(x) ≤ 1}. We next show that problem P is a convex optimization problem: Proposition 3.1. The minimum volume of a sublevel set {x : g(x) ≤ 1},

g ∈ P[x] 2d , that contains K ⊂ R n is ρ/Γ(1 + n/2d
) where ρ is the optimal value of the finite-dimensional convex optimization problem:

(3.1) P : ρ = inf g∈P[x] 2d R n exp(-g) dx : 1 -g ∈ C 2d (K) . Proof. From Theorem 2.2 vol ({x : g(x) ≤ 1}) = 1 Γ(1 + n/2d) R n exp(-g) dx.
Moreover, the sublevel set {x : g(x) ≤ 1} contains K if and only if 1 -g ∈ C 2d (K), and so ρ/Γ(1 + n/2d) in (3.1) is the minimum value of all volumes of sublevels sets {x :

g(x) ≤ 1}, g ∈ P[x] 2d , that contain K. Now since g → R n exp(-g)dx is strictly convex (see Corollary 2.5(b)) and C 2d (K) is a convex cone, problem P is a finite-dimensional convex optimization problem.
We also have the following characterization of an optimal solution of P when it exists. Let M (K) be the convex cone of finite Borel measures on K. Theorem 3.2. Let K ⊂ R n be compact and consider the convex optimization problem P in (3.1).

(a) Suppose that g * ∈ P[x] 2d is an optimal solution of P. Then there exists µ * ∈ M (K) such that

(3.2) R n x α exp(-g * )dx = K x α dµ * , ∀|α| = 2d; K (1 -g * ) dµ * = 0.
In particular, µ * is supported on the real variety V := {x ∈ K : g(x) = 1} and in fact, µ * can be substituted with another measure ν * ∈ M (K) supported on at most n+2d-1 2d

+ 1 points of V . (b) Conversely, if g * ∈ P[x] 2d and µ * ∈ M (K) satisfy (3.
2) then g * is an optimal solution of P.

Proof. (a) We may and will consider g * as an element of R[x] 2d with g * β = 0 whenever |β| < 2d. As K is compact, there exists θ ∈ P[x] 2d such that 1 -θ ∈ int C 2d (K), i.e., Slater's condition holds for P. Indeed, choose θ := M -1 x 2d for M > 0 sufficiently large so that 1 -θ > 0 on K. Hence with g 1 denoting the ℓ 1 -norm of the coefficient vector of g (in R[x] 2d ), there exists ǫ > 0 such that for every h ∈ B(θ, ǫ)(:

= {h ∈ R[x] 2d : θ -h 1 < ǫ}), 1 -h > 0 on K.
Therefore, the optimal solution g * satisfies the KKT-optimality conditions, which read:

(3.3) R n x α exp(-g * ) dx = y * α , ∀|α| = 2d; 1 -g * , y * = 0, for some y * = (y * α ), α ∈ N n 2d , an element of the dual cone C 2d (K) * ⊂ R s(2d) of C 2d (K). By Lemma 4.1 in §4.3, C 2d (K) * = {y ∈ R s(2d) : ∃µ * ∈ M (K) s.t. y α = K x α dµ * , α ∈ N n 2d },
and so (3.2) is just (3.3) restated in terms of µ * . Finally, the last statement follows from Anastassiou [1, Theorem 2.1.1, p. 39] applied to the n+2d-1 2d equality constraints of (3.2). (b) As Slater's condition holds for P, the KKT-optimality conditions (3.2) are sufficient to ensure that g * is an optimal solution on P. Theorem 3.2 states that when (3.2) holds, there is an optimal solution g * to P such that g(x) = 1 on (at most) n+2d-1 2d + 1 points of K, the analogue for d > 1 of the well-known property of the Löwner-John ellipsoid in the case d = 1.

Even though being convex and finite-dimensional, P is by no means easy to solve because there is no simple and computationally tractable way of describing the convex cone C 2d (K). However, there are cases where one may provide a sequence of inner or outer approximations that both converge to C 2d (K). One such case is when one knows all moments of a finite Borel measure whose support is exactly K, and another case is when K = {x : g k (x) ≥ 0, k = 1, . . . , m} for some polynomials (g k ) ⊂ R[x], i.e., K is a compact basic semi-algebraic set.

3.1. Lower bounds via inner approximations. Suppose that one knows all moments z = (z α ), α ∈ N n , of a finite Borel measure µ on K, i.e.,

z α = K x α dµ(x), ∀α ∈ N n ,
whose support is exactly K. For every k ∈ N and p ∈ R[x], let M k (p, z) be the localizing matrix with respect to the polynomial p and the moment sequence z, that is, M k (p, z) is the s(k) × s(k) real symmetric matrix with rows and columns indexed in the canonical basis (x α ), α ∈ N n k , of R[x] k , and with entries

M k (p, z)(α, β) = K p(x) x α+β dµ(x), ∀α, β ∈ N n k = γ p γ z α+β+γ
(when p(x) = γ p γ x γ ). We recall the following result. In view of Lemma 3.3, a natural idea is to relax the "difficult" constraint 1 -g ∈ C 2d (K) in (3.1) to M k (1 -g, z) 0 for fixed k, and then let k → ∞. Indeed, for every fixed k, the latter is much easier to handle as it defines a spectrahedron 3 ∆ k ⊂ R ℓ(2d) on the coefficients of the homogeneous polynomial g. And so one obtains a hierarchy of convex relaxations of P by minimizing the (strictly) convex function g → exp(-g) on the spectrahedra ∆ k , k ∈ N, which yields a monotone nondecreasing sequence of lower bounds ρ k ≤ ρ, k ∈ N, on ρ. Of course the larger k (i.e., the larger the size of the localizing matrix M k (1 -g, z)) the better the lower bound ρ k (but also the harder the problem).

Theorem 3.4. Let K ⊂ R n be compact with nonempty interior and consider the finite-dimensional convex optimization problem:

(3.4) P k : ρ k = inf g∈P[x] 2d R n exp(-g) dx : M k (1 -g, z) 0 .
The sequence (ρ k ), k ∈ N, is monotone nondecreasing with ρ k ≤ ρ and: for some M > 0, then ρ k → ρ and P has an optimal solution g * .

For a proof see §4.1. So problem P k amounts to minimize a strictly convex function on a spectrahedron of R ℓ(2d) . For instance, one may use interior point methods and minimize the standard log-barrier function

g → φ ν (g) := R n exp(-g)dx - 1 ν log det(M k (1 -g, z)),
3 A spectrahedron is a convex set that can be formed by intersecting the cone of positive semidefinite matrices Sn with a linear affine subspace.

with parameter ν, and let ν → ∞. For more details on log-barrier methods, the reader is referred to e.g. Wright [START_REF] Wright | On the convergence of the Newton/log-barrier method[END_REF].

3.2. Upper bounds via outer approximations. Let K ⊂ R n be the compact basic semi-algebraic set defined by K = {x : u j (x) ≥ 0, j = 1, . . . , m}, for some polynomials (u j ) ⊂ R[x]. As K is compact, with no loss of generality we may and will suppose that u 1 (x) = Mx 2 for some M large enough. Let

Q * = { m j=0
σ j u j : u j ∈ Σ[x], j = 0, . . . , m}, be the quadratic module of R[x] generated by the u j 's (with u 0 = 1). Q * is Archimedean because the quadratic polynomial x → Mx 2 belongs to Q * . If one defines

Q * k := { m j=0 σ j u j : u j ∈ Σ[x]; deg σ j u j ≤ 2k, j = 0, . . . , m}, k ∈ N,
(a subset of Q * with a degree bound on the SOS weights σ j ) then this time one may replace the "difficult" constraint 1 -g ∈ C 2d (K) by the stronger constraint 1

-g ∈ Q * k for fixed k ∈ N. The convex cone Q * k is the dual cone of the closed pointed convex cone (3.7) Q k := {y ∈ R s(2k) : M k (y) 0, M k-v j (u j , y) 0, j = 1, . . . , m},
where M k (y) (resp. M k-v j (u j , y)) is the moment matrix associated with the sequence y (resp. the localizing matrix associated with y and the polynomial u j ). Hence, Q * k has a nonempty interior; see e.g. Rockafellar [START_REF] Rockafellar | Convex Analysis[END_REF]. Then solving the problem (3.8) (a) The sequence

P ′ k : inf g∈P[x] 2d R n exp(-g)dx : 1 -g ∈ Q * k , k ∈ N, k ≥ d,
(ρ ′ k ), d ≤ k ∈ N, is monotone nonincreasing with ρ ′ k → ρ as k → ∞.
Moreover there exists an optimal solution g * k ∈ P[x] 2d . In addition, assume that there exists g 0

∈ P[x] 2d such that 1-g 0 ∈ int Q * k 0 for some k 0 ≥ d. Then: (b) If k ≥ k 0 and g * k ∈ P[x]
2d is an optimal solution of P ′ k , there exists a vector y k ∈ Q k such that:

(3.9) 0 = 1 -g * k , y k ; y k α = R n x α exp(-g * k ) dx, ∀ |α| = 2d. (c) Conversely, if k ≥ k 0 and (y k , 1 -g * k ) ∈ Q k × Q * k satisfy (3.9), then g * k is an optimal solution of P ′ k . (d) Let (g * k ) ⊂ P[x] 2d , k ≥ k 0 , be a sequence of optimal solutions. If sup k sup |α|=2d |g * kα | < N,
for some N > 0, then every accumulation point g * of the sequence (g * k ), k ∈ N, is an optimal solution of P.

For proof see §4.2. Proof. (a) That the sequence (ρ k ) is monotone non decreasing is straightforward since the constraints of P k are more and more restrictive as k increases. Next, as K is compact there exists M, δ > 0 such that Mx 2d > δ for all x ∈ K. With g 0 ∈ Σ[x] d being the polynomial x → M -1 x 2d , one has 1 -g 0 (x) > δ/M for all x ∈ K and so M k (1 -g 0 , z) ≻ 0. Indeed, for every 0

= f ∈ R[x] k , f , M k (1 -g 0 , z) f = K f 2 (1 -g 0 ) dµ ≥ δ M K f 2 dµ > 0.
where the last inequality is because K has nonempty interior and supp µ = K. Observe also that g 0 ∈ P[x] 2d and so g 0 is a strictly feasible solution of P k , that is, Slater's condition 4 holds for P k . So the Karush-Kuhn-Tucker (KKT) optimality conditions at a point g * k are necessary and sufficient for g * k to be a (global) minimizer of P k . Therefore, there exists 0 ∆ ∈ S s(k) such that:

(4.1) R n -x α exp(-g * k ) dx + ∆, M k (x α , z) = 0, ∀|α| = 2d, and
(4.2) M d (1 -g * k , z), ∆ = 0, where one has written M k (g * k z) = α g * kα M k (x α , z) (with M k (x α , z
) being the localizing matrix with respect to the polynomial x α and the moment sequence z). From its spectral decomposition, ∆ = j q j q T j for some vectors (q j ), which yields

∆, M k (x α , z) = K σ * k (x) x α dµ, ∀|α| = 2d, 4
For a convex optimization infx{f (x) : h k (x) ≥ 0, k = 1, . . . , m} Slater's condition holds if there exists x0 such that h k (x0) > 0 for all k. In this case the Karush-Kuhn-Tucker (KKT) optimality conditions are necessary and sufficient.

where σ * k = j (q T j v k (x)) 2 ∈ Σ k is a SOS polynomial. And so (4.1) yields (3.5), and the complementary condition (4.2) yields (3.6). Next, multiplying both sides of (4.1) with g * kα and summing up yields:

R n g * k exp(-g * k ) dx = n 2d R n exp(-g * k ) dx = ∆, M k (g * k , z) = K σ * k g * k dµ, = K σ * k dµ [by (4.2)] (b)
The converse is because under Slater's condition, the KKT optimality conditions are also sufficient for g * k to be a minimizer.

(c) Write g * k (x) = |α|=2d g * kα x α . As sup k sup |α|=2d |g * kα | < M
, there exists a subsequence (k i ), i ∈ N, and a vector g * ∈ R ℓ(2d) such that for every

α ∈ N n with |α| = 2d, g * k i α → g * α as i → ∞. Notice that x → g * (x) = lim sup i→∞ g * k i (x) = lim inf i→∞ g * k i (x).
Next, as ρ ≥ ρ k for every k, and exp(-g * k ) ≥ 0, by Fatou's Lemma (see e.g. Ash [START_REF] Ash | Real Analysis and Probability[END_REF])

ρ ≥ lim inf i→∞ ρ k i = lim inf i→∞ R n exp(-g * k i ) dx ≥ R n lim inf i→∞ exp(-g * k i ) dx = R n exp(-lim sup i g * k i ) dx = R n exp(-g * ) dx.
On the other hand, observe that for every k ∈ N, M k (1 -g * k , z) 0 implies M j (1 -g * k , z) 0 for all j ≤ k. Hence, let j be fixed arbitrary so that M j (1 -g * k , z) 0, for all sufficiently large k. For every f ∈ R[x] j , f 2 g k is uniformly bounded on K and so by Fatou's Lemma

0 ≤ lim sup i→∞ K f 2 (1 -g * k i ) dµ = -lim inf i→∞ K f 2 (g * k i -1) dµ ≤ K -lim inf i→∞ f 2 (g * k i -1) dµ = K f 2 (1 -g * ) dµ.
As f ∈ R[x] j was arbitrary, M j (1 -g * , z) 0; and as j was arbitrary, M j (1 -g * , z) 0, for all j = 0, 1, . . . But by Lemma 3.3, this implies 1 -g * ≥ 0 on K, and so g * is a feasible solution for P. Combining with ρ ≥ R n exp(-g * )dx yields that g * is an optimal solution of P.

4.2.

Proof of Theorem 3.5.

Proof. (a) The sequence (ρ

′ k ) is monotone non increasing because Q * k ⊂ Q * k+1
for every k ∈ N. Next, let ǫ > 0 be fixed, and let g ∈ P[x] 2d be such that

ρ ≤ R n exp(-g)dx ≤ ρ + ǫ. Observe that g := (1 + ǫ) -1 g ∈ P[x] 2d and R n exp(-g) dx = R n exp(-(1 + ǫ) -1 g) dx = (1 + ǫ) n/d R n exp(-g) dx ≤ (1 + ǫ) n/d (ρ + ǫ).
and 1 -g = 1+ǫ-g 1+ǫ > 0 on K. Therefore, as Q * is Archimedean, by Putinar's Positivstellensatz [START_REF] Putinar | Positive polynomials on compact semi-algebraic sets[END_REF]

, 1 -g ∈ Q * , that is, 1 -g ∈ Q * k 1 for some k 1 . Hence g is a feasible solution of P k 1 which implies ρ ≤ ρ ′ k 1 ≤ (1 + ǫ) n/d (ρ + ǫ).
As the sequence is monotone and ǫ > 0 was arbitrary, we obtain the desired convergence ρ ′ k → ρ. Next, let w = (w α ), α ∈ N n , be the sequence of moment of the Lebesgue measure on K, and let g ∈ R[x] 2d be an arbitrary feasible solution. Let M k (w) (resp. M k-v j (u j w)) be the moment (resp. localizing) matrix associated with w and u j . Hence, M d (w) ≻ 0 and M d-v j (u j w) ≻ 0, j = 1, . . . , m, because K has nonempty interior. The constraint 1-g ∈ Q k (u) which reads (1 -g) = σ 0 + m j=1 σ j u j for some SOS polynomials (σ j ) has the equivalent form

(4.3) (1 -g) α = X 0 , B α + m j=1 X j , C jα , ∀α ∈ N n 2k ,
for some appropriate real symmetric s(2(k -v j )) × s(2(k -v j )) matrices X j 0, (and where B α , C jα are given real symmetric matrices). The vector of coefficients of the SOS polynomial σ j ∈ Σ[x] is obtained from the eigenvectors of X j , j = 0, . . . , m. For more details see e.g. [START_REF] Lasserre | Moments, Positive Polynomials and Their Applications[END_REF]. Multiplying (4.3) by w α and summing up yield:

K (1 -g) dx = X 0 , α B α w α + m j=1 X j , α C jα w α = X 0 , M k (w) + m j=1 X j , M k-v j (u j w)
Observe that every feasible solution g ∈ R[x] 2d is nonnegative otherwise exp(-g)dx is not bounded. And so exp(-g) ≥ 1 -g on K because g ≥ 0. Therefore, for any minimizing sequence (g n , X n j ) of (3.8),

K (1 -g n ) dx ≤ K exp(-g n ) dx ≤ R n exp(-g n ) dx ≤ R n
exp(-g 0 ) dx.

And so sup

n    X n 0 , M k (w) + m j=1 X n j , M k-v j (u j w)    ≤ R n
exp(-g 0 ) dx.

As M k (w), M k-v j (u j w) ≻ 0 and X n j 0, j = 1, . . . , m, all matrices X n j are uniformly bounded. Hence one may extract a subsequence (n ℓ ) such that X n ℓ j → X * j 0, j = 0, . . . , n, as ℓ → ∞. And so for every α ∈ N n 2k ,

X * 0 , B α + m j=1 X * j , C jα = lim ℓ→∞ (1 -g n ℓ ) α =: (1 -g * ) α ,
for some homogeneous polynomial g * ∈ R[x] 2d (as all coefficients g * α with |α| = 2d vanish). In other words (g n ℓ ) α → g * α as ℓ → ∞, for all α ∈ N n 2k . As (1 -g n ℓ ) ≥ 0 on K one also has (1 -g * ) ≥ 0 on K. Finally, since we also have the pointwise convergence g n ℓ (x) → g * (x) for all x ∈ R n , invoking Fatou's lemma yields

ρ ′ k = lim ℓ→∞ R n exp(-g n ℓ ) dx ≥ R n lim inf ℓ→∞ exp(-g n ℓ ) dx = R n
exp(-g * ) dx, which proves that g * is an optimal solution of (3.8).

(b) 1 -g 0 ∈ int Q * k for all k ≥ k 0 because 1 -g 0 ∈ int Q * k 0 and Q * k ⊃ Q * k 0 . Hence Slater's condition holds for P ′ k whenever k ≥ k 0 . Therefore, if g * k is an optimal solution of P k , there exists y k ∈ Q k such that the KKT-optimality condition hold, which yields (3.9).

(c) Follows from the fact that under Slater's condition, the KKT-optimality conditions are sufficient for g * k to be an optimal solution of P ′ k . (d) If sup k sup |α|=2d |g * kα | < N , let g * be an accumulation point, i.e., a limit point of some subsequence (g * k i ), i ∈ N, i.e., such that for every α with |α| = 2d, g * k i α → g * α as i → ∞. Let g * ∈ R[x] 2d be the homogeneous polynomial with coefficients g * α , |α| = 2d. For every x ∈ K, (1 -g k i (x)) ≥ 0 because 1 -g * k i ∈ Q * k i for every i. Therefore, with x ∈ K fixed, 0 ≤ 1 -lim i→∞ g * k i (x) = 1 -g * (x), and so, as x ∈ K is arbitrary, 1 -g * ∈ C 2d (K), which implies that 1 -g * is a feasible solution of P. Next, from (a) and by Fatou's Lemma,

ρ = lim i→∞ ρ k i = lim inf i→∞ ρ k i = lim inf i→∞ R n exp(-g * k i ) dx ≥ R n lim inf i→∞ exp(-g * k i ) dx = R n exp(-g * ) dx,
and so as g * is feasible for P, it is an optimal solution of P. It is known that when K ⊆ R n is closed then ∆ * ∞ = C ∞ (K) and ∆ ∞ = C ∞ (K) * . See e.g. the proof in Laurent [START_REF] Laurent | Sums of squares, moment matrices and optimization over polynomials[END_REF]Prop. 4.5] for K = R n , which also works for any closed set K ⊂ R n . The following result is a truncated version when K is compact. In particular, the convergence y k 0 → y 0 implies that the sequence of measures (φ k ), k ∈ N, is bounded, that is, sup k φ k (K) < M for some M > 0. As K is compact, the unit ball of M (K) is sequentially compact in the weak ⋆ topology σ(M (K), C(K)) where C(K) is the space of continuous functions on K. Hence there is a finite Borel measure φ ∈ M (K) and a subsequence (k i ) such that K gdφ k i → K gdφ as i → ∞, for all g ∈ C(K). In particular, for every α ∈ N n d ,

y α = lim k→∞ y k α = lim i→∞ y k i α = lim i→∞ K x α dφ k i = K x α dφ,
which shows that y ∈ ∆ d , and so ∆ d is closed.

2. 1 .

 1 Notation and definitions. Let R[x] be the ring of polynomials in the variables x = (x 1 , . . . , x n ) and let R[x] d be the vector space of polynomials of degree at most d (whose dimension is s(d) := n+d n ). For every d ∈ N, let N n d := {α ∈ N n : |α| (= i α i ) = d}, and let v d (x) = (x α ), α ∈ N n , be the vector of monomials of the canonical basis (x α ) of R[x] d . Denote by S k the space of k × k real symmetric matrices with scalar product B, C = trace (BC); also, the notation B 0 (resp. B ≻ 0) stands for B is positive semidefinite (resp. positive definite).

  the convex cone of homogeneous polynomials of degree d with compact sublevel set {x : g(x) ≤ 1}. Hence, P[x] d ⊂ C d whenever d ∈ N.

  [x] d varies. Notice that the restriction of f h to P[x] d may be seen as a function f h : R ℓ(d) → R of the coefficient vector of the polynomial g ∈ P[x] d , where ℓ(d) = n+d-1 d .Before proceeding further we need the following result. Recall that the support supp µ of a Borel measure on R n is the smallest closed setA such that µ(R n \ A) = 0. Let C 0 d := {g ∈ C d : g is continuous} ⊂ C d . Lemma 2.4. Let µ be a non trivial σ-finite Borel measure on R n and let Θ µ ⊂ R[x] d be the convex cone of polynomials g of degree at most d such that exp(-g)dµ < ∞. Then: (a) With d ∈ R, the function f : C d → R, with g → f (g) := exp(-g)dµ, is convex (and strictly convex on C 0 d if supp µ = R n ). (b) If µ(O) > 0 for some open set O ⊂ R n and if d ∈ N, then f is strictly convex and twice differentiable on int(Θ µ ), with:

( 2 .Example 1 . 2 Example 2 .Example 3 .

 21223 27) is from Rockafellar[START_REF] Rockafellar | Convex Analysis[END_REF] Corollary 15.3.2] and (2.28) follows from Theorem 2.2 applied to the PHF g * and with y = 1/q. Let x → g(x) := |x| 3 if x > 0 and +∞ otherwise. Hence,d = 3, q = 3/2, G = [-1, 1], σ G (x) = |x|, and g * (x) = Let x → g(x) := |x| so that G = [-1, 1]. As g * (x) = 0 if x ∈ [-1,[START_REF] Anastassiou | Moments in Probability and Approximation Theory[END_REF] and +∞ otherwise (a PHF of degree 0) one may check that indeed G • = [-1, 1] = G and (2.27) holds although d = 1 and g is not strictly convex. Let x → g(x) := x 4 1 + x 4 2 and G = {x :

  1 2 v d (x) T Σv d (x))dx are still related in the following Gaussian-like manner. Let S ++ ℓ(d) ⊂ S ℓ(d) be the convex cone of ℓ(d) × ℓ(d) positive definite matrices, and let θ d : S ++ ℓ(d) → R be defined by:

Lemma 3 . 3 . ([ 15 ,

 3315 Theorem 3.2]) Let K ⊂ R n be compact and let µ be a finite Borel measure with support K and with moments z = (z α ), α ∈ N n . Then p ∈ R[x] is nonnegative on K if and only if M k (p, z) 0 for every k = 0, 1, . . ..

  (a) If P k has an optimal solution g * k ∈ P[x] 2d then there exists a SOS polynomial σ * k ∈ Σ[x] k such that R n x α exp(-g * k ) dx = Conversely if (3.5)-(3.6) holds for some g * k ∈ P[x] 2d and some σ * k ∈ Σ[x] k then g * k is an optimal solution of P k . (c) If in addition, sup k sup |α|=2d |g * kα | < M,

  now provides a monotone sequence of upper bounds ρ ′ k ≥ ρ, k ∈ N. Theorem 3.5. Let K ⊂ R n be compact with nonempty interior and consider the finite-dimensional convex optimization problem (3.8), k ≥ d.

4 . Proofs 4 . 1 .

 441 Proof of Theorem 3.4.

4. 3 .

 3 The dual cone of C d (K). Recall that M (K) is the space of finite Borel measures on K. Let C ∞ (K) ⊂ R[x] be the space of polynomials nonnegative on K ⊆ R n and let ∆ ∞ ⊂ R[x] * be the set:(4.4) ∆ ∞ := K x α dφ , α ∈ N n : φ ∈ M (K) .

Lemma 4 . 1 .f

 41 Let K ⊂ R n be compact. For every d ∈ N, the dual C d (K) * of C d (K) is the set:(4.5) ∆ d := K x α dφ , α ∈ N n d : φ ∈ M (K) . Proof. For every y = (y α ) ∈ ∆ d and f ∈ C d (K) with coefficient vector f ∈ R s(d) α x α dφ = K f dφ ≥ 0. Since (4.6) holds for all f ∈ C d (K) and y ∈ ∆ d , then necessarily ∆ d ⊆ C d (K) * and similarly, C d (K) ⊆ ∆ * d . Next, ∆ * d = f ∈ R s(d) : f , y ≥ 0 ∀y ∈ ∆ d = f ∈ R[x] d : K f dφ ≥ 0 ∀φ ∈ M (K) ⇒ ∆ * ⊆ C d (K),and so ∆ * d = C d (K). Hence the result follows if one proves that ∆ d is closed, because then C d (K) * = (∆ * d ) * = ∆ d , the desired result. So let (y k ) ⊂ ∆ d , k ∈ N, with y k → y as k → ∞. Equivalently, K x α dφ k → y α for all α ∈ N n d .

  , with same arguments, (2.7) also holds if we replace |h| with h + and h -and finiteness follows because -.Finally, from what precedes, finiteness of {x:g(x)≤1} |h|dx is equivalent to finiteness of R n |h| exp(-g)dx.

	0 ≤	{x : g(x)≤y}	h + dx ≤	{x : g(x)≤y}	|h| dx,
	and similarly for h

-. Hence (2.7) with h in lieu of |h| (i.e. (2.6)) follows by additivity since h = h + -h

  and so g ∈ int(Θ µ )). Moreover, h being non trivial, nonnegative and continuous, h > 0 on some open set O and so µ(O) > 0. Therefore, by Lemma 2.4(b), f h is twice differentiable and strictly convex on int(P[x] d ) and (2.16)-(2.17) yield (2.19) and (2.21), while (2.20) follows from (2.19) and Theorem 2.2. That ∇ 2 f h (g) is the matrix of 2d-moments of the measure dν = h exp(-g)dx follows from (2.

√ 3 |x| 3/2 . One retrieves that G • = G = [-1, 1] = {x : g * (x) ≤ (d -1)/d q }.


Ismael Bouassida Rodriguez

Aymen Kamoun
email: akamoun@laas.fr

Sakkaravarthi Ramanathan
email: sakkaravarthi@laas.fr

Khalil Drira
email: khalil@laas.fr

Ontology-based support for reconfigurable adaptive group communication architecture

In collaborative communication, adaptation is required to maintain the reliable connection. Within the framework of a wireless environment, it is very challenging for the host entities to handle a sudden/unexpected change in communication and available resources. This issue is addressed in the context of save and rescue missions carried out during natural disasters such as floods and forest fires by human and voluntary operators aided by ground mobile devices and autonomous robots. This paper proposes a multi level architecture that supports reconfigurable adaptive group communications using semantic model. Modeling follows the well-known SWRL (Semantic Web Rule Language) instructions which establish the degree of importance of each communication link or component. Adaptation is then achieved through initializing between various configurations and for a given entity, the latter is modified as need be. Providing generic and scalable solutions for automated self-reconfiguration is driven by rule-based reconfiguration policies using ontologies. Finally, we illustrate constraints capable of meeting evolving requirements and the solutions followed by implementation of this scenario.

Introduction

In autonomic communication, the design of self-configurable systems that can adapt dynamically is a major research topic. With respect to adaptive systems, the issue at hand is that of finding a way of enabling us to concentrate on the important service request and also on a specific aspect relevant to a particular situation only. An architecture that analyzes and manages the request in order to get the best possible decision in real-time needs to be designed. Many existing dynamic architectures provide just enough functionality to support simple distributed communications but in case of distributed software systems, more work has to focus on managing the group membership and the nodes deployment. In a group -wide communication under different situations, one has to ensure the tractability of the elaborated solutions when migrating from several users, node or components.

In our case study, we study issues related to Crisis Management Systems (CMS). In this kind of system, there exist different nodes communicating through various types of flows. For example, a set of smart devices communicate with robots to achieve a given mission and to self-adapt in a dynamic environment. Each device is embedded with software entities with properties, particularly in terms of reliability, self-healability, etc. The system relies on autonomous entities that manage the resources available in order to provide the communication. In this paper, we use semantic modeling approach to solve the different aspects related to communication challenges. Our goal is to provide a communication architecture that can manage the resources to ensure uninterrupted connectivity between mission participants according to the goals.

Section 2 presents related work and section 3 outlines the ROSACE project scenario. Section 4 details our multi-level architecture followed by context representation. Section 6 illustrates some constraints and the solutions while section 7 talks about the implementation. Conclusion and open issues are presented in section 8.

Related Work

After developing an information system, there might be many reasons for adaptation, e.g corrective, evolutional or perspective [START_REF] Ketfi | Adaptation dynamique, concepts et experimentations[END_REF]. If we consider the corrective adaptation, application does not behave as expected. The solution is to identify the module that causes the problem and replace it by a new module providing the same functionality. In case of evolutional adaptation, changing user needs is supported by adding new modules or modifying the existing functionality of the application architecture. Finally in perspective adaptation, the idea is to improve performance. For example, if a module receives numerous requests that result in performance degradation, it can be duplicated to share the workload.

Several studies focus on a model-based approach with automated management techniques for adaptation, i.e, introducing changes to a program or maintaining functionalities and, whenever possible, improving performance. In [START_REF] Nasser | Adaptive bandwidth framework for provisioning connection-level qos for next-generation wireless cellular networks[END_REF], an adaptive framework supporting multiple classes of multimedia services with different quality requirements in wireless cellular networks is proposed. This work focused on communication level but the migration of adaptive policies to higher level is not detailed. The work in [START_REF] Sun | Cme: a middleware architecture for networkaware adaptive applications[END_REF] envisions a middleware architecture for service adaptation based on network awareness to manage resources in an adaptive context. Though it highlights the necessary adaptation, there is no solution for context-aware problem such as resource constraint. Further research [START_REF] Exposito | FPTP: the XQoS aware and fully programmable transport protocol[END_REF] provides frameworks for designing transport protocols whose internal structure can be modified according to the application requirements and network constraints. In [START_REF] Welch | A distributed system reference architecture for adaptive qos and resource management[END_REF], a schema is described for dynamically managing distributed computing resources by continuously computing and assessing quality. Here, resource utilization metrics are determined a posteriori and an adaptive distributed system reference architecture is equally put forward but the authors didn't explain the adaptive policies.

Recently, ontology has been the subject of much attention as it allows semantics of the managed systems to be represented. As described in [START_REF] Lavinal | A multi-agent self-adaptative management framework[END_REF], ontology supports heterogeneous management entities by capturing semantics from the organizational, environmental and operational viewpoints of overall distributed environment. In [START_REF] Strassner | The use of context-aware policies and ontologies to facilitate business-aware network management[END_REF], the combined use of ontologies and policies is proposed to enforce, in a context-aware approach, the adaptive behavior of autonomous management entities.We also found that in the case of architecture adaptation, model-based strategies are widely used for system transformations that changes according to the environment and requirement. To cope with evolving constraints, behavioral and architectural adaptability is required at several levels. This entails coordination management without which performance would drop much below target.

Motivated by the above discussion, we present a novel ontology-based support for reconfigurable adaptive group communication architecture. Our framework is divided into many levels and the adaptive decision at higher level is migrated to lower for execution. Similarly, decision could be also at lower level thus overcoming the bottleneck associated with waiting for the response from the upper ones.

ROSACE Project Scenario

This scenario involves various types of mobile actors using a heterogeneous communication devices. Ground and aerial communicating robots along with human actors utilizing mobile devices operate within a wireless communication context. A distinction is made between the professionals with specific communication devices or occasional actors that carry a mobile device (e.g. PDAs, Phones). Likewise, robot actors such as Autonomous Aerial Vehicles (AAV) are differentiated from Autonomous Ground Vehicle (AGV). However in all cases, the communication system must deal with the evolution of user needs and with the changes due to device constraints. The idea is to exchange information between mobile participants to achieve a common mission. The three major roles assigned to this scenario are: mission supervisor, coordinators, and field investigators (Figure 1). Each participant plays his own role and is associated with an identifier and the devices he uses. The performed functions are as follows:

-The supervisor's function is to monitor, manage and authorize actions to coordinators and investigators. This entity supervises the whole mission. Once the data (information) has been received from the coordinators, the supervisor reviews the current situation. When he makes a decision, he then passes to the appropriate coordinators. He is equipped with permanent energy resources, high communication and CPU Capabilities; -Coordinator's job is to report to the supervisor. He manages the group of investigators during the mission and assign tasks to each one of them. The coordinator has to collect, interpret, summarize and diffuse information from and towards investigators. The coordinator has high software and hardware capabilities; -The investigator's role is to explore the operational field, observe, analyze, and report about the situation. They also take care of helping, rescuing and repairing.

To support this schema, network-oriented services should be dynamically activated to respond actor requests. They should provide ubiquitous access to peers and be technically transparent, taking into account different context requirements depending on the targeted activity, users' mobility, exchanged data flows (e.g. audio, video), and constraints such as variable communication and device resources. Moreover, changes in the cooperation structure need to be focussed in response to coordinator's decision or information acquired by the participants. There exist two major steps, i.e., "Exploration step" (localizing and identifying the crisis situation) and "Action step" (decision making and allocating tasks). To support this, there are two types of communication flows i.e. coordination and cooperation flows.

Coordination flows take place between investigators and their coordinator and between the coordinators and the supervisor. The investigators send D (Descriptive) data and P (Produced) data describing the situation to the coordinator. The following steps explain data exchanges for the coordination in the exploration step:

-Investigators continuously send type D data to their assigned coordinator and, periodically, type P data.

-Coordinators periodically send P data to the supervisor describing the current state of exploration. All coordinators under exploration step have the same priority to communicate with the supervisor. -When a critical situation is discovered by an investigator, exploration step ends. The architecture is then reconfigured and the mission moves to a new step called the "action step".

Cooperation flows occur between the investigators within the same group (A2A type: fireman2fireman, robot2robot, etc.) or between the investigators of different groups (A2B type: robot2fireman, AAV2fireman, etc.). In case of A2A, a distinction is made between the flows such as cooperation notifications, cooperation requests and cooperation suggestions. In the case of A2B cooperation flows, the flows are: cooperation notifications and cooperation requests.

Challenges are here to analyze and distribute requests equitably among a group of investigators and to actively monitor, control and allow the actors to make decision on their own. Although centralized control is simpler to design and implement, it introduces a single point of failure, which can impede service reliability and scalability. By distribution, we can retain the picture of current scenario thus influencing the results of future situation (a prior knowledge database is used). The following describes the messages between the different actors:

1. An investigator makes a request which is transparently intercepted by the coordinator.

The coordinator gets a decision from the knowledge database (supervisor)

and a solution is provided. 3. At times, decision has to be made by the coordinator itself. In this case, it is stored in the local database, then notified to the supervisor.

Multi-level architecture for CMS Scenario

In order to clearly separate different concerns in our approach, a multi-level architecture is proposed. Each level encapsulates issues into a specific model, thus abstracting complexity to a higher level. The modules at the higher level are abstract system representations, that tend to resemble human activities, while lower ones are much closer to real implementations of abstractions supporting these activities.

Relevant levels are identified and adaptation at the highest levels should be governed by changes in development of activity requirements. Adaptation at the lowest levels should be driven by execution context constraint changes. The levels retained are depicted in Figure 2 and detailed in the following paragraphs.

Here, we have chosen an ontology-based model because it constitutes a standard knowledge representation system, allowing reasoning and inference. Moreover, ontologies facilitate knowledge reuse and sharing through formal and real world semantics. Therefore, ontologies are high-level representations of business concepts and relations. These representations are close to developers' minds and therefore well suited to depict application level models. We have chosen to describe these models in OWL [START_REF] Smith | OWL Web ontology Langage Guide[END_REF], the Semantic Web standard for metadata and ontologies.

Application Level

The retained model in this level is a domain specific ontology that represents concepts and relations of the context of the application provided. It also represents the applications that need collaboration inside the group of users and/or devices (generically called entities). It includes software elements implementing the activity's business, as well as user interfaces, security modules, etc. Among these elements, (at least) those relevant to collaboration are represented in the architectural model corresponding to this level of abstraction. Only collaborationrelated elements of the activity level model will be taken into account in the refinement process. Nevertheless, other business elements (non collaborationrelated) are also included and can be used in order to represent the whole activity. The system designer will define the possible roles that can be playes by the defined entities.

In our previous work, we show an ontology that models the business concepts and its relations to the CMS activity. The main concept is the Participant featuring several properties. The different types of participant (Supervisor, Coordinator, Investigator) are modeled as sub-concepts, each one with his own additional properties. A Participant belongs to a Group that is operating under the authority of a Manager. The latter can be a Supervisor who manages a CoordinatorGroup or a Coordinator who handles an InvestigatorGroup. The other important idea is the Entity concept. Thus a Participant is also an Entity with two sub-concepts: Artificial and Human. Various human participants are represented as concepts e.g. Fireman, Pilot, or Walker or artificial entities e.g. robots or vehicles, for instance. The different types of robots (AmphibiousRobot, Drone, GroundRobot) are modeled as sub-concepts of Robot, each one with its own additional properties. This is related to the Generic Collaboration Ontology (GCO) developed by Sancho et al. [START_REF] Sancho | A Semantic-driven Auto-adaptive Architecture for Collaborative Ubiquitous Systems[END_REF], because GCO:CommunicationFlow is defined as a sub-concept of Flow and Entity as a sub-concept of GCO:Node. In other words the participants' roles is in agreement with the definition given in the collaboration ontology, and thus they inherit all their properties. For example, they have a related GCO:Node deployed on a GCO:Device, etc.

Collaboration Level

Here, session level abstraction is provided and the model details the structure of one or more sessions. It describes the way members in a group are organized within sessions, where they can send and receive data flows. The main issue is that of determining a high-level collaboration schema that meets the needs of application's collaboration. Hence, it supports collaborative sessions and can determine those elements needed to implement these sessions. Further explanations about this ontology and the associated choices can be found in [START_REF] Sancho | A Semantic-driven Auto-adaptive Architecture for Collaborative Ubiquitous Systems[END_REF]. In [START_REF] Sancho | A Semantic-driven Auto-adaptive Architecture for Collaborative Ubiquitous Systems[END_REF], an ontology model, containing generic collaboration knowledge as well as domainspecific knowledge, is proposed in order to enable architecture adaptation and to support spontaneous and implicit sessions inside groups of humans and devices.

Messaging Level

This level provides a communication model that masks low-level details (like TCP sockets, UDP datagrams, IP addresses, multicast, etc.) in order to simplify the representation of communication channels. It furnishes an abstract view of distributed systems, so that they become transparent for upper levels. For example, this model may be based on abstractions like Event-based Communications, Peer-to-Peer, Remote Procedure Calls or Remote Method Invocation. In this work, we have retained the Event-Based Communication (EBC) [START_REF] Meier | Taxonomy of distributed event-based programming systems[END_REF]. It represents a well established paradigm for interconnecting loosely coupled components and it provides one-to-many or many-to-many communication pattern. This model is a detailed graph containing a set of event producers (EP), event consumers (EC) and channel managers (CM) connected with push and pull links. Multiple producers and consumers may be associated through the same CM. Since this model represents a graph, it can also be expressed in the GraphML language.

Infrastructure Level

It contains all needed software components and hardware that enable to run the collaborative system and ensures communication.

Context Representation

In our work, adaptation is achieved through various context changes. Contextawareness does not depict the location/environment but represents resource constraints (e.g., connectivity, energy level, available memory, etc.) or changes in the activity and environment where participants can arrive/leave, change roles, etc.

The clear distinction of distributed adaptation at specific levels is the primary advantage of our architecture compared to previous existing ones. Policies are executed according to the context requirements, i.e, if there is a change in the mission, activity level policies are activated whereas in case of low energy level at the device, policies at low level are initiated without affecting the higher levels.

Reconfiguration Rules

This section presents our approach for modeling and implementation of the CMS activity. Also provided are examples of reconfiguration policies for activity evolution and resource context change using SWRL rules.

To make this system adaptable, reconfiguration rules are needed to adapt the ontology instance to the current situation. As events play a major role in our application, the transformation of entities needs to be triggered. Here, events could occur at activity level, i.e, addition of new participants, changing an action, transfer of a participant from one group to another, new connection between investigators from different groups, etc. The events could also take the form of resource context changes, e.g, parameters like the energy of a device, bandwidth range, CPU processing capacity, RAM availability etc. We use SWRL [START_REF] Horrocks | SWRL: A Semantic Web Rule Language Combining OWL and RuleML[END_REF] rules to define our adaptation policy. The application designer defines these rules according to context changes he wants to handle. As we explained earlier, each level executes their own rule when there is an appropriate need. Thanks to the decision model at every level, not all requests are passed to the supervisor. If there is no solution for an event at a particular level, then it triggers the higher level. In SWRL, the head points to the adaptation transformations whereas the body indicates the context of ontology elements. This reconfiguration rules are really useful for adapting the scenario dynamically.

In Figure 1, the supervisor controls the mission and manages the coordinator group, in our case, AAV and fireman coordinators. The flows are exchanged between these operators and each coordinator further manages his own group. Changes in the environment trigger adaptation events. When they occur, the supervisor executes the appropriate SWRL rules to the current instance of the CMS ontology. In our case, adaptation events are classified into 2 types:

High Level Event Reconfiguration Table 1 shows an example in the activity level evolution. Here, the supervisor is managing two coordinators (one for AAVs and other for firemen) and each coordinator has their own investigator group. To make it simpler for the reader, we presented here four investigators, two AAVs and two firemen. As we explained previously, actors communicated through coordination flows. If there is a need to establish a cooperation flow between the investigators among the different groups, the following rule is applied. The conditions for establishing the flow is explained in the scenario illustration.

Low Level Event Reconfiguration Table 2 lists an example of changes in the communication level constraints. if the energy level of a participant's device become low (e.g, below 20 %), a transformation is performed to move the channel manager from this device. In this case, the decision is made by the communication level of the coordinator that manages the participant having an energy problem by triggering the SWRL rule. More details can be found in the next section about the dischargeP articipantDevice.

Scenario Illustration

This section presents a top-down approach with respect to the architecture models used for the activity and communication levels. It also illustrates the refinement processes that exist between levels in the adaptation process. The architectural configuration of the activity level is captured and represented in the CMS ontology instance as shown in the upper part of Figure 3. Here, concepts and relations from the activity-specific and from the generic collaboration ontologies are instantiated. To refine this activity model, rules are processed over these ontological instances. Firstly, activity-specific rules are processed and then, generic collaboration rules are applied. Then, rules are processed for each instance of GCO:CommunicationFlow found, thus creating the corresponding channel man- agers, event consumers and event producers. The resulting set of ontological instances form a collaboration level graph. It is translated into GraphML language by means of XSLT transformation. In order to refine this collaboration level, a detailed graph grammar is used. This produces a valid configuration that contains terminal nodes only (i.e. nodes belonging to the EBC level). It is obtained by application of the sequence graph grammar production. This refinement creates a detailed deployment descriptor graph used by a deployment service in order to utilize the indicated components on each device, thus implementing the required activity level session.

Here two situations are considered. The first one deals with an activity-based event and second with resource change event.

Firstly, consider a situation where an investigator (AAV) intends to drop water in an area where another investigator (denoted as Fireman2) is already busy. The simplified version of this situation at activity and communication levels is shown in Figure 3. In this case, the AAV has to be notified as soon as possible, not to drop water. Another investigator (denoted as Fireman1), aware of the situation, establishes a coordination flow to its coordinator and subsequently the coordinator reports to the supervisor. The latter already knows the position of the approaching AAV and notifies the AAV not to drop any water in that area via the AAV coordinator.

As there is no connection between the AAV and the Fireman2, the latter has to obtain the supervisor's decision. The other simpler solution could be to establish a connection between the AAV and Fireman1 through use of a new cooperation flow obtained by running the SWRL rule (Table 1). Thus, Fireman1 can warn the AAV not to drop any water. After processing the rule, the activity level and the communication level have been changed as shown in Figure 4. With the new cooperation flow between Fireman1 and the AAV, a channel manager is needed in Fireman1's device and the corresponding event producer and event consumer must be deployed in the AAV (represented by dotted line on Figure 4). An example for context-awareness is illustrated in the following example. Consider an investigator (denoted as I2) sending messages to his coordinator via another investigator (denoted as I1). As each one is entrusted with his own task, may arise a situation where I1 could not handle this communication as he needs to move to another location or his power runs out. So he triggers the coordinator to make a decision because communication between I2 and coordinator is very important. In case of I1 moving to somewhere else, a solution could be another investigator establishing a new connection with I2.

In case of I1's power deficiency, a solution is to use the decision that is explained in Table 2. In the first part of the rule, the current situation is identified and due to the power deficiency, the procedure dischargeParticipantDevice is triggered. In our case, the channel manager moves from I1 either to coordinator or to I2. By doing so, I2 can send this prioritized flow for a sufficient time through I1 to coordinator. Clearly, we act only at the communication level, i.e, on EBC. Nothing has been modified in the activity level and the new EBC descriptor graph is shown in Figure 5.

Implementation

We implemented our work using FACUS (Framework for Adaptive Collaborative Ubiquitous Systems), an architecture that supports semantic adaptation enabling the awareness of the presence/absence, roles and tasks of collaborators. This framework is based on a generic multi-level modeling approach that ensures multi-level adaptation. A generic collaboration model, based on Semantic Web technologies is proposed in order to support real-time collaboration between groups of participants working together in different tasks. The framework defines common interfaces for collaborative systems to enable the management of cooperative actions.

In this framework, a node represents a communicating entity which takes part in a collaborative activity. Nodes may represent human users (i.e. humancontrolled software components) but also autonomous software components, agents, etc. Whether a node is an autonomous software component or it is a human-controlled component, it has to be executed on a physical machine. Such machines are represented by the concept Device (Node is linked to Device by the property hasHostingDevice). The execution context of the node will depend on the resources of the device that hosts it. At the present time, a minimal set of device properties is considered, containing IP addresses (hasIpAddress), operating system (hasOS), available memory (hasAvailableMemory), CPU load (hasAvailableMem) and battery level (hasBatteryLevel).

The concept Flow represents a communication link between two entities. Therefore, Flow is linked to Node by two properties: hasSource and hasDestination. In this ontology, flows are considered as being unidirectional, and thus if a bidirectional communication between two nodes is required, it will be represented by two instances of Flow with two opposite directions.

In order to handle data flows, nodes use external software components that are deployed on the same device as them. These external components are represented by the Tool concept. Tools are composed of several components, e.g., a sender component and a receiver component. Therefore the Tool concept is related to a concept called Component through the property hasComponent. Since components handle flows, a property called managesFlow links Component and Flow. Components have a data type (the same as the data type of the flow that they manage) and are deployed on a single device (isDeployedOn property which links Component and Device). SenderComponent and Receiver-Component are linked to Flow by two sub-relations of managesFlow: sendsFlow and receivesFlow, respectively.

Finally, the Session concept represents a set of flows belonging to the same collaborative activity. The hasFlow property relates a session to a flow. The inverse property, belongsToSession, is functional, i.e., a flow belongs to a single session. Since flows are related to nodes, nodes are indirectly related to one or more sessions depending on the flows that connect them to other entities.

In FACUS, we have chosen one group (fireman)to show the adaptation. Initially, the fireman1 and fireman2 are connected to firemancoordinator at WiFi infrastructure mode. This initial stage of this situation at application, collaboration and massaging levels are shown in Figure 6, Figure 7 and Figure 8.

Consider fireman1 lost the connection with its coordinator while searching for a victim. Once the connection is lost, the coordinator aware this situation and thanks to our policies, the coordinator and the other investigator will shift to adhoc mode. Also, the local decision of the lost investigator changes to adhoc mode automatically such that communication is established. Figure 9 and Figure 10 show the adaptive collaboration and middleware graph in the implementation.

Conclusion and Open Issues

In this paper, a multi-level modeling approach designed to support group communications has been detailed. For such a complex system, the whole scenario has been divided into different levels. Ontology has been used at the top two Fig. 6. Application instance at Initial phase Fig. 7. Initial Collaboration instance levels while we retained event-based communication at the third level to establish the flows between devices. The relations and transformations from top level to lower levels are presented. If a change arises in the environment, reconfiguration can be achieved by using SWRL rules. This is key for an appropriate management in case of changing resources in the environment. By using this approach, we could allow the architectural reconfigurations at run-time to handle the activity's evolving conditions. Unlike previous approaches, distributed decision model allows our architecture to make adaptive mechanism at lower levels and not influencing the higher ones.

Many open issues need to be discussed in our scenario. In case of environment change, triggers play an essential role to notify the decision components for initiating the adaptive policies. Even though the triggers are asynchronous and synchronous messages, a generic model for failure cases need to be analyzed. Also, assigning priorities to flows as well as monitoring resource deficiencies are our main challenges. Non-cooperative situations like low QoS and performance degradation are the topics worth for future work.

Fig. 1 .

 1 Fig. 1. The CMS Activity Description

Fig. 2 .

 2 Fig. 2. Abstraction Levels

  Supervisor(?S) ∧ CoordinatorGroup(?CG) ∧ managesGroup(S, CG)∧ Coordinator(?CO1) ∧ Coordinator(?CO2) ∧ hasM ember(CG, CO1)∧ hasM ember(CG, CO2) ∧ InvestigatorGroup(?IG1) ∧ InvestigatorGroup(?IG2)∧ managesGroup(CO1, IG1) ∧ managesGroup(CO2, IG2)∧ Investigator(?I1) ∧ Investigator(?I2) ∧ hasM ember(IG1, I1)∧ hasM ember(IG2, I2) ∧ CoordinationF low(?CF 1) ∧ hasReceiver(CF 1, CO1)∧ hasSender(CF 1, S) ∧ CoordinationF low(?CF 4) ∧ hasReceiver(CF 4, CO2)∧ hasSender(CF 4, S) ∧ CoordinationF low(?CF 3) ∧ hasReceiver(CF 3, I1)∧ hasSender(CF 3, CO1) ∧ CoordinationF low(?CF 5) ∧ hasReceiver(CF 5, I3)∧ hasSender(CF 5, CO2) → SW RLb : createOW LT hing (CooperationF low(?CP F 1))∧ hasSender(CP F 1, I2) ∧ hasReceiver(CP F 1, I1)

Fig. 3 .

 3 Fig. 3. EBC deployment: ontology and graph after initial refinement

Fig. 4 .

 4 Fig. 4. EBC Deployment: ontology and graph after Reconfiguration Event

Fig. 5 .

 5 Fig. 5. EBC Redeployment after Power Diminution Event

Fig. 8 .Fig. 9 .

 89 Fig. 8. Initial middleware graph

Fig. 10 .

 10 Fig. 10. Middleware graph after adaptation

Table 1 .

 1 Rule1-High Level Event Reconfiguration

	Supervisor(?S) ∧ CoordinatorGroup(?CG) ∧ managesGroup(S, CG)∧
	Coordinator(?CO1) ∧ hasM ember(CG, CO1) ∧ InvestigatorGroup(IG1)∧
	managesGroup(CO1, IG1) ∧ Investigator(?I1) ∧ Investigator(?I2)∧
	hasM ember(IG1, I1) ∧ hasM ember(IG1, I2) ∧ CoordinationF low(?CF 1)∧

hasReceiver(CF 1, CO1) ∧ hasSender(CF 1, I1) ∧ CooperationF low(?CP F 1)∧ hasReceiver(CP F 1, I1) ∧ hasSender(CP F 1, I2) ∧

Device(?device)∧ hasID(?device, ?idDevice) ∧ hasHardwareP rof ile(?device, ?hardware)∧ hasP arameter(?hardware, ?param) ∧ sameAs(?param, powerlevel)∧ hasV alue(?param, ?value) ∧ swrlb : lessT han(?value, 20) → adapt : dischargeP articipantDevice(?idDevice)

Table 2 .

 2 Rule2-Low Level Event Reconfiguration

Acknowledgments

This research is supported by the French project Rosace (RObots et Systémes Auto-adaptatifs Communiquants Embarquès).