Marek Tyburec
email: marek.tyburec@fsv.cvut.cz

Jan Zeman
email: jan.zeman@cvut.cz

Martin Kružík
email: kruzik@utia.cas.cz

Didier Henrion
email: henrion@laas.fr

Global optimality in

Keywords: discrete topology optimization, frame structures, shell structures, semidefinite programming, polynomial optimization, global optimality

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Introduction

Structural optimization is a research field developing concepts for the design of efficient structures. It was pioneered by [START_REF] Michell | The limits of economy of material in frame-structures[END_REF], who showed that minimum-weight truss structures under a single load case are fully-stressed, and the optimal trajectories of their bars align with the principal stress directions. Hence, optimal designs can contain an infinite number of bars in general. This drawback, which hinders their manufacturability, was overcome in the work of [START_REF] Dorn | Automatic design of optimal structures[END_REF] by introducing the ground structure approach, effectively discretizing the continuum into a finite set of potential nodes and their interconnections of finite elements. Because the dimensionality of these potential elements is lower than that of the continuum and the presence and sizing of each element are investigated, this setting is referred to as discrete topology optimization. Variables a 1 and a 2 stand for the cross-section areas of the two elements and c denotes the corresponding compliance (assuming moments of inertia I i = a 2 i , i ∈ {1, 2}).

A tremendous progress has been made for the case of trusses. While [START_REF] Dorn | Automatic design of optimal structures[END_REF] developed a linear programming formulation for the single-load-case plastic design, [START_REF] Bendsøe | New displacement-based methods for optimal truss topology design[END_REF] and [START_REF] Achtziger | Equivalent displacement based formulations for maximum strength truss topology design[END_REF] introduced a convex displacement-based elastic-design quadratic program that additionally allowed for multiple load cases. Its dual, which incorporates the cross-section variables explicitly, was shown by [START_REF] Lobo | Applications of second-order cone programming[END_REF] and [START_REF] Ben-Tal | Lectures on modern convex optimization[END_REF] to be a second-order conic program. This latter formulation handles multiple load cases with stress constraints efficiently (Tyburec et al., forthcoming). Convexity prevails even for fundamental free-vibration constraints, in which case semidefinite programming can be used [START_REF] Achtziger | Structural topology optimization with eigenvalues[END_REF][START_REF] Tyburec | Designing modular 3D printed reinforcement of wound composite hollow beams with semidefinite programming[END_REF].

A completely different situation holds for bendingresistant structures with continuous design variables. To the best of our knowledge, no convex formulation has been established so far, and, therefore, solely local optimization techniques have been used. Among these, [START_REF] Saka | Optimum design of rigidly jointed frames[END_REF] developed a sequential linear programming approach to design minimum-weight framed structures, and [START_REF] Wang | Alternative formulations for structural optimization: an evaluation using frames[END_REF] improved over its solution efficiency by using sequential quadratic programming instead. Another, relaxation-based sequential semidefinite programming method was proposed by [START_REF] Yamada | Relaxation approach to topology optimization of frame structure under frequency constraint[END_REF] to deal with vibration problems. Nonlinear programming [START_REF] Fredricson | Topology optimization of frame structures with flexible joints[END_REF], Optimality Criteria (OC) [START_REF] Khan | Optimality criterion techniques applied to frames having general cross-sectional relationships[END_REF][START_REF] Chan | Automatic optimal design of tall steel building frameworks[END_REF], the Method of Moving Assymptotes (MMA) [START_REF] Svanberg | The method of moving asymptotes-a new method for structural optimization[END_REF][START_REF] Fredricson | Topology optimization of frame structures-joint penalty and material selection[END_REF], and metaheuristics [START_REF] Huang | Topology and sizing optimization for frame structures with a two-level approximation method[END_REF] are other commonly used alternatives.

Except for our earlier conference paper (Tyburec et al., 2020a), which is a very preliminary version of this manuscript, the only global approach that can be found in the literature considers a discrete setting of the problem, in which case the cross-sections are selected from a predefined cata-log, allowing to use the branch-and-bound method [START_REF] Kanno | Mixed-integer second-order cone programming for global optimization of compliance of frame structure with discrete design variables[END_REF].

Motivation

This contribution investigates a conceptual design of bendingresistant structures with continuous design variables. As follows from the previous survey, only local optimization approaches have been adopted so far to tackle the frame/shell structure optimization problem. It is, therefore, not surprising that they fail to converge to globally optimal solutions even for toy problems, such as the one shown in Fig. 1a.

Consider a linear-elastic material with the dimensionless Young modulus E = 1, available material volume V = 1, element cross-sections parameterized by their area a i and the moment of inertia I i (a i ) = a 2 i , which corresponds to rectangular cross-sections with the height-to-width ratio of 12, for example. Our goal is to find the cross-section areas a i associated with the elements i = {1, 2} that induce minimum compliance c within all non-negative a i satisfying the volume bound V . Standard local optimization techniques such as OC, MMA, and MATLAB inbuilt optimizer fmincon all converge 1 to the optimized compliance c = 2.895 and the corresponding areas a 1 = 0.652 and a 2 = 0.242. However, the globally optimal design possesses compliance c * = 2.719, which requires a * 1 = 0.4 √ 5 and a * 2 = 0. This problem exhibits three local optima in particular, with the last one being a 1 = 0, a 2 = 0.4 √ 5 and c = 4.429, see Fig. 1b. Thus, the global optimum may be reached by examining a few different starting points in the optimizers. Such a procedure, however, cannot assure global optimality and cannot neither assess quality of 1 For OC and MMA, we adopted the commonly-used starting point of uniform mass distribution, i.e., a 1 = a 2 = 0.2 √ 5. For fmincon, the default starting point was used.

the optimized designs with respect to the global optimum. When the number of structural elements increases, designdependent loads are considered, and higher-order polynomials for the moments of inertia are used, finding globallyoptimal minimum-compliance designs becomes extremely challenging.

Aims and novelty

We address this problem by exploiting the simple fact that the constraints can be formulated as polynomial functions, hence forming a (basic) semi-algebraic feasible set. Using a polynomial objective function in addition, the momentsum-of-squares (Lasserre) hierarchy of convex outer approximations (relaxations) can be used to solve and extract all globally-optimal solutions. These relaxations provide a non-decreasing sequence of lower bounds, eventually attaining the optimal compliance, Figs 1c-1d. In addition, we show how to correct such obtained lower-bound designs, and hence generate feasible upper bounds. A comparison of these bounds then assesses the design quality, and their equality establishes a simple sufficient condition of global optimality. We further show that when a unique global optimum exists, we can expect the occurrence of a bound equality. Fortunately, this situation occurs quite often when the design domain lacks structural and boundary conditions symmetries.

This paper is organized as follows. In Section 2, we introduce polynomial optimization and the moment-sumof-squares hierarchy. Section 3.1 develops a non-linear semidefinite programming formulation for topology optimization of frame structures, which we modify subsequently for the moment-sum-of-squares hierarchy in Section 3.2. Section 3.3 reveals how to correct the lower-bound designs generated by the hierarchy to obtain feasible upper bounds, and Section 3.4 introduces the sufficient condition of global ε-optimality as well as a zero optimality gap for the case of a unique global optimum. Section 3.5 outlines the required changes in notation to allow for thickness optimization of shell structures. These results are then illustrated on five selected optimization problems in Section 4. We finally summarize our contributions in Section 5.

Moment-sum-of-squares hierarchy

In this section, we briefly outline the moment-sum-ofsquares hierarchy when applied to solution of problems with polynomial matrix inequalities. For more information, we refer the reader to an expository text [START_REF] Henrion | Convergent relaxations of polynomial matrix inequalities and static output feedback[END_REF] and to the excellent books [START_REF] Anjos | Handbook on semidefinite, conic and polynomial optimization[END_REF][START_REF] Lasserre | An introduction to polynomial and semi-algebraic optimization[END_REF].

Suppose we aim to solve an optimization problem of the form

f * = inf x f (x) (1a) s.t. G(x) 0, (1b)
where f (x) : R n → R is a real polynomial function and G(x) : R n → S m is a real polynomial mapping, so that ∀i, j : G i,j (x) = G j,i (x) are real polynomial functions of x. The degree of these polynomials is less than or equal to k ∈ N. The symbol S m denotes the space of real symmetric square matrices of size m, and " " establishes an ordering of fundamental eigenvalues, i.e., G(x) in ( 1b) is positive semidefinite. Hence, we call (1b) a polynomial matrix inequality (PMI) in what follows and denote its feasible set by K(G).

Clearly, the nonlinear semidefinite program (1) covers a variety of convex optimization problems as special cases, including linear and quadratic programming or linear semidefinite programming, see, e.g., (Ben-Tal and Nemirovski, 2001, Section 4.2). Although these instances can be solved in a polynomial time, hence efficiently, (1) exhibits N P-hardness in general. This can be seen, for example, by a reduction from binary programming, in which case the main diagonal of G(x) contains both x 2 i -x i and x i -x 2 i terms for all i ∈ {1 . . . n}. Despite the fact that the admissible set of x is generally non-convex, (1) admits an equivalent reformulation to a convex optimization problem over a finite-dimensional cone of polynomials C k (K(G)) of degree at most k which are nonnegative on K(G), i.e.,

f * = sup λ {λ : f (x) -λ ≥ 0, ∀x ∈ K(G)} = sup λ (f -λ), s.t. (f -λ) ∈ C k (K(G)).
(2)

Unfortunately, it is not known how to handle C k (K(G)) simply and in a tractable way.

To introduce an approach that allows to solve (2), we first adopt the following notation. Let x → b k (x) be the polynomial space basis of polynomials of degree at most k,

b k (x) = 1 x 1 x 2 . . . x n x 2 1 x 1 x 2 . . . x 1 x n x 2 2 x 2 x 3 . . . x 2 n . . . x 3 2 . . . x k n , (3)
Then, any polynomial p(x) of degree at most k can be written as

p(x) = q T b k (x), (4)
in which q denotes a vector of coefficients associated with the basis b k (x).

Definition 1 The polynomial matrix Σ(x) : R n → S m is a (matrix) sum-of-squares (SOS) if there exists a polynomial matrix H(x) : R n → R m×o , o ∈ N, such that

Σ(x) = H(x) [H(x)] T , ∀x ∈ R n . (5)
Let •, • denote the standard inner product on matrices, α ∈ N |b k (x)| with 1 T α ≤ k be the multi-index associated with the basis b k (x), and let y ∈ R |b k (x)| be the moments (of probability measures supported on K(G(x))) indexed in b k (x). In what follows, we adopt the following notation for the elements of y:

y α = y n i=1 x α i i is associated with n i=1 x αi i . (6)
For example, when α = (0 0 1 2) T , y 0012 = y x 1 3 x 2 4 corresponds to the polynomial

x 1 3 x 2 4 ∈ b k (x),
where k ≥ 3. Assumption 1. [START_REF] Henrion | Convergent relaxations of polynomial matrix inequalities and static output feedback[END_REF] Assume that there exist SOS polynomials x → p 0 (x) and x → R(x) such that the superlevel set {x ∈ R n :

p 0 (x) + R(x), G(x) ≥ 0} is compact.
Note that Assumption 1 is an algebraic certificate of compactness of the feasible set in problem (1). When Assumption 1 holds, then, the dual of (2) can be written equivalently as an infinite-dimensional generalized problem of moments, which is equipped with a finite-dimensional truncation:

f (r) = min y q T y (7a) s.t. y 0 = 1, (7b)
M k (y) 0, (7c)
M k-d (G(x)y) 0, (7d)
in which 2r ≥ k, r is the relaxation degree, and d stands for the maximum degree of polynomials in G(x). In addition, M k (y) and M k-d (G(x)y) are the truncated moment and localizing matrices associated with y and G(x). For a precise definition of M k (y) and M k-d (G(x)y), we refer the reader to [START_REF] Henrion | Convergent relaxations of polynomial matrix inequalities and static output feedback[END_REF]. These moment matrices are linear in y, hence (7) is a linear semidefinite program. Because (7) is a finite-dimensional convex relaxation of (1), we have f (r) ≤ f * , ∀r ∈ N. Moreover, these relaxations are tighter with increasing r, making the sequence f (r) ∞ r monotonically increasing and converging towards f * .

Theorem 1 [START_REF] Henrion | Convergent relaxations of polynomial matrix inequalities and static output feedback[END_REF], Theorem 2.2) Let Assumption 1 be satisfied. Then,

f (r) ↑ f * as r → ∞ in (7).
Moreover, all globally optimal solutions of (1) can be extracted from (7) based on the flat extension theorem of [START_REF] Curto | Solution of the truncated complex moment problem for flat data[END_REF]. Indeed, finite convergence occurs when

s = Rank(M k (y * )) = Rank(M k-d (y * )), (8)
where y * denotes the vector of optimal moments, and s stands for the minimum number of distinct global minimizers (Henrion and Lasserre, 2006, Theorem 2.4).

Example We illustrate the process of building the (Lasserre) moment-sum-of-squares hierarchy on an elementary example.

min a,c c (9a) s.t. c f f a 2 0, (9b)
V -a ≥ 0, (9c)

a ≥ 0. (9d)
In the first relaxation, y = y 00 y 10 y 01 y 20 y 11 y 02 T is indexed in the polynomial space basis b 1 (a, c) = 1 c a c 2 ca a 2 T . Then, the associated relaxation reads min y y 10 (10a)

s.t. y 10 f f y 02 0, (10b)
V -y 01 ≥ 0, (10c)

y 01 ≥ 0, (10d)
y 00 = 1, (10e)
        0. (11f)
When solved, this relaxation allows for extracting the global solution of a * = V and c * = f 2 /V 2 .

Methodology

Topology optimization of discrete structures provides a natural application for the ground structure approach [START_REF] Dorn | Automatic design of optimal structures[END_REF], a discretized design domain composed of a fixed set of n n ∈ N nodes and their subsets of admissible n e ∈ N finite elements. Here, we employ the simplest two-node Euler-Bernoulli frame elements that adopt linear shape functions to interpolate the longitudinal displacements and cubic shape functions to interpolate the lateral displacements and rotations. Another elements can be adopted though, see Section 4.4 for applications to the Timoshenko beam element and the MITC4 shell element. Each of these finite elements (indexed with i) must be supplied with the non-negative cross-section area a i ∈ R ≥0 and the area moment of inertia I i ∈ R ≥0 . These are to be found in the optimization process. In this contribution, we assume, for convenience, that the moment of inertia is a second-or third-order polynomial function of the crosssections,

I i (a i ) = c II a 2 i + c III a 3 i , (12)
with c II , c III ∈ R ≥0 being fixed constants. When I i (a i ) = 0 and a i = 0, the finite element vanishes and does not contribute to the load transfer. Different topology optimization formulations exist, accommodating specific needs of particular applications. Here, we consider the problem of searching the minimumcompliant design under multiple load cases (13a) while satisfying the linear-elastic equilibrium equation (13b) and limiting the material volume from above by V ∈ R >0 (13c). Physical admissibility of the resulting designs is ensured by the non-negative cross-section areas (13d). Combination of these ingredients establishes the basic elastic-design formulation

min a,u1,...,un lc n lc j=1 ω j f j (a) T u j (13a) s.t. K j (a)u j -f j (a) = 0, ∀j ∈ {1 . . . n lc }, (13b) V -T a ≥ 0, (13c) a ≥ 0, (13d)
in which ω ∈ R n lc >0 are positive weights associated with n lc load cases, and ∈ R ne ≥0 stands for the element lengths column vector. Further, f j (a) ∈ R n dof,j and u j ∈ R n dof,j denote the force and displacement column vectors of the j-th load case, n dof,j ∈ N stands for the associated number of degrees of freedom, and K j (a) ∈ R n dof,j ×n dof,j is the corresponding symmetric positive semidefinite stiffness matrix.

For these stiffness matrices, we require ∀a > 0 : K j (a) 0 to exclude rigid body motions. Using the finite element method, K j (a) is assembled as

K j (a) = K j,0 + ne i=1 K (1) j,i a i + K (2) j,i a 2 i + K (3) j,i a 3 i , (14)
with K j,0 0 standing for a design-independent stiffness (such as fixed structural elements), K

(1) j,i 0 being the unitcross-section-area membrane stiffness of the i-th element in the j-th load case, and

K (2) j,i 0 with K (3) j,i
0 are the corresponding bending stiffness counterparts associated with the unit cross-section area. The force column vector f j is assumed in the form

f j (a) = f j,0 + ne i=1 f (1) j,i a i , (15)
where f j,0 stands for the design-independent load and f

(1) j,i are the design-dependent loads such as self-weight. One can also add higher-order terms to (15) to handle non-zero displacement boundary conditions. The formulation ( 13) is nonlinear and lacks convexity in general. The non-convexity comes not only from the polynomial entries in the stiffness matrix ( 14), but also from its possible singularity caused by zero cross-section areas (13d).

Semidefinite programming formulation for topology optimization of frame structures

An approach to simplify (13) relies on eliminating the displacement variables u j from the problem formulation. In the nested approach, which is commonly used in topology optimization, the cross-section areas are bounded from below by a strictly positive ε ∈ R >0 , allowing for a computation of [K(a)]

-1 . Recall that K(a) 0 for all a > 0. The optimization procedure then traditionally adopts, e.g., the Method of Moving Asymptotes (MMA) [START_REF] Svanberg | The method of moving asymptotes-a new method for structural optimization[END_REF], or the Optimality Criteria (OC) method [START_REF] Rozvany | Structural design via optimality criteria[END_REF]. Notice that ε → 0 results in a high condition number of K(a) and that larger values of ε may impair quality of optimized designs, as sizing optimization is solved instead of the original topology optimization. In contrast, here, we eliminate u j and allow the cross sections to truly attain zero.

Because [K(a)] -1 may not exist in our case, we rely on the Moore-Penrose pseudo-inverse [K(a)] † instead. Its role in enforcing the equilibrium conditions is clarified in the next lemma.

Lemma 1 Consider the equation K j (a)u j = f j (a) -r j with u j = K j (a) † f j (a) and a residual vector r j ∈ R n dof . Then, r j = 0 if and only if f j (a) ∈ Im(K j (a)).

Proof. Let f j (a) = v j (a) + w j (a), in which v j (a) ∈ Im (K j (a)), and w j (a) ∈ Ker (K j (a)). Then, because K j (a)K j (a) † is an orthogonal projector onto the range of K j (a), we obtain K j (a)K j (a) † f j (a) = v j (a). Clearly, when f j (a) ∈ Im(K j (a)), we have v j (a) = f (a) with w j (a) = 0, implying that r j = 0. For the case of f j (a) / ∈ Im(K j (a)), w j (a) = 0, showing that r j = -w j (a).

Lemma 1 allows us to eliminate the displacement variables and write the optimization problem (13) only in terms of the cross-section areas a as

min a n lc j=1 ω j f j (a) T [K j (a)] † f j (a) (16a) s.t. V -T a ≥ 0, (16b)
a ≥ 0, (16c) ∀j ∈ {1 . . . n lc } : f j (a) ∈ Im(K j (a)). (16d)
Notice that (16d) essentially eliminates the nonphysical setup when

K j (a) = [K j (a)] † = 0 produces zero com- pliance. Because [K j (a)] † 0, ∀j ∈ {1 . . . n lc },
and ω > 0 by definition, (16a) is bounded from below by 0. Thus, we introduce slack variables c ∈ R n lc ≥0 comprising the to-beminimized upper bounds on compliances for the load cases and rewrite ( 16) equivalently as

min a,c ω T c (17a) s.t. c j -f j (a) T [K j (a)] † f j (a) ≥ 0, ∀j ∈ {1 . . . n lc },(17b) V -T a ≥ 0, (17c)
a ≥ 0, (17d) ∀j ∈ {1 . . . n lc } : f j (a) ∈ Im(K j (a)). (17e)
To derive a nonlinear semidefinite programming formulation, let us now recall the generalized Schur complement lemma:

Lemma 2 (Gallier, 2011, Theorem 16.1) Let A and C be symmetric square matrices, B have appropriate dimensions, and I denote an identity matrix. Then, the following conditions are equivalent:

1. A B T B C 0, 2. C 0, A -B T C † B 0, (I -CC † )B = 0.
Since we already have K j (a) 0 by definition and c j -

[f j (a)] T [K j (a)] † f j (a) ≥ 0 in (17b), to use Lemma 2 it
suffices to show that

(I -K j (a) [K j (a)] † )f j (a) = 0. ( 18
)
Proposition 1 The condition (18) is equivalent to f j (a) ∈ Im(K j (a)).

Proof. First, consider f j (a) ∈ Im(K j (a)). Then, f j (a) = K j (a)u j for some displacement vector u j . After inserting it into the left-hand-side of (18), we have

K j (a) -K j (a) [K j (a)] † K j (a) u j = 0, (19)
which holds for all such u j as K j (a) [K j (a)] † K j (a) = K j (a) by the definition of the Moore-Penrose pseudoinverse (Gallier, 2011, Lemma 14.1).

Otherwise, consider f j (a) / ∈ Im(K j (a)) and let ũ = [K j (a)]

† f j (a). Then, K j (a)ũ = f j (a) -r j for some r j ∈ Ker(K(a)), r j = 0 by Lemma 1. Thus, the left-hand-side of ( 18) simplifies to

f j (a) -K j (a) [K j (a)] † f j (a) = = f j (a) -K j (a)ũ = r j = 0, (20)
which completes the proof.

Finally, Proposition 1 and Lemma 2 facilitate an equivalent reformulation of the optimization problem (17) as a nonlinear semidefinite program

min a,c ω T c (21a) s.t. c j -f j (a) T -f j (a) K j (a) 0, ∀j ∈ {1 . . . n lc }, (21b)
V -T a ≥ 0, (21c)

a ≥ 0, (21d)
in which only the constraint (21b) lacks convexity. Importantly, all constraints are polynomial functions of a, forming therefore a semi-algebraic feasible set.

Efficient polynomial reformulation

The optimization problem (21) constitutes a minimization of a linear function over a semi-algebraic set, allowing for a solution using the moment-sum-of-squares hierarchy, as briefly discussed in Section 2. However, efficiency of the hierarchy can be improved after modifying (21) to provide a tighter feasible set of relaxed problems and to reduce numerical issues by scaling the design variables. These modifications are outlined in the following paragraphs.

Compactness of the feasible set

We start by enforcing compactness of the feasible set of the optimization problem (21) because of two reasons. First, compactness is required for Theorem 1, in the form of Assumption 1. Second, compactness also allows tightening the feasible sets of relaxed problems, notably improving numerical performance.

Proposition 2 Assume that a * and c * are optimal crosssection areas and compliances associated with the optimization problem (21). Then, ∀i ∈ {1 . . . n e } : 0 ≤ a * i ≤ a i with

a i = V / i and ∀j ∈ {1 . . . n lc } : 0 ≤ c * j ≤ c/ω j , where c = n lc j=1 ω j f (â) T K(â) -1 f (â) with â = 1V / ne i=1 i .
Proof. The cross-section areas are non-negative by definition (21d). Therefore, (21c) represents a conic combination and none of the structural elements can occupy a larger volume than the volume bound V , ∀i ∈ {1 . . . n e } :

a * i ≤ V / i .
The compliance variables are placed at the main diagonal of the polynomial matrix inequality (PMI) (21b) and are hence non-negative, c * j ≥ 0. Then, because ω > 0, the conic combination ω T c * is an upper bound for its summands, ω j c * j ≤ ω T c * . Moreover, since â determines uniquely the compliances ĉ, ĉj = f j (â) T K j (â) -1 f j (â), the pair (â, ĉ) is a feasible solution to (21b)-( 21d), so that we also have

ω T c * ≤ c = ω T ĉ. Consequently, ∀j ∈ {1 . . . n lc } : ω j c * j ≤ c.
Among the bounds in Proposition 2, only the compliance upper bounds are not enforced in the formulation (21). Indeed, for any fixed a > 0, c → ∞ is feasible to (21), so that Assumption 1 is not satisfied. To make the design space bounded, we add the (redundant) upper-bound compliance constraint from Proposition 2, or, eventually, an upper-bound obtained by solving the convex truss topology optimization problem instead, see Appendix A. Subsequently, we arrive at the optimization problem

min a,c ω T c (22a) s.t. c j -f j (a) T -f j (a) K j (a) 0, ∀j ∈ {1 . . . n lc }, (22b)
V -T a ≥ 0, (22c) c -ω T c ≥ 0, (22d)
a ≥ 0, (22e)
for which we have the following result:

Scaling and box constraints

After introducing box constraints in formulation ( 22), we can scale all variables domains to [-1, 1]. This scaling reduces numerical issues that may arise during the problem solution. To this goal, we have

c j = 1 2ω j (c s,j + 1) c, ∀j ∈ {1 . . . n lc }, (23a)
a i = 0.5 (a s,i + 1) a i , ∀i ∈ {1 . . . n e }, (23b)
where a s and c s are the scaled cross-section areas and compliance variables.

In addition, we explicitly insert the box constraints into the optimization problem formulation to tighten feasible sets of the relaxed problems. There are multiple options how to write these box constraints a s , c s ∈ [-1, 1], e.g.,

-1 ≤ a s,i ≤ 1, ∀i ∈ {1 . . . n b }, -1 ≤ c s,j ≤ 1, ∀j ∈ {1 . . . n lc }, (24a)
a 2 s,i ≤ 1, ∀i ∈ {1 . . . n b }, c 2 s,j ≤ 1, ∀j ∈ {1 . . . n lc }. (24b) a 4 s,i ≤ 1, ∀i ∈ {1 . . . n b }, c 4 s,j ≤ 1, ∀j ∈ {1 . . . n lc }. (24c)
Despite equivalent in what they enforce, their numerical performance in the moment-sum-of-squares hieararchy varies considerably; we refer the reader to the recent note of [START_REF] Anjos | A note on the Lasserre hierarchy for different formulations of the maximum independent set problem[END_REF]. Here, we use the quadratic bounds (24b) when the third-order terms K

(3) i,j a 3 i are absent, and quartic bounds (24c) otherwise. Then, the optimization problem reads min as,cs

n lc j=1 [0.5 (c s,j + 1) c] (25a) s.t. 1 2ωj (c s,j + 1) c -f j (a s ) T -f j (a s ) K j (a s ) 0, ∀j ∈ {1 . . . n lc }, (25b) 2 -n e -1 T a s ≥ 0, (25c) 1 -ω T c s ≥ 0, (25d)
bound constraints (24b) or (24c).

Because the feasible set of ( 25) is compact by Proposition 3, one may tempt to add a redundant polynomial inequality constraint to satisfy Assumption 1. However, the assumption is already satisfied in our case.

Proposition 4 The optimization problem (25) satisfies Assumption 1.

Proof. Let G(a s , c s ) be a block-diagonal matrix with the blocks (25b)-( 25e) and let H be a sparse matrix of the same dimensions with the structure

H = 0 0 0 I , ( 26
)
in which the identity matrix I ∈ S ne+n lc matches the positions of (25e) in G(a s , c s ). Clearly, H is a SOS because of

H = HH T , recall Definition 1. Then, if (24b) is used, p (24b) = HH T , G(a s , c s ) = n e + n lc - ne i=1 a 2 s,i - n lc j=1 c 2 s,j , so that the level set {a s ∈ R ne , c s ∈ R n lc | p (24b) ≥ 0} is compact. For (24c), p (24c) = n e + n lc - ne i=1 a 4 s,i - n lc j=1 c 4 s,j ,
showing that also the level set

{a s ∈ R ne , c s ∈ R n lc | p (24c) ≥ 0} is compact.
Remark 1 The higher-order constraints in (25e) are tighter in the moment representation, i.e., (24b) are tighter than (24a) and (24c) are tighter than (24b).

To see this, assume that (y 0 , y 1 , y 2 , y 3 , y 4 ) are the moments associated with the canonical basis of the vector space of polynomials of degree at most four, (1, x, x 2 , x 3 , x 4 ), where one can substitute x by any element of a s or c s . Then, in the first relaxation of the moment-sum-of-squares hierarchy, the fourth-order constraint 1 -x 4 ≥ 0 becomes

y 0 -y 4 ≥ 0, (27)
the quadratic constraint 1 -x 2 ≥ 0 yields

y 0 -y 2 ≥ 0, (28)
and the box constraint

-1 ≤ x ≤ 1 provides -y 0 ≤ y 1 ≤ y 0 , (29)
with y 0 = 1. Moreover, the localizing matrix of the entire optimization problem contains principal submatrices

y 0 y 1 y 1 y 2 0, (30a)
y 0 y 2 y 2 y 4 0, (30b)
y 2 y 3 y 3 y 4 0, (30c)
that must be positive semi-definite as the entire localizing matrix is. Notice that (30b) and (30c) are not present in the degree-one relaxation, which can, therefore, be used only if the bending stiffness is a polynomial of degree at most two. Thus, we omit them in the reasoning behind the relaxation tightness for (24a) and (24b).

In the lowest, degree-two, relaxation the fourth-order constraints (24c) provide y 4 ≤ 1 from Eq. ( 27) and y 2 ≥ 0 with y 4 ≥ 0 based on (30). Moreover, the determinants of (30) must be non-negative, implying that y 2 1 ≤ y 2 , y 2 3 ≤ y 2 y 4 , and y 2 2 ≤ y 4 . A combination of these inequalities then results in 0 ≤ y 4 1 ≤ y 2 2 ≤ y 4 ≤ 1 and 0 ≤ y 2 3 ≤ y 2 y 4 ≤ 1. For the quadratic constraints (24b), y 2 ≤ 1 from Eq. ( 28) and y 2 ≥ 0 because of Eq. ( 30). Writing the determinant of ( 30) then provides us with y 2 1 ≤ y 2 . Consequently, we observe that 0 ≤ y 2 1 ≤ y 2 ≤ 1. Notice that this inequality is automatically satisfied in the preceding case.

In the case of pure box constraints (24a), we only have 0 ≤ y 2 1 ≤ 1, Eq. ( 29), and y 2 1 ≤ y 2 , Eq. ( 30). Note that there is no upper bound for y 2 , which can attain arbitrarily large values in the first relaxation. From the mechanical point of view, this allows for an arbitrarily-large rotational stiffnesses K

(2) j,i a 2 i of the elements. These observations then allow us to show feasibility of the first-order moments for (25c)-(25e):

Proposition 5 Let y * c 1 and y * a 1 be the first-order moments associated with the variables c s and a s obtained from a solution to any relaxation of (25) using the moment-sum-ofsquares hierarchy. Then, these moments satisfy

2 -n e -1 T y * a 1 ≥ 0, (31a) 1 -ω T y * c 1 ≥ 0, (31b) 1 -y * a 1 i 2 ≥ 0 if (24b) is used, (31c) 1 -y * a 1 i 4 ≥ 0 if (24c) is used, (31d) 1 -y * c 1 j 2 ≥ 0 if (24b) is used, (31e) 1 -y * c 1 j 4 ≥ 0 if (24c) is used. (31f)
Proof. (31a) and (31b) hold trivially from construction of the hierarchy. (31c)-(31f) follow from Remark 1.

Recovering feasible upper-bound solutions

In Proposition 5, we have shown that the first-order moments obtained by solving any relaxation of the momentsum-of-squares hierarchy satisfy all the constraints of (25) except for (25b). This section is therefore devoted to the question how to "correct" these moments to produce feasible upper-bounds to the original problem (21) and provide a natural sufficient condition of global optimality. We start by proving the following essential result:

Proposition 6 Let y * c 1 and y * a 1 be the first-order moments associated with the variables c s and a s obtained from a solution to any relaxation of (25) using the moment-sum-ofsquares hierarchy and let ∀i ∈ {1 . . . n e } : ãi = 0.5(y * a 1 i + 1)a be the corresponding cross-section areas. Then,

f j,0 + ne i=1 f j,i ãi ∈ Im K j,0 + ne i=1 3 k=1 K (k) j,i ãk i . ( 32
)
Proof. In the lowest relaxation of the moment-sum-ofsquares hierarchy, the PMI constraint (25b) becomes

1 2ωj y * c 1 j + 1 c -f T j y * a 1 T -f j y * a 1 K j y * a 1 , y * a 2 , y * a 3 0, (33)
where y * a 2 and y * a 3 are the second-and third-order moments associated with a s and, with a slight abuse of notation, K j y * a 1 , y * a 2 , y * a 3 and f j y * a 1 are the stiffness matrix and force column vector constructed from the moments y. Using Lemma 2 and Proposition 1, we observe that

f (y * a 1 ) ∈ Im (K j (y * a 1 , y * a 2 , y * a 3 )) . (34)
Because we have considered solely degree-one moments in (32) and ∀a > 0 : K j (a) 0 was our initial assumption, we must show that the combination of a i = 0 with I i > 0 cannot occur for any i, because that would result in a lower rank of K j (ã) when compared with K j y * a 1 , y * a 2 , y * a 3 . To this goal, let a i = 0, which is equivalent to y *

a 1 i = -1.
Then, the non-negative determinant of (30a) and (30b) with the inequalities (24b) or (24c) imply that y *

a 2 i = y * a 4 i = 1.
Moreover, also the determinant of the principal submatrix

   1 y * a 1 i y * a 2 i y * a 1 i y * a 2 i y * a 3 i y * a 2 i y * a 3 i y * a 4 i    0 ( 35
)
of the moment matrix must be non-negative. Inserting y * 35) yields a unique feasible y * a 3 i = -1. Thus, we write the moment of inertia in terms of the scaled cross-section areas (23b). After inserting firstorder moments, we obtain

a 2 i = y * a 4 i = 1 with y * a 1 i = -1 into (
I i = 0.25c II a 2 y * a 2 i + 2y * a 1 i + 1 + 0.125c III a 3 y * a 3 i + 3y * a 2 i + 3y * a 1 i + 1 = 0. (36)
Using Proposition 6, we can correct c based on y * a 1 to provide a feasible solution to (21).

Theorem 2 Let y * c 1 and y * a 1 be the first-order moments associated with the variables c s and a s obtained from a solution to any relaxation of (25) using the moment-sum-ofsquares hierarchy. Then, ãi = 0.5(y *

a 1 i + 1)a i , ∀i ∈ {1 . . . n e }, ( 37a
) cj = [f j (ã)] T K † j (ã)f j (ã), ∀j ∈ {1 . . . n lc } (37b)
is feasible (upper-bound) to (21).

Proof. Based on Proposition 5, ã satisfies the constraints imposed on the cross-section areas. By correcting the compliance variables according to (37b), the equilibrium equation (and so the PMI (21b)) is satisfied due to Proposition 6. Consequently, all the constraints of ( 21) are feasible for the pair ã, c, showing that ω T c * ≤ ω T c < ∞.

We wish to emphasize that in Theorem 2, we have proved feasibility of the upper bounds to (21) and such upper bounds may violate the compliance bound constraints (25d). Thus, knowledge of ω T c * does not assure convergence of the lowest relaxation to the optimal cross-section areas.

Certificate of global ε-optimality

Because the hierarchy generates a sequence of lower bounds and we have just shown in Theorem 2 how to compute upper bounds in each relaxation, we naturally arrive at a simple sufficient condition of global ε-optimality.

Theorem 3 Let y * c 1 and y * a 1 be the first-order moments associated with the variables c s and a s obtained from a solution to any relaxation of (25) using the moment-sum-ofsquares hierarchy. Then,

ω T c - n lc j=1 0.5(y * c 1 j + 1)c ≤ ε (38)
is a sufficient condition of global ε-optimality.

Theorem 3 is very simple to verify computationally, significantly simpler than the traditional rank-based certificate of global optimality ( 8), e.g., [START_REF] Henrion | Convergent relaxations of polynomial matrix inequalities and static output feedback[END_REF]. However, (38) fails to be a necessary condition. Indeed, the optimality gap ε may remain strictly positive even when the hierarchy converged according to (8) in the case of multiple globally optimal solutions. Then, the optimal first-order moments y are not unique; for instance, they may correspond to any convex combination of the global optima, we refer to Section 4.2 for a specific example.

A stronger result holds, however, when the optimization problem possesses a unique global optimum. To show this, we first prove that, with an increasing relaxation degree r, the feasible space of relaxations converges to the convex hull of the initial (non-convex) problem.

Proposition 7 Let K (r) be the feasible set of the firstorder moments in the r-th relaxation of the moment-sum-ofsquares hierarchy of (25). Then, K (r) ↑ conv(K) as r → ∞, where K is the intersection of (25b)-(25e).

Proof. Let f (a s , c s ) be an arbitrary affine function. Based on Proposition 4, Assumption 1 holds for (25) independently of the objective function. Hence, optimization of f (a s , c s )

over K yields f (a s , c s ) ↑ f * (a s , c s ) as r → ∞ due to The- orem 1. Because f (a s , c s ) is arbitrary, K (r) ↑ conv(K) as r → ∞.
Finally, we can prove that the hierarchy eventually attains a zero optimality gap. Proof. Assuming r → ∞, optimization of (25a) over K (r) is equivalent to optimization of (25a) over conv(K) by Proposition 7. Because K is compact, its convex hull must be also compact. Hence, it can be equivalently expressed as the convex hull of the limit points of K that are denoted by

d 1 , d 2 , . . . , i.e., conv(K) = conv(∪ ∞ i=1 {d i }). ( 40
)
Because we assume there is the unique global optimum when optimizing over K, there must be a unique limit point d * associated with this optimum.

Remark 2 Although Theorem 4 relies on r → ∞, a finite (and fairly small) r is required in all our test cases to reach the zero optimality gap. Moreover, this bound equality has occurred when the hierarchy converged based on the rank test (8). It might be possible, therefore, to strengthen Theorem 4 to a finite termination result.

Global topology optimization of shell structures

Until now, solely frame structures have been considered. However, the optimization formulations ( 21) and ( 25) allow for simple modifications to optimize other discrete structures such as shells. Let t ∈ R ne ≥0 be the vector of shell element thicknesses. Then, the formulation (21) becomes

min t,c ω T c (41a) s.t. c j -f j (t) T -f j (t) K j (t) 0, ∀j ∈ {1 . . . n lc }, (41b) V -s T t ≥ 0, ( 41c
) t ≥ 0, (41d)
where s ∈ R ne >0 is a vector of the surface areas of individual shell elements, and K j (t) is assembled as

K j (t) = K j,0 + ne i=1 K (1) j,i t i + K (3) j,i t 3 i . ( 42
)
Because the design variables can be bounded very similarly to Proposition 2 and scaled, all proven results hold true.

Sample problems

This section investigates global topology optimization of selected small-scale structural design problems using the proposed strategy solved numerically by the MOSEK optimizer [START_REF] Mosek Aps | The MOSEK optimization toolbox for MATLAB manual[END_REF]. These examples demonstrate strengths and weaknesses of the presented approach: certificate of global ε-optimality using Theorem 3, extraction2 of all guaranteed globally optimal solutions based on the flat extension theorem [START_REF] Curto | Solution of the truncated complex moment problem for flat data[END_REF], but also higher computational demands when compared to selected local optimization techniques: OC and MMA adopting the nested approach, see, e.g., [START_REF] Bendsøe | Topology optimization: theory, methods, and applications[END_REF]), MATLAB's inbuilt optimizer fmincon solving (13) directly, and non-linear semidefinite programming (NSDP) formulation ( 21) solved by the PENLAB optimizer [START_REF] Fiala | A MATLAB solver for nonlinear semidefinite optimization[END_REF]. Except for the nested approaches, all optimization problems were modeled using the YALMIP toolbox [START_REF] Löfberg | YALMIP: a toolbox for modeling and optimization in MAT-LAB[END_REF]. Our implementation and the corresponding source codes written in MATLAB can be accessed at (Tyburec et al., 2020b).

The first three examples involve two finite elements only to allow visualization of the feasible sets and provide intuition about the solution approach. In the later part, we investigate the influence of finite element types on optimal design and increase the number of elements to evaluate scalability of the approach. All computations were performed on a personal laptop with 16 GB of RAM and Intel R Core TM i5-8350U CPU. Times of individual optimization approaches are measured to allow a simple comparison of the computational demands.

Structure possessing multiple global optima

As the first problem, we consider a frame structure composed of two Euler-Bernoulli frame elements, see Fig. 2a. Two loads are applied, each of them acting as a separate load case, and weighted equally by ω = 1. Both these frame elements posses the Young modulus E = 1, and their overall volume is bounded by V = 0.816597322 from above3 . Accordingly with Fig. 2b, the elements i = {1, 2} have rectangular cross-sections with areas a i = 0.3h i . Then, I i = 1 40 h 3 , which implies that c II = 0 and c III = 25/27 in Eq. ( 12).

The feasible domain of the optimization problem shown in Fig. 2c reveals that there are three global optima of the objective function value 7.738, corresponding to the following cases: (i) a * 1 = V / √ 2 and a * 2 = 0, (ii) a * 1 = 0 and Variables a 1 and a 2 stand for the cross-section areas of the two elements and ω T c denotes the corresponding weighted compliance of the two load cases (assuming the moments of inertia I i = 25/27a 3 i , i ∈ {1, 2}), and h i is the cross-section height.

1 2 3 1 2 1 1 1 1 1 (a) 0.3 h i (b) (c) (d) (e) (f)
a * 2 = V / √ 2, and (iii) a * 1 = a * 2 = V √ 2/4
. All these solutions are extracted in the fourth relaxation of the momentsum-of-squares hierarchy (Fig. 2f), which converged based on the rank condition ( 8) with the rank equal to s = 3 and also based on Theorem 3, ε = 6 × 10 -9 . Notice that in all the relaxations, the upper bounds recovered by Theorem 2 are global minima, Figs. 2d-2f.

Because all local minima are also global, all tested optimization algorithms converge to the optimal objective function value, see Table 1. Among these algorithms, OC and MMA exhibited the best performance in terms of computational time.

Irreducible positive optimality gap

Let us now modify the optimization problem described in the preceding section by fixing the volume bound to Table 1: Different optimization methods applied to the first optimization problem. LB denotes lower bound, UB abbreviates feasible upper bounds, and PO stands for polynomial optimization. The entries a i denote cross-section areas of the i-th element, or the areas constructed from the first-order moments in the case of PO. some V ∈ (0.816597322, 2.73603242). Whilst the boundary points of this open interval match the cases when three global optima occur, the interval interior removes a 1 = a 2 = V √ 2/4 from the set of globally optimal solutions. In what follows, we set V to the center of the interval.

Solving this modified optimization problem with the moment-sum-of-squares hierarchy produces the sequence of lower-and upper-bounds shown in Fig. 3. Although the hierarchy exhibited a finite convergence based on the rank condition (8) with s = 2, the corresponding optimality gap remains strictly positive (ε = 0.521) and cannot be reduced in the subsequent relaxations. Clearly, all outer convex approximations must contain the convex combination of their limit points. Hence, if the limit points denote the global optima, also their convex combinations attain the globally optimal objective function value. Therefore, they are also optimal for the associated relaxation, but may lack feasibility to the original problem. Depending on the optimization algorithm and its settings, one can either reach a lower bound that is actually feasible for the original problem (as was the case in Section 4.1), i.e., a zero optimality gap, or a positive optimality gap that cannot be further reduced, which is the case here.

I i = 25/27a 3 i , i ∈ {1, 2}).
For this particular problem, all local optimization techniques, using their default starting points and settings, missed the global optima, see Table 2. In fact, they approached the feasible upper-bound that was provided by Theorem 2. 3: Different optimization methods applied to the optimization problem with self-weight. LB denotes lower bound, UB abbreviates feasible upper bounds, and PO stands for polynomial optimization. The entries a i denote cross-sectional areas of the i-th element, or the areas constructed from the first-order moments in the case of PO.

In these cases, the volume inequality constraint could have been changed to equality, and, therefore, one design variable eliminated. However, such a procedure cannot be applied when design-dependent loads are present.

To visualize this, let our third illustration be the singleload-case frame structure in Fig. 4a, composed of two frame elements with E = 1 with I-shaped cross-sections, Fig. 4b, parametrized by the thickness t p,i . The overall volume is bounded from above by V = 1. The self-weight applies in the vertical direction and is parametrized by the material density ρ = 10. For the considered cross-sections, we have a i = 18t 2 p,i and I i = 246t 4 p,i . Hence, c II = 41/54 and c III = 0 in Eq. ( 12).

The feasible domain of this optimization problem, Fig. 4c the numerical accuracy of the optimizer. Also notice that the upper-bounds based on Theorem 2 are of very high qualities, see Table 3.

Using local optimization techniques, only OC and MMA were able to arrive at the global optimum, see Table 3. Among other formulations, NSDP approached the worst local optimum and fmincon failed even to find a feasible solution.

Different element types on a cantilever beam

A certain generality of the developed approach is illustrated on a cantilever beam/plate design problem, Fig. 5. The dimensions of the cantilever are 5 in length and 1 in width, and the thicknesses of 5 finite elements are to be found in the optimization. The beam is made of a linear-elastic material Fig. 6: Convergence of the moment-sum-of-squares hierarchy for the cantilever problem with three finite element types. Variable c denotes compliance, r stands for the relation degree, and LB with UB abbreviate lower-and upperbound. -9 × 10 -10 -2 × 10 -9 Eq. 8, s 1 1 1

Euler-Bernoulli

Table 4: Globally optimal thicknesses a * 1 , . . . , a * 5 and compliances c * for the cantilever problem for three element types: Euler-Bernoulli and Timoshenko frame elements, and the MITC4 shell element. Variables ε and s denote the optimality gap in Theorem 3 and the rank of the moment matrices according to (8), respectively. with the Young modulus E = 1 and Poisson ratio ν = 0.25. This structure is subjected to a tip distributed load of magnitude 1 induced under 30 • angle with respect to the midline/midsurface. We optimize the frame/shell thicknesses t i (of rectangular cross-sections) while satisfying V = 10. The shear correction factor is set to 5/6 where appropriate.

In what follows, we compare the optimization results of the cantilever problem for three finite element types: Euler-Bernoulli and Timoshenko frame elements, and the quadrilateral Mixed Interpolation Tensorial Component (MITC4) shell element [START_REF] Bathe | A formulation of general shell elements-the use of mixed interpolation of tensorial components[END_REF]. For both of the frame element types, we have c II = 0 and c III = I i (a)/a 3 i = 1/12 in Eq (12), whereas c II = 0 and c III = 1 for the MITC4 element.

The moment-sum-of-squares hierarchy required three steps (degree-four relaxation) to converge in all three cases, Fig. 6, and approached very similar optimal thicknesses, Table 4. As expected, the lowest compliance is provided by the Euler-Bernoulli frame elements, which neglect the shear effects. We account for these effects in the Timoshenko frame elements, increasing so the value of optimal compliance. Another generalization occurs when using the MITC4 shell elements, which not only consider the effects of shear, but further incorporate effects induced by bending about z axis, recall Fig. 5. Therefore, the optimal compliance associated with the MITC4 elements is the highest. These results thus reveal the importance of using an appropriate finite element type for particular problem, as neglecting a physical phenomenon may result in a suboptimal design. Moreover, this influence on optimal minimum-energy designs can be rigorously studied by the proposed approach. 4.5 22-elements frame structure

Our final example investigates topology optimization of a 22-element frame structure shown in Fig. 7a. Two loads are applied at nodes 3 and 6 in a single load case. In addition, we set E = 1 and V = 0.5. All the structural elements possess a thin-walled circular cross-section with the radius 5r i and the wall thickness r i , Fig. 7b. Hence, a i = 9πr 2 i and I i = 46.125πr 4 i , so that c II = 46.125/(81π) and c III = 0. The moment-sum-of-squares hierarchy requires two relaxations to achieve a guaranteed global optimum, both based on Theorem 3 with ε = 2 × 10 -5 and on Eq. ( 8) with s = 1. However, even the second relaxation is fairly computationally expensive (see

Conclusions

Our contribution has addressed the fundamental question in the structural design: how to find globally-optimal minimum-compliant bending-resistant structures in discrete topology optimization with continuous design variables. For the cases of frame and shell structures, multiple loading conditions and design-dependent loads, we have formulated this optimization problem as a (non-linear) semidefinite program constrained by a polynomial matrix inequality. The feasible space of this optimization problem forms a semialgebraic set; hence, powerful results on polynomial optimizationthe moment-sum-of-squares hierarchy-facilitate computation of the global solutions.

This hierarchy generates a sequence of tightening outer convex approximations in the space of moments of the probability measures, so that the first-order moments converge monotonically to the convex hull of the original problem. Therefore, a non-decreasing sequence of lower bounds is established. Using the first-order moments only, we have shown that a sequence of feasible upper bounds can be obtained by a simple correction. Consequently, because lower and upper bounds are available in each relaxation, the upperbound design quality can be assessed, establishing a sufficient condition of global ε-optimality. This condition is very simple to check and complements the traditional rank-based certificate of global optimality, e.g., [START_REF] Henrion | Convergent relaxations of polynomial matrix inequalities and static output feedback[END_REF].

Our condition fails to be necessary because the firstorder moments are not unique when considered optimization problem possesses multiple global optima, potentially leaving a strictly positive optimality gap. For the case of the unique global optimum, we have shown that the hierarchy eventually attains a zero optimality gap as the relaxation number approaches infinity. We note here that the possibility of the global minima multiplicity can almost be avoided in practice when the symmetry of structure and boundary conditions are exploited.

These theoretical results have been illustrated on five problems, which indicate the merits and weaknesses of the presented strategy. First, all of our test problems exhibited a rapid convergence of the hierarchy, allowing for extraction of all global solutions based on the flat extension theorem of [START_REF] Curto | Solution of the truncated complex moment problem for flat data[END_REF]. However, the computational complexity is currently fairly high, also when compared to investigated local optimization techniques, leaving the ability to compute proven global optima for smallscale optimization problems only. Yet, even for middle-scale problems, the hierarchy still provides a sequence of upper bounds of reasonable qualities, and, especially, the certificate of their ε-optimality.

In the future, we plan to extend these results in multiple directions. First, we believe that Theorem 4 can be strengthened to certify a zero optimality gap when convergence of the hierarchy in a finite number of steps occurs. Second, the computational demands could be decreased by exploiting the structural sparsity via clique-based chordal decomposition in the spirit of [START_REF] Kim | Exploiting sparsity in linear and nonlinear matrix inequalities via positive semidefinite matrix completion[END_REF][START_REF] Kočvara | Decomposition of arrow type positive semidefinite matrices with application to topology optimization[END_REF]. Another research directions may explore eigenvalue constraints and optimization [START_REF] Achtziger | Structural topology optimization with eigenvalues[END_REF][START_REF] Tyburec | Designing modular 3D printed reinforcement of wound composite hollow beams with semidefinite programming[END_REF], the minimum-weight setting, or, eventually, investigate performance of the hierarchy for different topology optimization formulations.

where K t,j (a) = K j,0 + ne i=1 K

(1) j,i a i . Suppose now that a * t are optimal cross-sections obtained from a solution to (21) with the terms K (2) j,i and K (3) j,i neglected, and ω T c * t is the associated optimal objective function value, which can be computed from a * t as

ω T c * t = nlc j=1 ω j [f j (a * t )] T [K t,j (a * t )] † f j (a * t ) . ( 44
)
When the optimal cross-sections of a truss structure, a * t , are reused in a frame structure, the resulting objective function value changes to

ω T c f = nlc j=1 ω j [f j (a * t )] T [K t,j (a * t ) + K b,j (a * t )] † f j (a * t ) (45)
with K b,j (a) = ne i=1 c II K

(2) j,i a 2 t,i + c III K (3) j,i a 3 t,i . To state a relation between ω T c * t and ω T c f we recall a useful lemma:

Lemma 3 [START_REF] Kovanic | On the pseudoinverse of a sum of symmetric matrices with applications to estimation[END_REF] Let A ∈ S n and B ∈ R n×q . Then,

A + BB T † = A † -A † B I + B T A † B -1 B T A † + B † ⊥ T B † ⊥ ( 46
)
with B ⊥ = I -AA † B.

Using this lemma, we prove that ω T ct provides an upper bound for ω T c f . Lemma 4 Suppose that ω T c * is the optimal objective function value of the frame structure design problem (21) and (43) holds. Then,

ω T c * ≤ ω T c f ≤ ω T c * t .
Proof. Because of f j (a * t ) ∈ Im (K t,j (a * t )), we clearly have f j (a * t ) ∈ Im (K t,j (a * t ) + K b,j (a * t )). Therefore, a * t is a feasible solution to the frame structure design problem (21) and the associated objective function is bounded from below by the global optimum ω T c * . Hence, ω T c * ≤ ω T c f .

For the other inequality, we express (45) using Lemma 3. To this goal, let K b,j (a * t ) = B j B T j , where B j is a real matrix because K b,j (a * t ) 0 by definition. Then, (45) can be written as

ω T c f = ω T c * t -ω T ca + ω T c b , (47)
where

c a,j = [f j (a * t )] T A † j B j I + B T j A † j B j -1 B T j A † j f j (a * t ), (48a)
c b,j = [f j (a * t )] T B † ⊥,j T B † ⊥,j f j (a * t ), (48b)
with A j = K t,j (a * t ). Clearly, Eq. ( 48a) is non-negative. For (48b),

B † ⊥ T B †
⊥ ∈ Ker(A j ), so that ω T c b vanishes. Hence,

ω T c f = ω T c * t -ω T ca ≤ ω T c * t .
Thus, when (43) holds true, the truss topology optimization produces an upper bound to the optimal objective of the frame structure topology optimization problem.

  Fig. 1: (a) Boundary conditions for the motivating problem, (b) the sublevel set c ≤ 5 of its feasible design space, and its (c) first and (d) second convex outer approximations constructed by the moment-sum-of-squares approach. Variables a 1 and a 2 stand for the cross-section areas of the two elements and c denotes the corresponding compliance (assuming moments of inertia I i = a 2 i , i ∈ {1, 2}).

Theorem 4

 4 If there is a unique global solution to (25), then ω T c -

Fig. 2 :

 2 Fig. 2: Frame structure composed of two elements: (a) boundary conditions, (b) the cross-section parametrization, and the sub-level set ω T c ≤ 10 of the (c) feasible space and of the (d) second, (e) third, and (f) fourth outer approximations with the associated lower-and upper-bounds. Variables a 1 and a 2 stand for the cross-section areas of the two elements and ω T c denotes the corresponding weighted compliance of the two load cases (assuming the moments of inertia I i = 25/27a 3 i , i ∈ {1, 2}), and h i is the cross-section height.

  PO (2) , Th. 3 0.289 0.289 5.065 7.738 0.023 PO (3) , Th. 3 0.289 0.289 7.647 7.738 0.071 PO (4) , Th. 3 0.289 0.289 7.738 7.738 0.438

Fig. 3 :

 3 Fig. 3: Frame structure possessing a non-zero optimality gap. The sub-level set ω T c ≤ 4 of the (a) feasible space and of the (b) second, and (c) third outer approximations with the associated lower-and upper-bounds. Variables a 1 and a 2 stand for the cross-section areas of the two elements and ω T c denotes the corresponding weighted compliance of the two load cases (assuming the moments of inertia I i = 25/27a 3 i , i ∈ {1, 2}).

4. 3

 3 Frame structure with self-weight For the previous examples, f (a) = f was constant, so that the optimum designs utilized the entire available volume V .

  method

Fig. 4 :

 4 Fig. 4: Frame structure with self-weight: (a) boundary conditions, (b) the cross-section parametrization, and the sub-level set c ≤ 140 of the (c) feasible space and of the (d) first, (e) second, and (f) third outer approximations with the associated lowerand upper-bounds. Variables a 1 and a 2 stand for the cross-section areas of the two elements and c denotes the corresponding compliance (assuming the moments of inertia I i = 41/54a 2 i , i ∈ {1, 2}), and t p,i stands for the flanges and web thickness.

Fig. 5 :

 5 Fig. 5: Boundary conditions of the cantilever beam design problem.

  Fig. 7: (a) Ground structure of the 22-elements frame optimization problem, (b) cross-section parameterized by r i , and optimized designs of compliances: (c) c = 3276.3 obtained by PO (1) , (d) c * = 1668.6 resulting from PO (2) and OC, (e) c = 1697.7 reached by MMA and fmincon, and (f) c = 1741.1 optimized by NSDP.

  Evaluation of the local optimization algorithms revealed that only OC converged to the global optimum (c * = 1668.6 shown in Fig.7d). The remaining optimization approaches reached local optima of comparable performance but considerably different topologies: MMA and fmincon converged to the design shown in Fig.7ewith c = 1697.7, and NSDP reached the design in Fig.7fwith c = 1741.1.

20 f y 10 y 11 f y 01 f y 02 f y 10 y 12 f y 01 y 03 y 20 f y 10 y 30 f y 20 y 21 f y 11 f y 10 y 12 f y 20 y 22 f y 11 y 13 y 11 f y 01 y 21 f y 11 y 12 f y 02

  we have y = (y 00 y 10 y 01 y 20 y 11 y 02 y 30 y 21 y 12 y 03 y 40 y 31 y 22 y 13 y 04 ) T indexed in b 2 (a, c) = (1 c a c 2 ca a 2 c 3 c 2 a ca 2 a 3 c 4 c 3 a c 2 a 2 ca 3 a 4 ) T . f y 01 y 03 f y 11 y 13 f y 02 y 04 01 V y 10 -y 11 V y 01 -y 02V y 10 -y 11 V y 20 -y 21 V y 11 -y 12 V y 01 -y 02 V y 11 -y 12 V y 02 -y 03 10 y 01 y 20 y 11 y 02 y 10 y 20 y 11 y 30 y 21 y 12 y 01 y 11 y 02 y 21 y 12 y 03 y 20 y 30 y 21 y 40 y 31 y 22 y 11 y 21 y 12 y 31 y 22 y 13 y 02 y 12 y 03 y 22 y 13 y 04

		 y 00 y 10 y 01	
		 y 10 y 20 y 11	 0.	(10f)
		y 01 y 11 y 02
	For r = 2, The
	corresponding relaxation is written as
	min y	y 10		(11a)
				y 10 f y 
	s.t.		      		      	0, (11b)
			V -y 
					 0, (11c)
						y 01 y 11 y 02	
						y 11 y 21 y 12	 0, (11d)
					y 02 y 12 y 03
					y 00 = 1, (11e)
					 y 00 y
					     
					

Table 2 :

 2 Different optimization methods applied to the second optimization problem. LB denotes lower bound, UB abbreviates feasible upper bounds, and PO stands for polynomial optimization. The entries a i denote cross-sectional areas of the i-th element, or the areas constructed from the first-order moments in the case of PO.

	method	a 1	a 2	LB	UB time [s]
	OC	0.628 0.628	-	2.161 0.003
	MMA	0.628 0.628	-	2.161 0.004
	fmincon	0.628 0.628	-	2.161 0.049
	NSDP	0.628 0.628	-	2.161 0.200
	PO (2) , Th. 3	0.628 0.628 0.936 2.161 0.022
	PO (3) , Th. 3	0.628 0.628 1.640 2.161 0.063
	PO (3) , Eq. (8)	1.256 0.000 1.640 1.640 0.063 0.000 1.256 1.640 1.640 0.063

Table 5

 5

			), prohibiting solution of
	higher relaxations of similarly-sized problems on standard
	hardware.
	method	LB	UB	time [s]
	OC	-	1668.585 2.454
	MMA	-	1697.749 13.816
	fmincon	-	1697.665 0.650
	NSDP	-	1741.062 4.406
	PO (1) , Th. 3, Eq. (8) 1062.105 3276.294 0.103
	PO (2) , Th. 3, Eq. (8) 1668.584 1668.584 1492.842

Table 5 :

 5 Different optimization methods applied to the 22frame structure design problem. LB denotes lower bounds, UB abbreviates feasible upper bounds, and PO stands for polynomial optimization.

For rank computation we considered the eigenvalues with the absolute value smaller than 10 -8 to be singular.

3 Fewer digits may prevent the solver from reaching all three global optima. Although an analytical formula for this specific V can be derived, we omit it for the sake of brevity.

f j (a) ∈ Im (K t,j (a)) , ∀j ∈ {1 . . . n lc },(43)

Acknowledgements We thank Edita Dvořáková for providing us with her implementation of the MITC4 shell elements (Dvořáková, 2015).

Marek Tyburec, Jan Zeman, and Martin Kružík acknowledge the support of the Czech Science Foundation project No. 19-26143X.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest. Replication of results Source codes are available at (Tyburec et al., 2020b).

A Relation to truss topology optimization

The problem formulation (21) has already been known in the context of truss topology optimization [START_REF] Vandenberghe | Semidefinite programming[END_REF], for which the constraints (21b) reduce to linear matrix inequalities (LMI). Consequently, the feasible set is convex, allowing for an efficient solution of ( 21) by interior point methods, for example.

A natural question then arises: What happens when the rotational degrees of freedom are neglected, solving truss topology optimization problem instead of the frame one? To this goal, however, we must first satisfy the rather restrictive assumption that the truss ground structure is capable of carrying the loads f j (a), i.e.,


Theorizing Relational Work in the Collaborative Economy: An Application to Peer-To-Peer Renting

Dominique Roux, Université de Reims Champagne-Ardenne, France Russ Belk, York University, Canada

EXTENDED ABSTRACT

Why, to whom and how do people rent out their possessions? Previous research explains why people prefer renting rather than owning [START_REF] Durgee | An Exploration into Renting as Consumption Behavior[END_REF][START_REF] Moeller | The burdens of ownership: reasons for preferring renting[END_REF] and how peer-to-peer rental systems organize balanced short-term exchanges between strangers [START_REF] Ikkala | Monetizing network hospitality: Hospitality and sociability in the context of Airbnb[END_REF][START_REF] Philip | Examining temporary disposition and acquisition in peer-to-peer renting[END_REF][START_REF] Von Richthofen | Airbnb and hybridized logics of commerce and hospitality[END_REF]. Considered as 'pseudo-sharing' [START_REF] Belk | Sharing Versus Pseudo-Sharing in Web 2.0[END_REF], renting illustrates a growing tendency toward commodification whereby individuals marketize personal assets, including private goods/spaces [START_REF] Belk | Marketization: Theory and Evidence from Emerging Economies[END_REF][START_REF] Eckhardt | The Relationship between Access Practices and Economic Systems[END_REF][START_REF] Harvey | A Brief History of Neoliberalism[END_REF]. Previous literature highlights the crucial role of platforms in "hybridizing" conflicting institutional logics between commerce and hospitality [START_REF] Scaraboto | Selling, Sharing, and Everything In-Between: The Hybrid Economies of Collaborative Networks[END_REF][START_REF] Von Richthofen | Airbnb and hybridized logics of commerce and hospitality[END_REF]) and fostering 'market empathization' [START_REF] Giesler | Designing a Sharing Economy through the Process of Empathization[END_REF]. In addition, extant literature on liquid relationships to possessions [START_REF] Bardhi | Liquid Consumption[END_REF][START_REF] Bardhi | Liquid Relationship to Possessions[END_REF] argues that access-based modes of consumption fuel a new culture of circulation [START_REF] Aronczyk | Introduction: Cultures of circulation[END_REF]Bardhi and Eckhardt 2012), which helps both renters to avoid unnecessary purchases and owners to make extra money from the temporary disposition of their personal possessions [START_REF] Horton | Owning, Using and Renting: Some Simple Economics of the 'Sharing Economy[END_REF]. However, previous studies on renting present three gaps that our research intends to fill. First, they mostly concentrate on users and thus fail to examine how owners become renters, when and why (Belk, Bardhi, and Eckhardt 2019;[START_REF] Philip | Examining temporary disposition and acquisition in peer-to-peer renting[END_REF]. Second, they pay little attention to what is rented, and in particular to the relationships owners have with their possessions before, during and after rental. Third, because they concentrate on short-term rental with Airbnb as a favorite focus of study [START_REF] Ikkala | Monetizing network hospitality: Hospitality and sociability in the context of Airbnb[END_REF][START_REF] Von Richthofen | Airbnb and hybridized logics of commerce and hospitality[END_REF], they also limit our understanding of rental activity as a whole. In short, examining "networked hospitality businesses" only [START_REF] Oskam | Airbnb: The future of networked hospitality businesses[END_REF] prevents us from understanding other contexts such as managing long term rentals or providing neighbors with everyday items.

Our approach to renting draws on [START_REF] Zelizer | Economic Lives: How Culture Shapes the Economy[END_REF][START_REF] Zelizer | How I Became a Relational Economic Sociologist and What Does That Mean[END_REF] contribution on relationality, i.e., the way social relationships shape economic lives [START_REF] Zelizer | Pricing the Priceless Child. The Changing Social Value of Children[END_REF][START_REF] Zelizer | The Social Meaning of Money. Pin money, paychecks, poor relief and other currencies[END_REF][START_REF] Zelizer | The purchase of intimacy[END_REF]. Rather than considering sociality as an extra touch of soul brought into economic activities, we contend that distinctive social ties with renters result in specific economic transactions, meanings and media of exchange, what Zelizer (2012) terms "relational packages." We theorize renting as a set of "relational packages." In addition, we extend Zelizer's (2012) approach to objects. In particular, we consider the attachment and social bonds owners have with their possessions as the cornerstone of "relational packages." Through 'relational work', including the emotional labor [START_REF] Hochschild | The Managed Heart: Commercialization of Human Feeling[END_REF]) that owners undertake both with relinquishing their items and with renters, we examine how various forms of economic exchange are articulated with specific moral valuation, meanings, communication channels and currencies that typify particular object-person and person-person-relationships [START_REF] Zelizer | How I Became a Relational Economic Sociologist and What Does That Mean[END_REF]. With renting as a relevant but overlooked practice in the field of collaborative consumption (Albinsson and Perera 2018), we raise three research questions: How do possession-owner relationships result in assessing the rent-ability of private objects/spaces? Once a possession is deemed rent-able, what conditions make objects/spaces enter into an economic relation with others? And what are the appropriate means of exchange, associated meanings and social ties that owners develop with people whom they distinctively term tenants, guests or simply neighbors?

To answer these questions, we conducted a 3-year qualitative study with 43 French informants who rent dwelling, cars and other objects. Informants were sequentially recruited through social networks using a "snowballing" technique. The interviews were mostly conducted in homes where pictures and videos were taken with their owner's consent. Although no claim of representativeness is made, informants were almost equally distributed by gender (22 F and 21 M), ranging in age from 22 to 79 (average 35), and represented a mix of different types of occupation, levels of income and rented goods, namely dwellings (19), various equipment, tools and kitchen appliances (16), cars (9), washing machines (3) and a bicycle (1).

Our findings first focus on how possessions enter the realm of 'sharing out' [START_REF] Belk | Sharing[END_REF]. We show how informants assess the 'rent-ability' of what they make available to others. This relational work with one's private goods consists of both shaping the boundaries of what is 'rent-able' and of sometimes excluding others from using parts of these things. While Philip et al. (2015, 8) argue that rented goods are preferably "unbreakable, fairly durable and not precious," our findings show that "rent-ability" is a feature attached to the closeness and secrecy of exchanges individuals have with their possessions [START_REF] Simmel | The sociology of secrecy and of secret societies[END_REF]. Hence rented goods never belong to informants' territory of intimacy [START_REF] Goffman | Relations in Public[END_REF], although this varies according to people's biographical sensitivity.

Next, we examine the circumstances under which possessions actually come to be rented and what relational work is performed both with objects and tenants/guests/neighbors as informants variously term them. We highlight three different circumstances whereby private goods/spaces come to be rented. For some informants, expensive goods such as dwellings, cars and luxurious objects (e.g., wedding dresses) are acquired from the beginning for a regular, long-term rental activity. These semi-professionals owners develop a quite intensive secondary occupation that competes with marketplace offering by providing other consumers with better prices and services. In such cases, relational work with objects/spaces appears to be demanding. It consists of selecting "interesting" goods in terms of investment such as well-situated houses, comfortable cars or highly demanded goods. Once goods are dedicated and adapted to renting, relational work with tenants requires finding the "proper renter" (i.e., liable, solvable and respectful) through various exchange media such as social networks, C2C websites and word-of-mouth. The second category of owners comes to rent goods usually after prior experiences as users of online short-term rental platforms. We call this 'generalized reciprocal sharing out' because it occurs outside the family circle [START_REF] Belk | Sharing[END_REF], but within a system of generalized reciprocity [START_REF] Sahlins | On the Sociology of Primitive Exchange[END_REF] or mutuality [START_REF] Arnould | Mutuality: Critique and substitute for Belk's "sharing[END_REF] whereby owners welcome guests as they expect to be welcomed when they are themselves travelling. While shared meanings entangle commercial/hospitality logics that are shaped by platforms (von Richthofen and Fischer 2019), our findings demonstrate high variability in how hospitality is understood and performed. Likewise, we show that relational work with objects/places results in various approaches, from leaving everything in place to restricting access to some areas and sometimes excluding oneself from one's own place of living. Finally, the third category of informants comes to rent goods because othersmostly neighbors or friends -request some things that they possess. What we call "local trade" includes lending a vacant garage, garden or DIY tools, or kitchen equipment whose request may be free before it turns into regular and paid rental. As the object is offered "as is," relational work with possessions is limited. Preference for paid rental gradually appears within existing relationships and is intended to tell renters that objects must be returned. Fees also serve to maintain distanced relationships that distinguish such exchanges from sharing 'in' [START_REF] Belk | Sharing[END_REF] and to enable owners to avoid renters' indebtedness.

We first contribute by extending the approach of P2P activities beyond single-field approaches thus showing that rental is a multifaceted activity. Second, we emphasize that relational work with possessions is required prior to making them available to others. In so doing, we extend [START_REF] Zelizer | How I Became a Relational Economic Sociologist and What Does That Mean[END_REF] work on relationality to objects. Third, we detect three different types of relational packages whereby different social ties, economic transactions, exchange media and negotiated meanings are articulated. Though limited to rental and an idiosyncratic cultural context, our study extends current research on the collaborative economy (Belk et al. 2019;[START_REF] John | The Age of Sharing[END_REF]Widlock 2017) by offering a broader and theorized approach to an overlooked access-based mode of consumption (Albinsson and Perera 2018). It shows that using a unique action verb-here renting out but elsewhere sharing or accessing-may be problematic to embrace heterogeneous realities. Rather, we assume that renting, sharing and other collaborative activities can better understood by examining the various relationships owners have with their possessions and how these define the process whereby private goods/spaces are rented, shared, mutualized or sold (Lastovicka and Fernandez 2005), with whom and how. Finally, our research sketches fruitful ways of understanding how marketplace dynamics affect and is affected by particular institutionalizing processes of entanglement of economic and relational exchanges [START_REF] Findlay | Precursors to the sharing economy: cooperatives, in The Rise of the Sharing Economy[END_REF][START_REF] Giesler | Designing a Sharing Economy through the Process of Empathization[END_REF].


Introduction

For K/F a quadratic extension of p-adic fields, let σ be the conjugation relative to this extension, and η K/F be the character of F * with kernel norms of K * . If π is a smooth irreducible representation of GL(n, K), and χ a character of F * , the dimension of the space of linear forms on its space, which transform by χ under GL(n, F ) (with respect to the action [(L, g) → L • π(g)]) , is known to be at most one (Proposition 11, [F]). One says that π is χ-distinguished if this dimension is one, and says that π is distinguished if it is 1-distinguished. In this article, we give a description of distinguished principal series representations of GL(n, K). The result (Theorem 3.2) is that the the irreducible distinguished representations of the principal series of GL(n, K) are (up to isomorphism) those unitarily induced from a character χ = (χ 1 , ..., χ n ) of the maximal torus of diagonal matrices, such that there exists r ≤ n/2, for which χ σ i+1 = χ i -1

for i = 1, 3, .., 2r -1, and χ i|F * = 1 for i > 2r. For the quadratic extension C/R, it is known (cf. [P]) that the analogous result is true for tempered representations.

For n ≥ 3, this gives a counter-example (Corollary 3.1) to a conjecture of Jacquet (Conjecture 1 in [A]). This conjecture states that an irreducible representation π of GL(n, K) with central character trivial on F * is isomorphic to πσ if and only if it is distinguished or η K/F -distinguished (where η K/F is the character of order 2 of F * , attached by local class field theory to the extension K/F ).

For discrete series representations, the conjecture is verified, it was proved in [K]. Unitary irreducible distinguished principal series representations of GL(2, K) were described in [H], and the general case of distinguished irreducible principal series representations of GL(2, K) was treated in [F-H]. We use this occasion to give a different proof of the result for GL(2, K) than the one in [F-H]. To do this, in Theorems 4.1 and 4.2, we extend a criterion of Hakim (th.4.1,[START_REF] Hakim | Distinguished p-adic Representations[END_REF]) characterising smooth unitary irreducible distinguished representations of GL(2, K) in terms of γ factors at 1/2, to all smooth irreducible distinguished representations of GL(2, K).

Preliminaries

Let φ be a group automorphism, and x an element of the group, we sometimes note x φ instead of φ(x), and x -φ the inverse of

x φ . If φ = x → h -1 xh for h in the group, then x h designs x φ .
Let G be a locally compact totally disconnected group, H a closed subgroup of G. We note ∆ G the module of G, given by the relation d

G (gx) = ∆ G (g)d G (x), for a right Haar measure d G on G.
Let X be a locally closed subspace of G, with H.X ⊂ X. If V is a complex vector space, we note D(X, V ) the space of smooth V -valued functions on X with compact support (if V = C, we simply note it D(X)). Let ρ be a a smooth representation of H in a complex vector space V ρ , we note D(H\X, ρ, V ρ ) the space of smooth V ρ -valued functions f on X, with compact support modulo H, which verify

f (hx) = ρ(h)f (x) for h ∈ H and x ∈ X (if ρ is a character, we note it D(H\X, ρ)). We note ind G H (ρ) the representation by right translations of G in D(H\G, (∆ G /∆ H ) 1/2 ρ, V ρ ).
Let F be a non archimedean local field of characteristic zero, and K a quadratic extension of F . We have K = F (δ) with δ 2 in F * . We note | | K and | | F the modules of K and F respectively. We note σ the non trivial element of the Galois group G(K/F ) of K over F , and we use the same letter to design for its action on M n (K). We note N K/F the norm of the extension K/F and we note η K/F the nontrivial character of F * which is trivial on N K/F (K * ). Whenever G is an algebraic group defined over F , we note G(K) its K-points and G(F ) its F -points.

The group GL(n) will be noted G n , its standard Borel subgroup will be noted B n , its unipotent radical U n , and the standard maximal split torus of diagonal matrices T n .

We note S the space of matrices M in G n (K) satisfying M M σ = 1.

Everything in this paragraph is more or less contained in [F1], we give detailed proofs here for convenience of the reader.

Proposition 2.1. ( [S], ch.10, prop.3) We have a homeomorphism between G n (K)/G n (F ) and S given by the map S n : g → g σ g -1 .

Proposition 2.2. For its natural action on S, each orbit of B n (K) contains one and only one element of S n of order 2 or 1.

Proof. We begin with the following:

Lemma 2.1. Let w be an element of S n ⊂ G n (K) of order at most 2. Let θ ′ be the involution of T n (K) given by t → w -1 t σ w, then any t ∈ T n (K) with tθ ′ (t) = 1 is of the form a/θ ′ (a) for some a ∈ T n (K).

Proof of Lemma 2.1. There exists r ≤ n/2 such that up to conjugacy, w is (1, 2)(3, 4)...(2r -1, 2r).

We write t =               z 1 z 1 ′ . . . z r z r ′ z 2r+1 . . . z n               , hence for i ≤ r, we have z i σ(z ′ i ) = 1,
and z j σ(z j ) = 1 for j ≥ 2r + 1. Hilbert's Theorem 90 asserts that each z j , j ≥ 2r + 1 is of the form u j-2r /σ(u j-2r ), for some

u j-2r ∈ K * .
We then take a =

              z 1 1 . . . z r 1 u 1 . . . u n-2r               . Lemma 2.2. Let N be an algebraic connected unipotent group over K. Let θ be an involutive automorphism of N (K). If x ∈ N (K), verifies xθ(x) = 1 N then, there is a ∈ N such that x = θ(a -1 )a.
Proof of Lemma 2.2: The group N (K) has a composition series 1 K), such that each quotient N i+1 /N i is isomorphic to (K, +), and each commutator subgroup [N, N i+1 ] is a subgroup of N i . Now we prove the Lemma by induction on n:

N = N 0 ⊂ N 1 ⊂ ...N n-1 ⊂ N n = N (
If n = 1, then N (K) is isomorphic to (K, +), one concludes taking a = x/2. n → n + 1 :
suppose the Lemma is true for every N (K) of length n.

Let N (K) be of length n + 1, we note x the class of x in N (K)/N 1 .

By induction hypothesis, one gets that there exists an element in h ∈ N 1 , and an element u in N (K) such that x = θ(u -1 )uh.

Here h lies in the center of

N (K), because [N (K), N 1 ] = 1 N . As xθ(x) = 1, we get hθ(h) = 1. By induction hypothesis again, we get h = θ(b -1 )b for b ∈ N 1 .
We then take a = ub.

We get back to the proof of the Proposition 2.2.

For w in S n , one notes U w the subgroup of U n generated by the elementary subgroups U α , with α positive, and wα negative, and U w ′ the subgroup of U n generated by the elementary subgroups U α , with α positive, and wα positive. Then U n = U w ′ U w . Let s be in S. According to Bruhat's decomposition, there is w in S n , and a in T n (K), n 1 in U n (K) and n 2 + in U w , such that s = n 1 awn 2 + , with unicity of the decomposition.

Then s = s -σ = n 2 + -σ w -1 a -σ n 1 -σ . Thus we have aw = (aw) -σ , i.e. w 2 = 1 and a w = a -σ . Now we write n 1 -σ = u -u + with u -∈ U w ′ and u + ∈ U w , comparing s and s -σ , u + must be equal to n 2 + . Hence s = n 1 awu --1 n 1 -σ , thus we suppose s = awn, with n in U ′ w . From s = s -σ , one has the relation awn(aw) -1 = n -σ , applying σ on each side, this becomes (aw) -1 n σ aw = n -1 . But θ : u → (aw) -1 u σ aw is an involutive automorphism of U ′ w , hence from Lemma 2.2, there is u ′ in U ′ w such that n = θ(u -1 )u. This gives s = u -σ awu, so that we suppose s = aw. Again wa σ w = a -1 , and applying Lemma 2.1 to θ ′ : x → wx σ w, we deduce that a is of the form yθ ′ (y -1 ), and s = ywy -σ .

Let u be the element 1 -δ

1 δ of M 2 (K); one has S 2 (u) = 0 1 1 0 (cf. Proposition 2.1).
We notice for further use (cf. proof of Proposition 3.1), that if we note T the subgroup

z 0 0 z σ ∈ G 2 (K)|z ∈ K * , then u -1 T u = T = x ∆y y x ∈G 2 (F )|x, y ∈ F .
For r ≤ n/2, one notes U r the n × n matrix given by the following block decomposition:

     u . . . u I n-2r      If w is an element of S n naturally injected in G n (K), one notes U w r = w -1 U r w.
Corollary 2.1. The elements U w r for 0 ≤ r ≤ n/2, and w ∈ S n give a complete set of representatives of classes of B n (K)\G n (K)/G n (F ).

Let G n = w∈Sn B n wB n be the Bruhat decomposition of G n . We call a double-class BwB a Bruhat cell.

Lemma 2.3. One can order the Bruhat cells C 1 , C 2 ,..., C n! so that for every

1 ≤ i ≤ n!, the cell C i is closed in G n - i-1 k=1 C i . Proof. Choose C 1 = B n . It is closed in G n . Now
let w 2 be an element of S n -Id, with minimal length. Then from 8.5.5. of [Sp], one has that the Bruhat cell Bw 2 B is closed in G n -B n with respect to the Zariski topology, hence for the p-adic topology, we call it C 2 . We conclude by repeating this process.

Corollary 2.2. One can order the classes A 1 ,...,

A t of B n (K)\G n (K)/G n (F ), so that A i is closed in G n (K) - i-1 k=1 A i . Proof. From the proof of Proposition 2.2, we know that if C is a Bruhat cell of G n , then S n ∩C is ei- ther empty, or it corresponds through the homemorphism S n to a class A of B n (K)\G n (K)/G n (F ).
The conclusion follows the preceeding Lemma.

Corollary 2.3. Each A i is locally closed in G n (K) for the Zariski topology.

We will also need the following Lemma: Lemma 2.4. Let G, H,X, and (ρ, V ρ ) be as in the beginning of the section, the map Φ from D(X) ⊗ V ρ to D(H\X, ρ, V ρ ) defined by Φ :

f ⊗ v → (x → H f (hx)ρ(h -1 )vdh) is surjective. Proof. Let v ∈ V ρ , U an open subset of G that intersects X, small enough for h → ρ(h)v to be trivial on H ∩ U U -1 . Let f ′ be the function with support in H(X ∩ U ) defined by hx → ρ(h)v.
Such functions generate D(H\X, ρ, V ρ ) as a vector space. Now let f be the function of D(X, V ρ ) defined by

x → 1 U∩X (x)v, then Φ(f ) is a multiple of f ′ . But for x in U ∩ X, Φ(f )(x) = H ρ(h -1 )f (hx)dh because h → ρ(h)v is trivial on H ∩ U U -1 , plus h → f (hx)
is a positive function that multiplies v, and f (x) = V , so F (f )(x) is v multiplied by a strictly positive scalar.

Corollary 2.4. Let Y be a closed subset of X, H-stable, then the restriction map from D(H\X, ρ, V ρ ) to D(H\Y, ρ, V ρ ) is surjective.

Proof. This is a consequence of the known surjectivity of the restriction map from D(X) to D(Y ), which implies the surjectivity of the restriction from D(X, V ρ ) to D(Y, V ρ ) and of the commutativity of the diagram:

D(X) → D(Y ) ↓ Φ ↓ Φ D(H\X, ρ) → D(H\Y, ρ)
.

Distinguished principal series

If π is a smooth representation of G n (K) of space V π , and χ is a character of F * , we say that π is χ-distinguished if there exists on V π a nonzero linear form L such that L(π(g)v) = χ(det(g))L(v) whenever g is in G n (F ) and v belongs to V π . If χ is trivial, we simply say that π is distinguished.

We first recall the following:

Theorem 3.1. ([F], Proposition 12)
Let π be a smooth irreducible distinguished representation of G n (K), then π σ ≃ π.

Let χ 1 , ..., χ n be n characters of K * , with none of their quotients equal to | | K . We note χ the character of B n (K) defined by χ

   b 1 ⋆ ⋆ . . . ⋆ b n    = χ 1 (b 1 )...χ n (b n ).
We note π(χ) the representation of G n (K) by right translation on the space of functions D(B n (K)\G n (K), ∆ Bn -1/2 χ). This representation is smooth, irreducible and called the principal series attached to χ. If π is a smooth representation of G n (K), we note pi its smooth contragredient.

We will need the following Lemma: Lemma 3.1. (Proposition 26 in [F1]) Let m = (m 1 , . . . , m l ) be a partition of a positive integer m, let P m be the corresponding standard parabolic subgroup, and for each

1 ≤ i ≤ l, let π i be a smooth distinguished representation of G mi (K), then π 1 × • • • × π l = ind Gm(K) P m(K ) (∆ -1/2 P m (K) (π 1 ⊗ • • • ⊗ π l )) is distinguished.
We now come to the principal results: Proposition 3.1. Let χ = (χ 1 , . . . , χ n ) be a character of T n (K), suppose that the principal series representation π(χ) is distinguished, there exists a re-ordering of the χ i 's, and r ≤ n/2, such that χ σ i+1 = χ i -1 for i = 1, 3, .., 2r -1, and that χ i|F * = 1 for i > 2r.

Proof. We write B = B n (K), G = G n (K). We have from Corollary 2.2 and 2.4 the following exact sequence of smooth G n (F )-modules:

D(B\G -A 1 , ∆ -1/2 B χ) ֒→ D(B\G, ∆ -1/2 B χ) → D(B\A 1 , ∆ -1/2 B χ).
Hence there is a non zero distinguished linear form either on

D(B\A 1 , ∆ -1/2 B χ), or on D(B\G -A 1 , ∆ -1/2 B χ).
In the second case we have the exact sequence

D(B\G -A 1 ⊔ A 2 , ∆ -1/2 B χ) ֒→ D(B\G -A 1 , ∆ -1/2 B χ) → D(B\A 2 , ∆ -1/2 B χ).
Repeating the process, we deduce the existence of a non zero distinguished linar form on one of the spaces 1, prop.3.4). From this we deduce that χ ′ is positive on

D(B\A i , ∆ -1/2 B χ). From Corollary 2.1, we choose w in S n and r ≤ n/2 such that A i = BU w r G n (F ). The applica- tion f → [x → f (U w r x)] gives an isomorphism of G n (F )-modules between D(B\A i , ∆ -1/2 B χ) and D(U -w r BU w r ∩ G n (F )\G n (F ), ∆ ′ χ ′ ) where ∆ ′ (x) = ∆ -1/2 B (U w r xU -w r ) and χ ′ (x) = χ(U w r xU -w r ). Now there exists a nonzero G n (F )-invariant linear form on D(U -w r BU w r ∩ G n (F )\G n (F ), ∆ ′ χ ′ ) if and only if ∆ ′ χ ′ is equal to the inverse of the module of U -w r BU w r ∩ G n (F ) (cf.[B-H], ch.
U -w r BU w r ∩ G n (F ) or equivalently χ is positive on B ∩ U w r G n (F )U -w r .
Let Tr be the F -torus of matrices of the form

              z 1 z 1 σ . . . z r z r σ x 1 . . . x t               with 2r + t = n, z i ∈ K * , x i ∈ F * , then one has Tr w ⊂ B ∩ U w r G n (F )U -w
r , so that χ must be positive on Tr w .

We remark that if χ is unitary, then χ is trivial on Tr w , and π(χ) is of the desired form.

For the general case, we deduce from Theorem 3.1, that there exists three integers p ≥ 0, q ≥ 0, s ≥ 0 such that up to reordering, we have

χ 2i = χ -σ 2i-1 for 1 ≤ i ≤ p, we have χ 2p+k |F * = 1 for 1 ≤ k ≤ q
4 Distinction and gamma factors for GL(2)

As said in the introduction, in this section we generalize to smooth infinite dimensional irreducible representations of G 2 (K) a criterion of Hakim (cf. [H], Theorem 4.1) characterising smooth unitary irreducible distinguished representations of G 2 (K). In proof of Theorem 4.1 of [H], Hakim deals with unitary representations so that the integrals of Kirillov functions on F * with respect to a Haar measure of F * converge. We skip the convergence problems using Proposition 2.9 of chapter 1 of [J-L].

We note M (K) the mirabolic subgroup of G 2 (K) of matrices of the form a x 0 1 with a in K * and x in K, and M (F ) its intersection with G 2 (F ). We note w the matrix 0 -1 1 0 .

Let π be a smooth infinite dimensional irreducible representation of G 2 (K), it is known that it is generic (cf. [Z] for example). Let K(π, ψ) be its Kirillov model corresponding to ψ ([J-L], th. 2.13), it contains the subspace D(K * ) of functions with compact support on the group K * . If φ belongs to K(π, ψ), and x belongs to K, then φ -π 1 x 0 1 φ belongs to D(K * ) ([J-L], prop.2.9, ch.1), from this follows that K(π, ψ) = D(K * ) + π(w)D(K * ).

We now recall a consequence of the functional equation at 1/2 for Kirillov representations (cf. [B], section 4.7). Forall φ in K(π, ψ) and χ character of K * , we have whenever both sides converge absolutely:

K * π(w)φ(x)(c π χ) -1 (x)d * x = γ(π ⊗ χ, ψ) K * φ(x)χ(x)d * x (1)
where d * x is a Haar measure on K * , and c π is the central character of π. Proof. In fact, using a Fourier inversion in functional equation 1 and the change of variable x → x -1 , we deduce that forall φ in D(K * ) ∩ π(w)D(K * ), we have

c π (x) F * π(w)φ(tx -1 )d * t = F * φ(tx)d * t
(d * t is a Haar measure on F * ) which for x = 1 gives

F * π(w)φ(t)d * t = F * φ(t)d * t.
Now we define on K(π, ψ) a linear form λ by:

λ(φ 1 + π(w)φ 2 ) = F * φ 1 (t)d * t + F * φ 2 (t)d * t
for φ 1 and φ 2 in D(K * ), which is well defined because of the previous equality and the fact that

K(π, ψ) = D(K * ) + π(w)D(K * ).
It is clear that λ is w-invariant. As the central character of π is trivial on F * , λ is also F *invariant. Because GL 2 (F ) is generated by M (F ), its center, and w, it remains to show that λ is

M (F )-invariant. Since ψ is trivial on F , one has if φ ∈ D(K * ) and m ∈ M (F ) the equality λ(π(m)φ) = λ(φ) . Now if φ = π(w)φ 2 ∈ π(w)D(K * ), and if a belongs to F * , then π a 0 0 1 π(w)φ 2 = π(w)π 1 0 0 a φ 2 = π(w)π a -1 0 0 1 φ 2 because the central character of π is trivial on F * , and λ(π a 0 0 1 φ) = λ(φ). If x ∈ F , then π 1 x 0 1 φ-φ is a function in D(K * ), which vanishes on F * , hence λπ( 1 x 0 1 φ- φ) = 0. Eventually λ is M (F )-invariant, hence G 2 (F )-invariant, it is clear that its restriction to D(K * ) is non zero. Corollary 4.1. Let µ be a character of K * , then π(µ, µ -σ ) is distinguished. Proof. indeed, first we notice that the central character µµ -σ of π(µ, µ -σ ) is trivial on F * . Now let χ be a character of K * /F * , then γ(π(µ, µ -σ )⊗χ, ψ) = γ(µχ, ψ)γ(µ -σ χ, ψ) = γ(µχ, ψ)γ(µ -1 χ σ , ψ σ ),
and as ψ|F = 1 and χ |F * = 1, one has ψ σ = ψ -1 and χ σ = χ -1 , so that γ(π(χ, χ -σ ), ψ) = γ(µχ, ψ)γ(µ -1 χ -1 , ψ -1 ) = 1. The conclusion falls from Proposition 4.1.

Assuming Theorem 1.2 of [A-G], the converse of Theorem 4.1 is also true: Theorem 4.2. Let π be a smooth irreducible representation of infinite dimension of G 2 (K) with central character trivial on F * and ψ a non trivial character of K/F , it is distinguished if and only if γ(π ⊗ χ, ψ) = 1 for every character χ of K * trivial on F * .

Proof. It suffices to show that if π is a smooth irreducible distinguished representation of infinite dimension of G 2 (K), and ψ a non trivial character of K/F , then γ(π, ψ) = 1. Suppose λ is a non zero G 2 (F )-invariant linear form on K(π, ψ), it is shown in the proof of the corollary of Proposition 3.3 in [H], that its restriction to D(F * ) must be a multiple of the Haar measure on F * . Hence for any function φ in D(K * ) ∩ π(w)D(K * ), we must have F * φ(t)d * t = F * π(w)φ(t)d * t.

Theorem 4. 1 .

 1 Let π be a smooth irreducible representation of G 2 (K) of infinite dimension with central character trivial on F * , and ψ a nontrivial character of K trivial on F . If γ(π ⊗ χ, ψ) = 1 for every character χ of K * trivial on F * , then π is distinguished.

From

  this one deduces that for any function inD(K * ) ∩ π(w)D(K * ): -1 )d * tda = K * /F * F * φ(ta)d * tda = K * φ(x)d * x .This implies that either γ(π, ψ) is equal to one, or K * φ(x)d * x is equal to zero on D(K * ) ∩ π(w)D(K * ). In the second case, we could define two independant K * -invariant linear forms onK(π, ψ) = D(K * ) + π(w)D(K * ), given by φ 1 + π(w)φ 2 → K * φ 1 (x)d * x, and φ 1 + π(w)φ 2 → K * φ 2 (x)d * x.This would contradict Theorem 1.2 of[A-G].

Acknowledgements

I would like to thank Corinne Blondel and Paul Gérardin, for many helpfull comments and suggestions.

and these χ 2p+k 's are different (so that χ 2p+k = χ -σ 2p+k ′ for k = k ′ ), and χ 2p+q+j |F * = η K/F for 1 ≤ j ≤ s, these χ 2p+q+j 's being different. We note µ k = χ 2p+k for q ≥ k ≥ 1, and ν ′ k = χ 2p+q+k ′ for s ≥ k ′ ≥ 1. We show that if such a character χ is positive on a conjugate of Tr by an element of S n , then s = 0. Supose ν 1 appears, then either ν 1 is positive on F * , but that is not possible, or it is coupled with another χ i , and (ν 1 , χ i ) is positive on elements (z, z σ ), for z in K * . Suppose χ i = ν j for some j = 1, then (ν 1 , χ i ) is unitary, so it must be trivial on couples (z, z σ ), which implies ν 1 = ν j -σ = ν j , which is absurd. The character χ i cannot be of the form µ j , because it would imply ν 1|F * = 1. The last case is i ≤ 2p, then ν 1 -σ = ν 1 must be the unitary part of χ i because of the positivity of (ν 1 , χ i ) on the couples (z, z σ ). But χ i -σ also appears and is not trivial on F * , hence must be coupled with another character χ j with j ≤ 2p and j = i, such that (χ i -σ , χ j ) is positive on the elements (z, z σ ), for z in K * , which implies that χ j has unitary part ν 1 -σ = ν 1 . The character χ j cannot be a µ k because of its unitary part. If it is a χ k with k ≤ 2p, we consider again χ k -σ . But repeating the process lengthily enough, we can suppose that χ j is of the form ν k , for k = 1. Taking unitary parts, we see that ν k = ν 1 -σ = ν 1 , which is in contradiction with the fact that all ν i 's are different. We conclude that s = 0.

Theorem 3.2. Let χ = (χ 1 , . . . , χ n ) be a character of T n (K), the principal series representation π(χ) is distinguished if and only if there exists r ≤ n/2, such that χ σ i+1 = χ i -1 for i = 1, 3, .., 2r -1, and that χ i |F * = 1 for i > 2r.

Proof. There is one implication left. Suppose χ is of the desired form, then π(χ) is parabolically (unitarily) induced from representations of the type π(χ i , χ i -σ ) of G 2 (K), and distinguished characters of K * . Hence, because of Lemma 3.1 the Theorem will be proved if we know that the representations π(χ i , χ i -σ ) are distinguished, but this is Corollary 4.1 of the next paragraph.

This gives a counter-example to a conjecture of Jacquet (conjecture 1 in [A]), asserting that if an irreducible admissible representation π of G n (K) verifies that π is isomorphic to π σ , then it is distinguished if n is odd, and it is distinguished or η K/F -distinguished if n is even.

Corollary 3.1. For n ≥ 3, there exist smooth irreducible representations π of G n (K), with central character trivial on F * , that are neither distinguished, nor η K/F -distinguished, but verify that π is isomorphic to π σ .

Proof. Take χ 1 , . . . , χ n , all different, such that χ 1|F * = χ 2|F * = η K/F , and χ j |F * = 1 for 3 ≤ j ≤ n. Because each χ i has trivial restriction to N K/F (K * ), it is equal to χ -σ i , hence π is isomorphic to π σ . Another consequence is that if k and l are two different integers between 1 and n, then χ k = χ -σ l , because we supposed the χ i 's all different. Then it follows from Theorem 3.2 that π = π(χ 1 , . . . , χ n ) is neither distinguished, nor η K/Fdistinguished, but clearly, the central character of π is trivial on F * and π is isomorphic to π σ .