Matthieu Barreau

Alexandre Seuret

Frédéric Gouaisbaut

Exponential Lyapunov Stability Analysis of a Drilling Mechanism

This article deals with the stability analysis of a drilling system which is modelled as a coupled ordinary differential equation / string equation. The string is damped at the two boundaries but leading to a stable open-loop system. The aim is to derive a linear matrix inequality ensuring the exponential stability with a guaranteed decay-rate of this interconnected system. A strictly proper dynamic controller based on boundary measurements is proposed to accelerate the system dynamics and its effects are investigated through the stability theorem and simulations. It results in an efficient finite dimension controller which subsequently improves the system performances.

I. INTRODUCTION

Many physical situations like string-payloads [START_REF] He | Adaptive control of a flexible crane system with the boundary output constraint[END_REF] or drilling systems [START_REF] Bresch-Pietri | Output-feedback adaptive control of a wave PDE with boundary anti-damping[END_REF] are modeled by infinite dimensional systems. They are, in their fundamentals, related to a Partial Differential Equation (PDE) and consequently, their stability analysis and control are not straightforward and has been under active research during the last decade.

A drilling mechanism is within this class of systems. It is used in the industry to pump oil deep in the soil. This physical system is subject to torsion and radial deformation due to the torque applied on one boundary of the pipe. This system is usually modeled by a coupled Ordinary Differential Equation (ODE) / string equation. These heterogeneous equations appear naturally when the torsional motion of the pit is coupled with the axial deformation of the pipe [START_REF] Challamel | Rock destruction effect on the stability of a drilling structure[END_REF]. Moreover, as there is friction all along the pipe, it leads to a complex system made up of two non-linear equations. The commonly used methodology to control this system is the backstepping.

The aim is to use a control to transform the problem into a target system with the desired properties. Then, using a Lyapunov approach for example, the stability can be proven. This has been widely used in [START_REF] Bresch-Pietri | Output-feedback adaptive control of a wave PDE with boundary anti-damping[END_REF], [START_REF] Krstic | Delay compensation for nonlinear, adaptive, and PDE systems[END_REF], [START_REF] Krstic | Boundary control of PDEs: A course on backstepping designs[END_REF], [START_REF] Wu | Static output feedback control via PDE boundary and ODE measurements in linear cascaded ODE-beam systems[END_REF]. There are many advantages because it provides a Lyapunov functional useful for a robustness analysis for example but it also provides a very accurate control as it mostly depends on the target system. But the calculations are tedious and lead to an infinite dimension control law which may be subjected to implementation issues.

Coming from the stability analysis of time-delay systems, a new method based on Linear Matrix Inequalities (LMIs) seems to be promising. As time-delay systems are a particular case of infinite dimension systems [START_REF] Fridman | Introduction to Time-Delay Systems[END_REF], it is possible to extend the methodology described in [START_REF] Seuret | Hierarchy of LMI conditions for the stability analysis of time delay systems[END_REF] to other systems. It relies on a Lyapunov functional and a state extension using projections of the infinite dimensional state on a basis of orthonormal polynomials. The key result is based on an extensive use of Bessel inequality. It has been successfully applied to transport equations in [START_REF] Safi | Tractable sufficient stability conditions for a system coupling linear transport and differential equations[END_REF], to the heat equation [START_REF] Baudouin | Lyapunov stability analysis of a linear system coupled to a heat equation[END_REF] and to the wave equation also [START_REF] Barreau | Lyapunov stability analysis of a string equation coupled with an ordinary differential system[END_REF]. M. Barreau, A. Seuret, F. Gouaisbaut are with LAAS -CNRS, Université de Toulouse, CNRS, UPS, France e-mail: (mbarreau,aseuret,fgouaisb@laas.fr).

In this paper, we focus on the exponential stability analysis of a linearized drilling mechanism as described in [START_REF] Saldivar | Analysis and Control of Oilwell Drilling Vibrations: A Time-Delay Systems Approach[END_REF] with the previous methodology. First, we explain the problem and discuss the existence of a solution. Then, an exponential stability result is provided. The theorem ensures the exponential stability with a guaranteed decay-rate. Some necessary conditions are drawn from the LMI condition and then, an example using physical values is provided. A control law is also derived to show the effectiveness of the method.

Notations:

In this paper, R + = [0, +∞) and (x, t) → u(x, t) is a multi-variable function from [0, 1] × R + to R.
The notation u t stands for ∂u ∂t . We also use the notations L 2 = L 2 ((0, 1); R) and for the Sobolov spaces:

H n = {z ∈ L 2 ; ∀m n, ∂ m z ∂x m ∈ L 2 }. The norm in L 2 is z 2 = Ω |z(x)| 2 dx = z, z .
For any square matrices A and B, the operations 'He' and 'diag' are defined as follow:

He(A) = A+A and diag(A, B) = [ A 0 0 B ].
A positive definite matrix P ∈ R n×n belongs to the set S n + and P 0.

II. PROBLEM STATEMENT

A. Modeling of the drilling process

A drilling mechanism was first modeled in [START_REF] Fridman | Bounds on the response of a drilling pipe model[END_REF] using the work of [START_REF] Challamel | Rock destruction effect on the stability of a drilling structure[END_REF]. This system described in Figure 1 is the result of a coupling between a radial deformation and an axial movement. This coupling was later modeled in [START_REF] Saldivar | Analysis and Control of Oilwell Drilling Vibrations: A Time-Delay Systems Approach[END_REF], [START_REF] Saldivar | The control of drilling vibrations: A coupled PDE-ODE modeling approach[END_REF] by the following nonlinear model for x ∈ (0, 1) and t > 0:

               z tt (x, t) = c 2 z xx (x, t) -dz t (x, t), z x (0, t) = g (z t (0, t) -ũ1 (t)) , z x (1, t) = -hz tt (1, t) -kz t (1, t) -qT nl (z t (1, t)), Ẏ (t) = AY (t) + B ũ2 (t) + E 1 z t (1, t) + E 2 T nl (z t (1, t)), (1)
with initial condition z(•, 0) = z 0 , z t (•, 0) = z 0 t on (0, 1) and Y (0) = Y 0 . In this model, z is the twist angle and it propagates along the pipe following a damped wave equation of speed c and internal damping d. Since the internal damping stabilizes the system, in this study, we consider the worst case scenario with d = 0 like in [START_REF] Fridman | Bounds on the response of a drilling pipe model[END_REF]. A similar work can be done with d > 0 but leads to more tedious calculation and is then omitted. There are two boundary conditions at x = 0 and x = 1. At x = 0, a rotary table whose speed is controlled by the input ũ1 allows to twist the pipe. Furthermore, the boundary damping with a coefficient g at x = 0 represents a viscous friction torque. The drilling pit is located at x = 1. When drilling, an external torque applies at this boundary and the momentum equation leads to a second order in time boundary condition. The term T nl is a non-linear function related to the change of torque and given below. To simplify the system as done in [START_REF] Fridman | Bounds on the response of a drilling pipe model[END_REF], we

Rotary table

Kelly

Winch

From/to mud pumps

Drill pipes

Drill collars Bit Fig. 1. Schematic of a drilling mechanism originally taken from [START_REF] Saldivar | The control of drilling vibrations: A coupled PDE-ODE modeling approach[END_REF]. Data corresponding to physical vaues are given in Table I.

consider the equation at the bottom of the pipe to be only a first order boundary damping, then h = 0. The axial deformation is modeled by a finite dimensional equation as noted in [START_REF] Challamel | Rock destruction effect on the stability of a drilling structure[END_REF]. This equation is related to the axial deformation of the pipe. In [START_REF] Saldivar | The control of drilling vibrations: A coupled PDE-ODE modeling approach[END_REF], a second order damped harmonic oscillator is used because it models a mass subject to a force for small vibrations. The control at x = 0 for the axial position is t → ũ2 (t) and corresponds to the force needed in the system to drill. Denoting by y the axial bit position and by Γ 0 the rate of penetration, Y (t) = [y(t) -Γ 0 t ẏ(t) -Γ 0 ] ∈ R 2 represents the axial position error and axial velocity error, leading to the last equation in [START_REF] Baudouin | Lyapunov stability analysis of a linear system coupled to a heat equation[END_REF].

Remark 1: Note that this model does not take into account a coupling between torsion and axial deformation but more a cascaded effect between them. The parameters c, g, k, q, A 21 , A 22 , b, e 1 and e 2 are physical parameters given in [START_REF] Saldivar | The control of drilling vibrations: A coupled PDE-ODE modeling approach[END_REF] and reported in Table I. The matrices have the following structure:

A = 0 1 A21 A22 , B = [ 0 b ] , E 1 = 0 e1 , E 2 = 0 e2 .
The aim is to design control laws ũ1 and ũ2 such that the angular speed z t (1, t) in system (4) converges to the desired angular velocity Ω e and Y to 0. Without loss of generality, we assume Ω e > 0.

In [START_REF] Challamel | Rock destruction effect on the stability of a drilling structure[END_REF], [START_REF] Saldivar | The control of drilling vibrations: A coupled PDE-ODE modeling approach[END_REF], the nonlinear part of the torque is described by the following equations for θ ∈ R:

   T nl (θ) = W ob R b µ b (θ) sign(θ), µ b (θ) = µ cb + (µ sb -µ cb ) e -γ b |θ| . (2)
Considering Ω e 0, then e -γ b Ωe is small and T nl is linearized around Ω e as follows:

T nl (z t (1, t)) W ob R b µ cb = T e . ( 3
)
Remark 2: This approximation prevents from the stick-slip effect which is the main problem that occurs when dealing with drilling pipes for small Ω e . This work can be seen as a preliminary version of an extended one considering the nonlinearity. That leads to an approximated linear system defined for t 0 with the same initial conditions and x ∈ (0, 1):

               w tt (x, t) = c 2 w xx (x, t), w x (0, t) = g (w t (0, t) -ũ1 (t)) , w x (1, t) = -kw t (1, t) -qT e Ẏ (t) = AY (t) + B ũ2 (t) + w t (1, t)E 1 -T e E 2 . (4)
It is possible to use the Riemann coordinates to simplify the writing of this system using the following variable: χ(x, t) = wt(x,t)+cwx(x,t) wt(1-x,t)-cwx(1-x,t) . The system becomes for t 0:

         χt (x, t) = c χx (x, t),
x ∈ (0, 1),

1-cg 0 0 1-ck χ(0, t) = 0 1+cg 1+ck 0 χ(1, t) + -2cg ũ1(t) 2cqT e , Ẏ (t) = AY (t) + B ũ2 (t) + Ẽ1 χ(0,t) χ(1,t) -T e E 2 , (5)
with Ẽ1 = 1 2 E 1 [ 0 1 1 0
]. The stability of system ( 5) implies the stability of ( 4) and then the study focuses on system [START_REF] Baudouin | Stability analysis of a system coupled to a transport equation using integral inequalities[END_REF].

Assuming ( χe , Y e ) is an equilibrium point of system ( 5), it satisfies χe t = 0, w e t = Ω e and Ẏ e = 0. Therefore, a feedforward open-loop control is introduced as:

ũe 1 = Ω e 1 + k g + q g T e , ũe 2 = T e e 2 -Ω e e 1 b . (6)
Introducing the error variables χ(x, t) = χ(x, t) -χe (x), u 1 (t) = ũ1 (t)-ũe 1 and u 2 (t) = ũ2 (t)-ũe 2 , the aim is to show the exponential stability of χ to 0 in order to get w t → Ω e and Y → 0. The inputs u 1 and u 2 are assumed to be the results of a strictly proper dynamic controller whose inputs are w t (0, t), w t (1, t) and Y . That means that the measurements are these three variables but it is not possible to apply exactly w t (1) or w t (0), corresponding to the situation where the actuator is bandwidth limited for instance. This assumption is important as the wave can be seen as a neutral system [START_REF] Barreau | Input / output stability of a damped string equation coupled with ordinary differential system[END_REF] and using directly w t means that we can affect directly the neutral part. This phenomena is known to be absolutely non-robust [START_REF] Hale | Effects of small delays on stability and control[END_REF] to small delay for example. Assuming the controller is of order n, it is written for t 0: 2 and K ∈ R 1×2 . The closed-loop system in Riemann coordinates can be rewritten as:

         Ẋc (t) = A c X c (t) + B c1 Y (t) + B c2 wt(0,t) wt(1,t) , u 1 (t) = C 1 Xc(t) Y (t) , u 2 (t) = C 2 X c (t) + KY (t). with C 1 , C 2 ∈ R 1×(n+2) , A c ∈ R n×n , B c1 , B c2 ∈ R n,
         χ t (x, t) = cχ x (x, t), 1-cg 0 0 1-ck χ(0, t) = 0 1+cg 1+ck 0 χ(1, t) -2cgC1X(t) 0 , Ẋ(t) = ÃX(t) + B χ(0,t) χ(1,t) , (7) with initial conditions χ(x, 0) = χ 0 (x), X(0) = X 0 , X = [ X c Y ] and à =   A c B c1 BC 2 A+BK   , B = 1 2   B c2 E 1 0 2,1   1 0 0 1 0 1 1 0
.

Remark 3: A similar control law is proposed in [START_REF] Terrand-Jeanne | Regulation of the downside angular velocity of a drilling string with a P-I controller[END_REF] but the stability is dealt using another Lyapunov functional.

Remark 4: From now on, to ease the reading, the parameter t may be omitted and χ refers to a solution of [START_REF] Challamel | Rock destruction effect on the stability of a drilling structure[END_REF].

B. Existence and uniqueness

The existence and uniqueness follows the same lines than in [START_REF] Barreau | Lyapunov stability analysis of a string equation coupled with an ordinary differential system[END_REF]. Define the following set:

H m = R n+2 × H m × H m with m ∈ N.
The space H = H 0 can be equipped with the following norm:

∀(X, χ) ∈ H, (X, χ) H = |X| 2 + 1 2 χ 2 = |X| 2 + c 2 w x 2 + w t 2 .
Using the operator notation [START_REF] Tucsnak | Observation and control for operator semigroups[END_REF], system ( 7) is formulated as follows:

T X χ =   ÃX + B χ(0) χ(1) cχ x   , and
T : D(T ) → H, with
D(T ) = (X, χ) ∈ H 1 , 1-cg 0 0 1-ck χ(0) = 0 1+cg 1+ck 0 χ(1) -2cgC1X 0 .
The existence of a continuous solution for (X 0 , χ 0 ) ∈ D(T ) is ensured by applying Lumer-Philips theorem (for example in [23, p.103]) whose conditions are recalled below:

1) there exists a function V : H → R + such that its derivative along the trajectories of ( 7) is negative; 2) there exists λ sufficiently small such that D(T ) ⊆ R(λI -T ) where R is the range operator. The first condition relies on the existence of a Lyapunov functional and is therefore the subject of the following part. The second statement needs some calculations very similar to the one conducted in [START_REF] Barreau | Lyapunov stability analysis of a string equation coupled with an ordinary differential system[END_REF] or [START_REF] Morgül | A dynamic control law for the wave equation[END_REF]. For a given λ > 0, let (r, f ) ∈ D(T ), the aim is to prove the existence of (X, χ) ∈ D(T ) satisfying the following for x ∈ (0, 1):

   λX -ÃX -B χ(0) χ(1) = r, λχ(x) -cχ x (x) = f (x). That leads to χ(x) = k 1 e λ x c + F (x) with F (x) = c -1 x 0 e λ x-s c f (s)ds ∈ H 1 and k 1 = diag(k 11 , k 12 ), k 11 , k 12 ∈ R.
Using the boundary conditions, we get a system of two equations:

(1 -cg)k 1 = k 2 e λ c (1 + cg)(A + F (1)) -2cg λ C 1 X, (1 -ck)k 2 = k 1 e λ c (1 + ck)(A + F (1))
Since there exists a λ such that à + B χ(0) χ( 1) is not the null matrix, then this system has a unique solution for a given X that ends the proof of existence.

III. EXPONENTIAL STABILITY OF THE DRILLING PIPE A. Main result

The main result of this paper is the α-stability criterion for system [START_REF] Challamel | Rock destruction effect on the stability of a drilling structure[END_REF] expressed in terms of LMIs, therefore easily tractable. Let us first define the α-stability.

Definition 1: System ( 7) is α-stable (or exponentially stable with a decay-rate of at least α) with respect to the norm • H if there exists γ 1 such that the following holds for (X 0 , χ 0 ) the initial condition:

(X(t), χ(•, t)) H γ (X 0 , χ 0 ) H e -αt .
Considering this definition, we propose a stability theorem for system [START_REF] Challamel | Rock destruction effect on the stability of a drilling structure[END_REF].

Theorem 1: Let N > 0. Assume there exists

P N ∈ S n+2+2(N +1) + , R, S ∈ S 2
+ such that the following LMI holds:

Ψ N,α -cR N ≺ 0, (8)
with

Ψ N,α = He((Z N + αF N ) P N F N ) -cG N SG N +cH N (S + R) H N e 2α c , F N = I n+2+2(N +1) 0 n+2+2(N +1),2 , Z N = N N Z N , N N = Ã 0 n+2,2(N +1) B , Z N = c1 N H N -c 1N G N -0 2(N +1),n+2 L N 0 2(N +1),2 , (9)
G N = -cgC1 01,n+2 0 2,2(N +1) G , G = 0 1+cg 1+ck 0 , H N = 01,n+2 cgC1 0 2,2(N +1) H , H = 1-ck 0 0 1-cg , R N = diag(0 n , R, 3R, • • • , (2N + 1)R, 0 2 ), L N = 0,0 I2 ••• 02 . . . . . . . . . N,0 I2 ••• N,N I2 , 1 N = I2 . . . I2 , 1N = I2 . . . (-1) N I2
, and k,j = (2j + 1)(1 -(-1) j+k ) if j k and 0 otherwise. Then system (7) is α-exponentially stable. The proof of this theorem relies on the construction of a Lyapunov functional described in the following subsections.

Remark 5: A necessary condition for (8) to be fulfilled is that the last 2 × 2 diagonal block of (8) must be definite negative corresponding to the following inequality:

H (S + R)He 2 α c -G SG ≺ 0.
This condition implies:

α α max = max c 2 log (ck + 1)(cg + 1) (ck -1)(cg -1) , 0 . ( 10
)
Setting g = 0 or k = 0 leads to the same maximal decayrate than in [START_REF] Barreau | Input / output stability of a damped string equation coupled with ordinary differential system[END_REF], [START_REF] Bastin | Stability and boundary stabilization of 1-d hyperbolic systems[END_REF], [START_REF] Datko | Two questions concerning the boundary control of certain elastic systems[END_REF]. This condition is also related to the τ -stabilization which is a common phenomenon when considering a wave equation [START_REF] Olgac | A practical method for analyzing the stability of neutral type LTI-time delayed systems[END_REF]. One can notice that for g > 0 and k > 0, the PDE system itself is asymptotically stable, because the two boundary conditions are adding damping. Notice that if one of them is negative, there exist also values of the other coefficient making the system asymptotically stable. Note also that for g = c -1 or k = c -1 leads to α max = +∞ meaning there is no neutral part and the system resumes to a time-delay system. For d > 0, the neutral part is not modified and the same limit can be observed.

Remark 6 (Hierarchy): Define the following set:

C N = {α 0 | Ψ N,α -R N ≺ 0, P N 0, R 0, S 0} ,
and assume this set is not empty. Then, denote α N = sup C N .

The hierarchy property states that α N +1 α N . This can be proved using the same strategy than in [START_REF] Barreau | Lyapunov stability analysis of a string equation coupled with an ordinary differential system[END_REF], [START_REF] Safi | Tractable sufficient stability conditions for a system coupling linear transport and differential equations[END_REF].

B. Proof of Theorem 1 1) Preliminaries:

The main contribution of this paper relies on the extensive use of Bessel inequality to encompass traditional results. Before stating this inequality, we need to introduce an orthonormal family. The definition is as follows:

Definition 2 (Legendre polynomials): Let N ∈ N, the family of Legendre polynomials of degree less than or equal to N is denoted by {L } ∈[0,N ] with

L (x) = (-1) l=0 (-1) l l +l l x l with l = ! l!( -l)! .
The sequence {L k } is made up of "shifted"-Legendre polynomials on [0, 1]. As seen in [START_REF] Baudouin | Stability analysis of a system coupled to a transport equation using integral inequalities[END_REF], [START_REF] Courant | Methods of mathematical physics[END_REF], [START_REF] Seuret | Hierarchy of LMI conditions for the stability analysis of time delay systems[END_REF], this family is orthonormal in L 2 with the canonical inner product. That leads to the following definition.

Definition 3: Let χ ∈ L 2 . The projection of χ on the th Legendre polynomials is defined as follows:

X := 1 0 χ(x)L (x)dx.
The Bessel inequality is obtained considering the previous definitions and the orthogonal property of the shifted-Legendre family.

Lemma 1 (Bessel Inequality): For any function χ ∈ L 2 and symmetric positive matrix R ∈ S 2 + , the following Bessel-like integral inequality holds for all N ∈ N:

1 0 χ (x)Rχ(x)dx N =0 (2 + 1)X RX . ( 11
)
This lemma and its short proof can be seen in [START_REF] Barreau | Lyapunov stability analysis of a string equation coupled with an ordinary differential system[END_REF]. The derivation of X along time is needed in the sequel. Lemma 3 from [START_REF] Barreau | Lyapunov stability analysis of a string equation coupled with an ordinary differential system[END_REF] deals with this issue.

Lemma 2: For any function χ ∈ L 2 , the following expression holds for any N in N using notations (9):

Ẋ0 . . . ẊN = c1 N χ(1) -c 1N χ(0) -cL N X0 . . . X N .
The link between α-exponential stability and a Lyapunov functional is made by the following lemma.

Lemma 3: Let V be a Lyapunov functional for system (7) and α ≥ 0. Assume there exist ε 1 , ε 2 , ε 3 > 0 such that the following holds for all t 0:

   ε 1 (X, χ) 2 H V (X, χ) ε 2 (X, χ) 2 H , V (X, χ) + 2αV (X, χ) -ε 3 (X, χ) 2 H , (12)
then system ( 7) is α-exponentially stable.

Proof. Inequalities [START_REF] Hale | Effects of small delays on stability and control[END_REF] bring the following: V (X, w) + α + ε3 ε2 V (X, χ) 0. Then integrating this inequality between 0 and t leads to:

(X(t), χ(t)) 2 H ε 2 ε 1 (X 0 , χ 0 ) 2 H e -2αt .
Once these useful lemmas reminded, a Lyapunov functional can be defined.

2) Lyapunov functional candidate: The aim of this subpart is to build a Lyapunov functional candidate for system [START_REF] Challamel | Rock destruction effect on the stability of a drilling structure[END_REF]. Following the same methodology than introduced in [START_REF] Barreau | Lyapunov stability analysis of a string equation coupled with an ordinary differential system[END_REF], a first Lyapunov functional V α for the PDE part is defined with S, R ∈ S 2 + :

V α (χ) = 1 0 e 2 αx c χ (x)(S + xR)χ(x)dx,
The Lyapunov functional candidate is then the summation of a quadratic term and V α . This quadratic term contains the stability of state X but also some terms merging the ODE and the PDE. This is done to enlarge the stability analysis, enabling the study of stability of the whole interconnected system and not of each subsystem independently. This technique, as shown in [START_REF] Barreau | Lyapunov stability analysis of a string equation coupled with an ordinary differential system[END_REF], is well-suited for the study of an unstable ODE coupled with a PDE for instance. The total Lyapunov function of order N ∈ N is then:

V N,α (X, χ) = X N P N X N + V α (χ) (13)
with

P N ∈ S n+2+2(N +1) +
and X N = X X 0 . . . X N . The aim now is to prove the existence of ε 1 , ε 2 and ε 3 > 0 to apply Lemma 3 on the functional V N,α and then conclude the proof.

3) Existence of ε 1 : Conditions P N 0 and S, R ∈ S 2 + mean that there exists ε 1 > 0, such that for all x ∈ [0, 1]:

P N ε 1 diag (I n+2 , 0 2 ) , S + xR S ε1 2 I 2 .
These inequalities imply:

V N,α (X, w) ε 1 |X| 2 + 1 2 χ 2 + 1 0 χ (x) S + xR -ε1 2 I 2 χ(x)dx ε 1 |X| 2 n + 1 2 χ 2 ε 1 (X, χ) 2 H .
4) Existence of ε 2 : Since P N , S and R are definite positive matrices, there exists ε 2 > 0 such that:

P N diag ε 2 I n+2 , ε2 4 diag {(2 + 1)I n } ∈(0,N ) , (S + xR) S + R ε2 4 e -2 α c I 2 , ∀x ∈ (0, 1)
. Then, we get:

V N,α (X, χ) ε 2 |X| 2 + ε 2 4 N =0 (2 +1)X X + χ 2 ε 2 |X| 2 + 1 2 χ 2 = ε 2 (X, χ) 2
H . The inequality comes from Bessel inequality [START_REF] Fridman | Bounds on the response of a drilling pipe model[END_REF]. 5) Existence of ε 3 : This part is the most important and shows that system (7) is dissipative [START_REF] Barreau | Lyapunov stability analysis of a string equation coupled with an ordinary differential system[END_REF], [START_REF] Tucsnak | Observation and control for operator semigroups[END_REF]. Differentiating with respect to time [START_REF] He | Adaptive control of a flexible crane system with the boundary output constraint[END_REF] along the trajectories of system [START_REF] Challamel | Rock destruction effect on the stability of a drilling structure[END_REF] leads to:

VN,α (X, w) = He        Ẋ Ẋ0 . . . ẊN    P N   X X0 . . . X N       + Vα (w).
The goal here is to find an upper bound of VN,α using the extended state: ξ N = X N w t (1) w t (0) . The first step is to derive an expression of Vα . Similarly to [START_REF] Barreau | Lyapunov stability analysis of a string equation coupled with an ordinary differential system[END_REF], we get: Using the previous equation, Lemma 2 and equation ( 4), we note that [START_REF] Datko | Two questions concerning the boundary control of certain elastic systems[END_REF]. Then we can write:

Vα (χ) = 2c
X N = F N ξ N , ẊN = Z N ξ N , χ(0) = G N ξ N , χ(1) = H N ξ N where matrices F N , Z N , H N , G N are given in
VN,α (X, χ) = ξ N Ψ N,α ξ N + c N =0 X (2 + 1)RX -c 1 0 χ (x)Rχ(x)e 2 αx c dx -2αV N,α (X, χ).
Denoting by W N,α (X, χ) = VN,α (X, χ) + 2αV N,α (X, χ), the previous equality implies the following upper bound:

W N,α (X, χ) ξ N Ψ N,α ξ N + c N =0 (2 + 1)X RX -c 1 0 χ (x)Rχ(x)dx. (14)
Since R 0 and Ψ N,α ≺ 0, there exists ε 3 > 0 such that:

R ε3 2 I 2 , Ψ N,α -ε 3 diag I n+2 , 1 2 I 2 , 3 2 I 2 , . . . , 2N+1 2 I 2 , 0 2 . (15)
Using ( 15) and Bessel's inequality, equation ( 14) becomes:

W N,α (X, χ) -ε 3 |X| 2 + 1 2 χ 2 -ε 3 (X, χ) 2 H ,
and that concludes the proof.

IV. EXAMPLES AND DISCUSSION

In this section, we illustrate the proposed theorem by using values taken from [START_REF] Saldivar | Analysis and Control of Oilwell Drilling Vibrations: A Time-Delay Systems Approach[END_REF], [START_REF] Saldivar | The control of drilling vibrations: A coupled PDE-ODE modeling approach[END_REF] and shown in Table I. The simulation is based on a finite-difference method of order 2. The two cases under study here are summarized below:

1) the feedforward control with n = 0 (using only u e 1 and u e 2 in ( 6)) and

C 1 = [ 0 0 ] , C 2 = 0, K = [ 0 0 ] . (16)
2) a dynamic control with the following parameters:

à = -800 0 0 -150 , B c1 = 0 2,2 , B c2 = I 2 , C 1 = [ 800 0.015 0.01 -0.1 ] , C 2 = [ 0 -0.0718 ] , K = [ -82.2 10.4 ] . (17)
The dynamic controller is obtained considering two lowpass filters. Denote by s ∈ C the Laplace variable, the two transfer functions for the low-pass filters are u1 wt(0) = 1 1+sωc1 and u1 wt(1) = 1 1+ωc2 with the cut-off frequencies ω c1 = 800 and w c2 = 150. Gain K has been chosen such that the eigenvalues of A + BK are -2.4603 ± 0.1230i. C 2 has been chosen to cancel the dependence on w t (1, •) in the ODE.

With the feedforward controller only, it is possible to estimate the decay-rate of the solution. Indeed, there is no real coupling between the ODE and the PDE and the decayrate of the interconnected system will be the smallest between their respective ones. Here, the PDE has a decay-rate given by equation (10) of 1.2302 and the ODE is 0.2159. The results of Theorem 1 is given in Table II. The maximum decay-rate for the feedforward case is obtained for N 1 and is, as expected, the decay-rate of the ODE.

Figure 2 shows the time response of system [START_REF] Challamel | Rock destruction effect on the stability of a drilling structure[END_REF] in the two cases. The initial state for this computation is X 0 = 0, w(x, 0) = 2-Ω e x and w t (x, 0) = Ωe-qT e k x-u e 1 -Ωe g

(1-x) for x ∈ (0, 1). Of course states X 1 and X 2 are much faster, which results from the direct influence of static feedback gain K but also the speed w t (1), which is more regular and converges faster to 0. Indeed, as shown in Table II, the speed is much faster in the situation with the dynamic control. The hierarchy of Remark 6 is clearly visible and reaches its maximum value (up to three a 3 digits precision) at N = 2. If d > 0, one can notice a slightly higher decay rate but the limit remains the same. One of the drawback of such a system is the angular speed w t (x, •) for x ∈ (0, 1), which increases significantly compared to the first case as it is possible to see on Figure 3.

Remark 7: A backstepping control law could have been considered with a target system of arbitrary large decayrate. Compared to this method, the price to pay for a finite dimension controller is seen by equation [START_REF] Fridman | Introduction to Time-Delay Systems[END_REF]. Indeed, it is not possible to accelerate the system with an arbitrary large decayrate. Other differences are that there is no design methodology using LMI yet and the control is a finite-dimension statefeedback using the knowledge of only Y , w t (0) and w t (1) with strictly proper controllers. [START_REF] Olgac | A practical method for analyzing the stability of neutral type LTI-time delayed systems[END_REF]. αmax IS CALCULATED USING [START_REF] Fridman | Introduction to Time-Delay Systems[END_REF].

V. CONCLUSION

We have studied the stability of a drilling mechanism, which dynamics can be modeled as a coupled ODE/PDE. Approximating this model around a desired equilibrium point leads to an interconnected ODE / damped wave equation. Therefore, the stability of this coupled system is studied using a Lyapunov approach and the stability condition of such a system has been expressed in terms of LMI. Using Bessel inequality, we provided a hierarchy of LMI conditions for this kind of interconnected system with linear feedback controllers. Using only strictly proper hand-designed controllers, a control law has been derived improving subsequently the decayrate of the system. Further studies would investigate how to automatically design such controllers.

1 0 1 0 1 0

 111 χ x (x)(S + xR)χ(x)e 2 αx c dx = 2c χ (1)(S + R)χ(1)e 2 α c -χ (0)Sχ(0) -χ (x)Rχ(x)e 2 αx c dx -4αV α (χ) -Vα (χ) = c χ (1)(S + R)χ(1)e 2 α c -χ (0)Sχ(0)χ (x)Rχ(x)e -2 αx c dx -2αV α (χ).

Fig. 2 . 1 - 8 .

 218 Fig. 2. Simulation on the feedforward and dynamic controlled system.

Fig. 3 .

 3 Fig. 3. Angle velocity wt in the situation with dynamic control.

TABLE I COEFFICIENT

 I VALUES TAKEN FOR THE SIMULATIONS.

TABLE II MAXIMUM

 II DECAY-RATE α USING THEOREM 1 AT AN ORDER N . THE FEEDFORWARD CONTROLLER REFERS TO (16) WHILE THE DYNAMIC CONTROLLER IS WITH

VI. ACKNOWLEDGMENTS

The authors gratefully acknowledge anonymous reviewers' comments. This work is supported by the ANR project SCIDiS contract number 15-CE23-0014.


Aline Sarradon-Eck

Sylvain

The chronicization

teaching and research institutions in France or abroad, or from public or private research centers.

The chronicization of cancer and the struggle to survive: a patient's case study

Aline Sarradon-Eck (SESSTIM -Centre Norbert Elias) and Sylvain Besle (SESSTIM -Institut Gustave Roussy) asarradon@gmail.com besle.sylvain@gmail.com

SHORT ABSTRACT

The new therapeutics available in oncologymostly through clinical trialsallow new hopes for patients. But the risk of death is still present at all steps of the disease, especially for advanced cancer. So the chronicization can sometimes looks like a struggle to survive much more than an integration to the daily life. We explore this point by the presentation of a patient case study.

LONG ABSTRACT

The development of new therapeutics perspectives in oncology, such as targeted therapies and immunotherapies, extends the survival time for certain patients. If this lengthening is sometimes described as the transition of cancer to a chronic disease, the comparison stands only in a limited way. First, some of cancers' patients still experiences a progression of their disease and will die from it before the first five years. Second, the representation of the cancer is directly associated to the risk of death and prevents a stable integration of the disease in the patient's live. Third, an important part of those new drugs are still in development and their outcomes remain uncertain. The benefits of these therapeutics innovations in terms of survival extent and quality of life often depend on patients' inclusion in a clinical trial. If the survival time increases for patients with advanced cancer, their trajectories are more complex and hardly comparable to other chronic disease as diabetes.

To explore this complexity and the role of clinical trials in what some people call a "chronicization of the cancer", we present a case study of an unusual patient. After a diagnostic of lung cancer with a poor prognosis, this man actively fights again the disease during seven years particularly by breaking down barriers to access to clinical trials or to get new therapies. The analysis of his care trajectory reveals the ambiguities of the cancer chronicization. We argue that the chronicization can take a marginal way that it occurs at the border of the healthcare system. This case study is built on direct interviews with the patient, his family, physicians and relatives involved in the treatments. We also use the content of the patient's website and videos of meeting where the course of his disease is related.


Enrique Juan

Martinez-Legaz

Dominikus Noll
email: dominikus.noll@math.univ-toulouse.fr.

Wilfredo Sosa
email: sosa@ucb.br.

J.-E Martinez-Legaz
email: juanenrique.martinez.legaz@uab.es.

MINIMIZATION OF QUADRATIC FUNCTIONS ON CONVEX SETS WITHOUT ASYMPTOTES

Keywords: Quadratic optimization problem, asymptotes, conic asymptotes, Motzkin decomposition, Frank and Wolfe theorem, complementarity problem

The classical Frank and Wolfe theorem states that a quadratic function which is bounded below on a convex polyhedron P attains its infimum on P . We investigate whether more general classes of convex sets F can be identified which have this Frank-and-Wolfe property. We show that the intrinsic characterizations of Frank-and-Wolfe sets hinge on asymptotic properties of these sets.

Introduction

The classical Frank and Wolfe theorem [START_REF] Frank | An algorithm for quadratic programming[END_REF][START_REF] Blum | Direct proof of the existence theorem in quadratic programming[END_REF] states that a quadratic function q which is bounded below on a convex polyhedron P attains its infimum on P . It is known that this result has consequences with regard to the existence of solutions to linear complementarity problems [START_REF] Gowda | Minimising quadratic functionals over closed convex cones[END_REF]. Here we investigate ways in which the Frank and Wolfe theorem can be extended.

A first line is to go beyond polyhedra and ask whether there are more general classes of Frank-and-Wolfe sets, that is, convex sets F with the property that every quadratic function q which is bounded below on F attains its infimum on F . What one would like to obtain is an internal characterization of Frank-and-Wolfe sets via geometric properties, or likewise, verifiable sufficient conditions for the Frank-and-Wolfe property. In response we will characterize Frank-and-Wolfe sets as those convex sets which do not admit conic asymptotes in a sense to be made precise here.

A variant of the same question concerns the larger class of convex sets F with the property that every quadratic function q which is bounded below on F , and which is in addition convex or quasiconvex on F , attains its infimum on F . It turns out that this class has a nice internal characterization. It consists of those convex sets that do not have affine asymptotes in the sense of Klee [START_REF] Klee | Asymptotes and projections of convex sets[END_REF].

A second idea to extend the Frank and Wolfe theorem would be to go beyond quadratics and look for more general classes of functions f attaining their finite infimum on polyhedra P . For instance, do higher degree polynomials f have this property? It turns out that without convexity this line has little hope for success, as shown by the quartic function f (x) = x 2 1 +(1-x 1 x 2 ) 2 , which has infimum 0 on the plane, but does not attain its infimum there. Positive results can at best be expected for convex polynomial functions f . For instance, Rockafellar [START_REF] Rockafellar | Convex Analysis[END_REF]Cor. 27.3.1] shows that a convex polynomial f which is bounded below on a polyhedron P attains its infimum on P . Other variations of this theme are for instance Perold [START_REF] Perold | A generalization of the Frank and Wolfe theorem[END_REF], Hirsch and Hoffman [START_REF] Hirsch | Fixed charge problems[END_REF], or Belousov and Klatte [START_REF] Belousov | A Frank-and-Wolfe theorem for convex polynomial programs[END_REF].

The structure of the paper is as follows. In Section 2 we define Frank-and-Wolfe sets and variants and obtain first basic properties. Section 3 establishes the link between the Frank-and-Wolfe property and the absence of f -asymptotes in the sense of Klee. In Section 2 we consider the Frank-and-Wolfe property within the class of Motzkin decomposable sets, where one expects key information to be provided by the recession cone. Section 5 characterizes Frank-and-Wolfe sets by the absence of q-asymptotes, a geometric notion we define in Section 5. In the final Section 6 we obtain an application to generalized complementarity problems.

Notations

We generally follow Rockafellar's book [START_REF] Rockafellar | Convex Analysis[END_REF]. The closure of a set F is F . The Euclidean norm in R n is ∥ • ∥, and the Euclidean distance is dist(x, y) = ∥x -y∥. For subsets M, N of R n we write dist(M, N ) = inf{∥x -y∥ : x ∈ M, y ∈ N }. A direction d with x + td ∈ F for every x ∈ F and every t ≥ 0 is called a direction of recession of F , and the cone of all directions of recession is denoted as 0 + F .

A function q(x)

= 1 2 x T Ax + b T x + c with A = A T ∈ R n×n , b ∈ R n , c ∈ R is called quadratic. The quadratic q : R n → R is quasiconvex on a convex set F ⊂ R n if the sublevel sets of q |F : F → R are convex. Similarly, q is convex on the set F if q |F is convex.

Frank and Wolfe sets

We call a convex set F in R n Frank-and-Wolfe if every quadratic function q : R n → R which is bounded below on F attains its infimum on F . For short we say that F is a FW -set. In the same vein we call the convex set F quasi-Frank-and-Wolfe if the property holds for every quadratic q which is in addition quasiconvex on F . For short, such sets are called qFW -sets.

Formally we may also consider convex sets F where the property holds for every quadratic q which is convex on F . We temporarily call those cFW -sets. Ultimately this class will turn out equivalent to quasi-Frank-and-Wolfe sets, i.e., cFW = qFW.

Clearly every bounded closed convex set is Frank-and-Wolfe, so the disquisition is only useful in studying unbounded convex sets. Trivially FW -sets are qFW, and qFW -sets are cFW. The classical theorem of Frank and Wolfe [START_REF] Frank | An algorithm for quadratic programming[END_REF] says that every convex polyhedron P is a FW -set. Our first observation is the following. Lemma 1. Every cFW-set is closed, hence so are qFW-and FW-sets. 2 is quadratic convex and its infimum on F is 0. Since by hypothesis this infimum is attained, we must have x ∈ F . □

Proof: Consider x ∈ F , then q(•) = ∥ • -x∥
Another useful property of Frank-and-Wolfe sets is the following.

Proposition 1. Affine images of cFW-sets are cFW-sets. Similarly, affine images of qFW-sets are qFW, and affine images of FW-sets are FW. In particular, affine images of cFW-sets, qFW-sets and FW-sets are closed.

Proof: Closedness of the affine image of a cFW -set F under an affine image follows from the first part of the statement in tandem with Lemma 1. To prove the first part let F be a cFW -set and T : R n → R m an affine operator. Let F = T (F ). We show that F is cFW. Let q : R m → R be quadratic and convex on F , and suppose it is bounded below on F with infimum γ. Then q = q • T is quadratic and convex on F , and bounded below on F with the same infimum γ. By the hypothesis on F the infimum γ of q is attained at x 0 ∈ F , and then clearly q attains its infimum γ on F at T x 0 .

Similarly, if q is quasiconvex on F , then q is quasiconvex on F . Therefore the other two statements follow as well. □

Yet another elementary property of FW -sets is the following Proposition 2. Suppose F is a FW-set, and let F ′ be a closed convex set containing F such that F ′ \F is bounded. Then F ′ is FW. The analogous statement holds for qFW-sets.

Proof: Suppose q is a quadratic function with finite infimum γ ′ on F ′ . Then q has also a finite infimum γ on F , where obviously γ ≥ γ ′ . There are two cases. If γ = γ ′ , then we choose x ∈ F where γ is attained, and then γ ′ is also attained at x. On the other hand, if

γ ′ < γ, then inf x∈F ′ q(x) = inf x ′ ∈F ′ \F q(x ′ ). Since F ′ \ F is bounded, there exists x ′ ∈ F ′ \ F ⊂ F ′
where the infimum γ ′ is attained. □

f -asymptotes

Following Klee [START_REF] Klee | Asymptotes and projections of convex sets[END_REF], an affine manifold M in R n is called an f -asymptote of the closed convex set F if F ∩ M = ∅ and dist(F, M ) = 0. The link between f -asymptotes and the Frank-and-Wolfe property is given by the following Theorem 1. Let F be a convex set in R n . Then the following statements are equivalent:

(i) Every quadratic function q which is quasiconvex on F and bounded below on F attains its infimum on F . That is, F is qFW. (ii) Every quadratic function q which is convex on F and bounded below on F attains its infimum on F . That is, F is cFW. (iii) F is closed and has no f -asymptotes.

Proof:

The implication (i) =⇒ (ii) is clear. Consider (ii) =⇒ (iii). We have to show that F is closed and has no f -asymptotes. Closedness follows readily from Lemma 1. Now let M be an affine manifold with dist(F, M ) = 0. We have to show that M is not an f -asymptote of F . Suppose M = y + U for its direction space U and some y ∈ U ⊥ . Let P be the orthogonal projection on U ⊥ , then P (M ) = {y} and M = P -1 (y). Since dist(F, M ) = 0, there exist sequences

x k ∈ F , z k ∈ M , such that dist(x k , z k ) → 0. Then dist(P x k , P z k ) ≤ dist(x k , z k ) → 0, but P z k =
y for every k, hence dist(P x k , y) → 0, so the sequence P x k converges to y. Now since F has property (ii), its affine image P (F ) is closed by Proposition 1, so y ∈ P (F ). Pick x ∈ F with y = P x, then x ∈ F ∩ P -1 (y) = F ∩ M , so that F ∩ M ̸ = ∅. This shows that F does not have f -asymptotes.

It remains to prove the implication (iii) =⇒ (i). We will prove this by induction on the dimension n of F . For dimension n = 1 the implication is clearly true, because any quadratic function q : R → R which is bounded below on a closed convex set F ⊂ R attains its infimum on F . Suppose therefore that the result is true for dimension n -1, and consider a quadratic function q : R n → R which is quasiconvex on F and bounded below on F . Assume without loss that the dimension of F is n, i.e., F has nonempty interior, as otherwise the result follows directly from the induction hypothesis. Let γ = inf{q(x) : x ∈ F } > -∞, and fix α > γ. If the sublevel set S α := {x ∈ F : q(x) ≤ α} is bounded, then by the Weierstrass extreme value theorem the infimum of q over S α is attained. But this infimum is also the infimum of q over F , so in that case we are done. Assume therefore that S α is unbounded. Since q is quasiconvex on F , the set S α is closed convex, which means S α has a direction of recession d, that is, a direction with x+td ∈ S α for every t ≥ 0 and every x ∈ S α (see e.g. [START_REF] Rockafellar | Convex Analysis[END_REF]Theorem 8.4]). Fix x ∈ S α . Expanding q at x + td ∈ S α gives

γ ≤ q(x + td) = 1 2 x T Ax + b T x + c + td T (Ax + b) + 1 2 t 2 d T Ad ≤ α
for every t ≥ 0, and this implies d T Ad = 0. Substituting this back gives

γ ≤ q(x + td) = 1 2 x T Ax + b T x + c + td T (Ax + b) ≤ α, for every t ≥ 0. That implies d T (Ax + b) = 0. But the argument is valid for every x ∈ S α .
By assumption F has dimension n, so S α has nonempty interior, meaning x + ϵB ⊂ S α for some ϵ > 0, with B the unit ball. That shows Ad = 0. Going back with this into d T (Ax + b) = 0 shows d T b = 0, too. Altogether we have shown [START_REF] Banks | Non-linear parametric optimization[END_REF] q(x + td) = q(x) for every x ∈ S α and every t ≥ 0.

Since q is a quadratic function and S α has nonempty interior, this implies q(x + td) = q(x) for every x ∈ R n and every t ∈ R. Now let P be the orthogonal projection onto the hyperplane H = d ⊥ . Then q := q |H is quadratic on the (n -1)-dimensional space H and takes the same values as q due to (1). In particular, q = q |H is bounded below on the qFW -set F = P (F ). Since q is quasiconvex on F , q is quasiconvex on F . Therefore q attains its infimum on F by the induction hypothesis, since dim( F ) = n -1, and then q, having the same values, also attains its infimum on F . □ Remark 1. From the implication (iii) =⇒ (i) it is clear that for a quadratic function q bounded below on F to attain its infimum on F , it is sufficient to have just one of its sublevel sets S α with α > γ = inf x∈F q (x) convex, a condition which is weaker than quasiconvexity on F . An even weaker condition suffices, namely, the existence of a not necessarily convex sublevel set S α and a direction d ∈ R n with the following property: For every x ∈ S α there exists t x ∈ R such that x + td ∈ S α for every t ≥ t x .

Remark 2. Yet another equivalent condition which we could add to the above list is (iv) P (F ) is closed for every orthogonal projection P . Indeed (ii) =⇒ (iv) is Proposition 1, and (iv) =⇒ (ii) is implicit in the proof of (ii) =⇒ (iii) above. For the equivalence of (iii) and (iv) see also [START_REF] Klee | Asymptotes and projections of convex sets[END_REF].

Corollary 1. Frank-and-Wolfe sets have no f -asymptotes. □

We end this section by indicating that the converse of Corollary 1 is not true. Put differently, the absence of f -asymptotes does not characterize Frank-and-Wolfe sets. Or put again differently, there exist quasi-Frank-and-Wolfe sets, which are not Frank-and-Wolfe.

Example 3.1. We construct a closed convex set F without f -asymptotes, which is not Frank-and-Wolfe. We use Example 2 of [START_REF] Luo | On extensions of the Frank-Wolfe theorem[END_REF], which we reproduce here for convenience. Consider the optimization program

minimize q(x) = x 2 1 -2x 1 x 2 + x 3 x 4 subject to c 1 (x) = x 2 1 -x 3 ≤ 0 c 2 (x) = x 2 2 -x 4 ≤ 0 x ∈ R 4
then as Lou and Zhang [START_REF] Luo | On extensions of the Frank-Wolfe theorem[END_REF] show the constraint set F = {x ∈ R 4 : c 1 (x) ≤ 0, c 2 (x) ≤ 0} is closed convex, and the quadratic function q has infimum γ = -1 on F , but this infimum is not attained.

Let us show that F has no f -asymptotes. Note that F = F 1 ×F 2 , where

F 1 = {(x 1 , x 3 ) ∈ R 2 : x 2 1 -x 3 ≤ 0}, F 2 = {(x 2 , x 4 ) ∈ R 2 : x 2 2 -x 4 ≤ 0}. Observe that F 1 ∼ = F 2 ,
and that F 1 does not have asymptotes, being a parabola. Therefore, F does not have f -asymptotes either. This can be seen from the following

Proposition 3. Suppose F 1 , F 2 do not have f -asymptotes. Then neither does F 1 × F 2 have f -asymptotes. Proof: We write F 1 × F 2 = (F 1 × R n ) ∩ (R n × F 2 ). Suppose M is an f -asymptote of F 1 × F 2 , then by Klee [9, Theorem 4] the flat M contains either an f -asymptote N 1 of F 1 × R n , or it contains an f -asymptote N 2 of R n × F 2 .
Assume without loss that M contains N 1 . Let P be the projection on the first coordinate, then P (N 1 ) is an affine manifold, and it is easy to see that it is an f -asymptote of

F 1 . □ Example 3.2. Let F be the epigraph of f (x) = x 2 +exp(-x 2 ) in R 2 . Then q(x, y) = y-x 2
is bounded below on F , but does not attain its infimum, so F is not FW. However, F has no f -asymptotes, so it is qFW. □

Motzkin decomposable sets

The proof of the classical Frank-and-Wolfe theorem [START_REF] Frank | An algorithm for quadratic programming[END_REF] exploits the fact that a polyhedron P can be decomposed as P = C + D, where C is a polytope, and D a convex polyhedral cone. This rises the question whether the Frank and Wolfe theorem may be extended to other classes of convex sets F with this type of decomposition. We recall the following

Definition 1. A nonempty closed convex set F in R n is called Motzkin decomposable if there exists a compact convex set C and a closed convex cone D such that F = C + D. We call (C, D) a Motzkin decomposition of F . □
We start with a disclaimer. Not all Motzkin decomposable sets are Frank-and-Wolfe.

Example 4.1. We put D = {x ∈ R 3 :

x 1 ≥ 0, x 2 ≥ 0, x 1 x 2 -x 2 3 ≥ 0}, then D is a closed convex cone,
hence is trivially Motzkin decomposable. But D is not Frank-and-Wolfe. In fact, it is not even quasi-Frank-and-Wolfe, as we now show. Indeed, define q : R 3 → R by q(x) = x 2 1 + (x 3 -1) 2 , then q is quadratic convex and bounded below by 0. In fact, γ = 0 is the infimum of q on D, because q

( 1 k , (k+1) 2 k , 1 + 1 k ) = 2 k 2 → 0, but 0 is not attained on D. In view of Theorem 1, the cone D must have f -asymptotes. □ Example 4.2.
In the same vein consider the quadratic function q : R 3 -→ R defined as q (x, y, z) := (x -1) 2 -y+z and the ice-cream cone

F := { (x, y, z) ∈ R 3 : z ≥ √ x 2 + y 2 } . Clearly q ≥ 0 on F since z ≥ y for every (x, y, z) ∈ F. But the infimum of q on F is 0, since ( 1, k, √ 1 + k 2 ) ∈ F and q ( 1, k, √ 1 + k 2 ) = √ 1 + k 2 -k -→ 0,
and this infimum is not attained, as for (x, y, z) ∈ F, one has either x ̸ = 1 or z ≥ √ 1 + y 2 > y, which both imply q (x, y, z) > 0.

The orthogonal projection of F onto the hyperplane

H := { (x, y, z) ∈ R 3 : y + z = 0 }
is not closed. To see this, notice that the orthogonal projection P on H is given by

P (x, y, z) = ( x, y-z 2 , z-y 2 ) . Consider again ( 1, k, √ 1 + k 2 ) ∈ F , then P ( 1, k, √ 1 + k 2 ) = ( 1, k- √ 1+k 2 2 , √ 1+k 2 -k 2
) ∈ P (F ), but its limit (1, 0, 0) does not belong to P (F ), because

P -1 (1, 0, 0) = {(x, y, z) ∈ R 3 : x = 1, y = z} does not intersect F. □
These examples raise the question whether a Motzkin decomposable set F is Frankand-Wolfe as soon as its recession cone 0 + F is Frank-and-Wolfe. A similar question can be asked for quasi-Frank-and-Wolfe sets. For the latter class things have been simplified due to Theorem 1, and we have the following answer. Proposition 4. Let F be a Motzkin decomposable set. Then F is quasi-Frank-and-Wolfe if and only if its recession cone 0 + F is quasi-Frank-and-Wolfe.

Proof: 1) Suppose 0 + F is qFW. Assume contrary to what is claimed that F has an f -asymptote M . Write M = y + U for the direction space U of M and y ∈ U ⊥ . Let P be the orthogonal projection onto U ⊥ . Then M = P -1 (y). Observe that P (F ) is not closed. Indeed, there exist

x k ∈ F , y k ∈ M , with dist(x k , y k ) → 0. Therefore P x k → y. But y ̸ ∈ P (F ), because if y = P x for some x ∈ F , then x ∈ F ∩ P -1 (y) = F ∩ M , which is impossible due to F ∩ M = ∅.
Since F is Motzkin decomposable, there exist a compact convex C with F = C + 0 + F . Then P (F ) = P (C) + P (0 + F ), while P (F ) = P (C) + P (0 + F ). Since P (F ) ̸ = P (F ), we deduce that P (0 + F ) cannot be closed, and that means 0 + F has an f -asymptote parallel to U , contradicting the fact that 0 + F is a qFW -set.

2) Conversely, suppose F is qFW, but that 0 + F is not qFW. Then 0 + F must have an f -asymptote L by Theorem 1. Suppose L = y + W with W the direction space of L and y ∈ W ⊥ . Let P be the orthogonal projection on W ⊥ , then again P (0 + F ) is not closed. Now by [START_REF] Iusem | Motzkin predecomposable sets[END_REF]Proposition 5] F has an f -asymptote parallel to W , and by Theorem 1 this contradicts the fact that F is qFW. □ Remark 3. The statement of Proposition 4 is no longer correct if one drops the hypothesis that F is Motzkin decomposable. We take F = {(x, y) ∈ R 2 : x > 0, y > 0, xy ≥ 1}, then F , being a hyperbola, has f -asymptotes, but 0 + F is the positive orthant, which does not have f -asymptotes.

An immediate consequence of Proposition 4 is the following Corollary 2. Let F be a Motzkin decomposable qFW-set. Let T : R n → R m be a linear operator. Then T (0

+ F ) is closed in R m . □
Proposition 4 is a strong incentive to look for similar criteria for the Frank-and-Wolfe property in terms of 0 + F . A first partial answer is the following generalization of the classical Frank and Wolfe theorem.

Theorem 2. Let F be a Motzkin decomposable convex set, and suppose its recession cone 0 + F is polyhedral. Then F is Frank-and-Wolfe.

Proof: Write F = C + 0 + F for C compact convex. Now consider a quadratic function q(x) = 1 2 x T Ax + b T x bounded below by γ on F . Hence (2) inf x∈F q(x) = inf y∈C inf z∈0 + F q(y + z) = inf y∈C ( q(y) + inf z∈0 + F y T Az + q(z) ) ≥ γ.
Now observe that for fixed y ∈ C the function q y : z → y T Az + q(z) is bounded below on 0 + F by η = γ -max y ′ ∈C q(y ′ ). Indeed, for z ∈ 0 + F we have

y T Az + q(z) ≥ ( q(y) + inf z ′ ∈0 + F y T Az ′ + q(z ′ ) ) -q(y) ≥ inf y∈C ( q(y) + inf z ′ ∈0 + F y T Az ′ + q(z ′ ) ) -max y ′ ∈C q(y ′ ) ≥ γ -max y ′ ∈C q(y ′ ) = η.
Since q y is a quadratic function bounded below on the polyhedral cone 0 + F , the inner infimum is attained at some z = z(y). This is in fact the classical Frank and Wolfe theorem on a polyhedral cone. In consequence the function f :

R n → R ∪ {-∞} defined as f (y) = inf z∈0 + F y T Az + q(z),
satisfies f (y) = y T Az(y) + q(z(y)) > -∞ for y ∈ C, so the compact set C is contained in its domain. But now a stronger result holds, which one could call a parametric Frank and Wolfe theorem, and which we shall prove in Lemma 2 below. We show that f is continuous relative to its domain. Once this is proved, the infimum (2) can then be written as

inf x∈F q(x) = inf y∈C q(y) + f (y),
and this is now attained by the Weierstrass extreme value theorem due to the continuity of q + f on the compact C. Continuity of f on C is now a consequence of the following Lemma 2. Let D be a polyhedral convex cone and define

f (c) = inf x∈D c T x + 1 2 x T Gx,
where G = G T . Then dom(f ) is a polyhedral convex cone, and hence f is continuous relative to dom(f ).

Proof: If x T Gx < 0 for some x ∈ D, then dom(f ) = ∅, so we may assume for the remainder of the proof that x T Gx ≥ 0 for every x ∈ D. Clearly then, Now observe that if u ≥ 0 satisfies u T Z T GZu = 0, then it is a minimizer of the quadratic function u T Z T GZu on the cone u ≥ 0, hence Z T GZu ≥ 0 by the Kuhn-Tucker conditions. Therefore we can write the set P = {u ∈ R p : u ≥ 0, u T Z T GZu = 0} as

dom(f ) = { c : c T x ≥ 0 for every x ∈ D such that x T Gx = 0 } .

Now by the

P = ∪ I⊂{1,...,p} P I ,
where the P I are the polyhedral convex cones

P I = {u ≥ 0 : Z T GZu ≥ 0, u i = 0 for all i ∈ I, (Z T GZu) j = 0 for all j ̸ ∈ I}.
For every I ⊂ {1, . . . , p} choose m I generators u I1 , . . . , u Im I of P I . Then,

dom(f ) = { c : c T Zu ≥ 0 for every u ∈ P } (3) = { c : c T Zu ≥ 0 for every u ∈ ∪ I⊂{1,...,p} P I } = ∩ I⊂{1,...,p} { c : c T Zu ≥ 0 for every u ∈ P I } = ∩ I⊂{1,...,p} { c : c T Zu Ij ≥ 0 for all j = 1, . . . , m I } .
Since a finite intersection of polyhedral cones is polyhedral, this proves that dom(f ) is a polyhedral convex cone. To conclude, continuity of f relative to its domain now follows from [12, Thm. 10.2], since f is clearly concave and upper semicontinuous. □ Remark 4. The proof includes the case when x T Gx > 0 for every x ∈ D \ {0}. In that case one has P I = {0} for every I ⊂ {1, ..., p}, and therefore {c : c T Zu ≥ 0 for every u ∈ P I } = R n , so that the equality (3) still holds and reduces to dom(f ) = R n .

Remark 5. We refer to Banks et al. [1, Thm. 5.5.1 (4)] or Best and Ding [START_REF] Best | On the continuity of the minimum in parametric quadratic programs[END_REF] for a related result in the case where G ⪰ 0. For the indefinite case see also Tam [START_REF] Tam | Continuity of the optimal value function in indefinite quadratic programming[END_REF]. Remark 6. The example in Remark 3 shows that Theorem 2 is no longer true if F is not Motzkin decomposable.

A second partial answer to the question whether the Frank-and-Wolfe property of 0 + F implies that of F is given in the following Proposition 5. Let F have a Motzkin decomposition of the form F = P + 0 + F with P a polytope. If 0 + F is Frank-and-Wolfe, then so is F .

Proof: Consider a quadratic q which is bounded below on F . Splitting the infimum according to (2), we see as in the proof of Theorem 2 that every q y : z → y T Az + q(z) is quadratic and bounded below on 0 + F , and since 0 + F is Frank-and-Wolfe by hypothesis, the inner infimum in ( 2) is attained at z(y) ∈ 0 + F . As in the proof of Theorem 2 define f (y) = inf z∈0 + F q y (z) = q y (z(y)), then f is the infimum of the family of affine functions y → y T Az +q(z) on the polytope P , hence is lower semi-continuous on P by [12, Theorem 10.2]. But then y → q(y) + f (y) is lower semi-continuous on P , and by compactness of P the outer infimum y ∈ P in ( 2) is therefore attained. □

q-Asymptotes

The discussion in Section 3 shows that the absence of f -asymptotes is only a necessary condition for the Frank-and-Wolfe property. In this section we shall develop a related concept of asymptotes, where we replace affine (flat) surfaces by quadratic surfaces. We start with the following Definition 2. A quadric in R n , also called a quadratic surface or a conic, is a set of the form Klee [9] and in the sense of Section 3. Now we can give the central definition of this section.

Q = {x ∈ R n : 1 2 x T Ax + b T x + c = 0} with A = A T ̸ = 0. □ Definition 3. A nonempty closed set A is said to be asymptotic to the nonempty closed convex set F if A ∩ F = ∅ and dist(F, A) = 0. □ If A is an affine subspace of R n , then A is asymptotic to F iff it is an f -asymptote in the sense of
Definition 4. The quadric Q = {x ∈ R n : q(x) = 1 2 x T Ax + b T x + c = 0} is a q-asymptote of the closed convex subset F of R n if F ∩Q = ∅ and dist(Q×{0}, {(x, q(x)) : x ∈ F }) = 0. □
The condition means F ∩Q = ∅, and that there exist x k ∈ F and y k with q(y k ) = 0 such that x k -y k → 0 and q(x k ) → 0. This shows that the notion of a q-asymptote is invariant under an affine change of coordinates in R n , hence is a concept of affine geometry. The condition could also be expressed as follows: The quadric Q × {0} in R n+1 is asymptotic to graph F (q) := graph(q) ∩ (F × R) in the sense of Definition 3.

Remark 7.

If Q is a q-asymptote of F , then Q is clearly asymptotic to F , but the converse is not true in general. To see this consider the following example. Let F = {(x, y) ∈ R 2 : x ≥ 0, y ≥ 0} be the positive orthant, and let q(x, y)

= xy + 1, then Q = {(x, y) ∈ R 2 : q(x, y) = 0} = {(x, y) : xy = -1} is a hyperbola with F ∩ Q = ∅. We have dist(F, Q) = 0, because (-1 n , n) ∈ Q and (0, n) ∈ F , so Q is asymptotic to F in the sense of definition 3. But Q is not a q-asymptote of F , because the sets Q × {0} = {(x, -1 x , 0) : x ̸ = 0} and graph(q) ∩ (F × R) = {(x, y, xy + 1) : x ≥ 0, y ≥ 0} cannot be close. □ Remark 8. Consider the quadric Q : q(x) = x 2 1 + • • • + x 2 n-1 = 0, then Q = {x ∈ R n : q(x) = 0}
is the x n -axis. Suppose the x n -axis is an f -asymptote of a closed convex set F . This is equivalent to Q being asymptotic to F . However, we argue that Q is then even a q-asymptote of F in the sense of definition 4. Namely, we have

Q × {0} = {(0 n-1 , ξ, 0) : ξ ∈ R} ⊂ R n+1 and graph(q) ∩ (F × R) = {(x, q(x)) : x ∈ F } = {( x, n-1 ∑ i=1 x 2 i ) : x ∈ F } . Now given ϵ > 0 choose x ∈ F and t ∈ R such that ∥x -(0 n-1 , t)∥ 2 < ϵ 2 , which is possible because dist(Q, F ) = 0. (Naturally, we could take t = x n ). Then q(x) = x 2 1 + • • • + x 2 n-1 < ϵ 2 and (x n -t) 2 < ϵ 2 . Therefore ∥(0, t, 0) -(x, q(x))∥ 2 ≤ ∥x -(0 n-1 , t)∥ 2 + q(x) 2 ≤ ϵ 2 + q(x) 2 < ϵ 2 + ϵ 4 .
This shows the claim. We can generalize this to a proof that any flat M which is an f -asymptote is also a q-asymptote when interpreted as a quadric: Proposition 6. Let F be closed convex in R n , and let Q = {x ∈ R n : q(x) = 0} be a quadric. Suppose Q is flat, that is, degenerates to an affine subspace. Then Q is a qasymptote of F in the sense of Definition 4 if and only if it is an f-asymptote of F in the sense of [START_REF] Klee | Asymptotes and projections of convex sets[END_REF]. Moreover, for any f -asymptote M of F there exists a quadric representation of M as M = {x ∈ R n : q(x) = 0}, and then M is also a q-asymptote of F .

Proof:

The fact that Q is an affine subspace of dimension k ≤ n -1 means that one can find affine coordinates in R n such that Q has the form

Q = {x ∈ R n : x 2 k+1 + • • • + x 2 n = 0} = R k × {0 n-k }.
Since being a q-asymptote implies being asymptotic, and since for an affine subspace this coincides with being an f -asymptote, we have but to prove the opposite implication.

Assume therefore that Q is an f -asymptote of

F , i.e., F ∩ Q = ∅ and dist(F, Q) = 0. We have to show that Q × {0} = {(x, 0) : x ∈ Q} is asymptotic to graph(q) ∩ (F × R) = {(y, q(y)) : y ∈ F }. Clearly the two sets are disjoint. Splitting x = (x ′ , x ′′ ), y = (y ′ , y ′′ ) ∈ R k × R n-k , we have q(y) = y ′′2 k+1 + • • • + y ′′2 n = ∥y ′′ ∥ 2 . Now pick x r ∈ Q, y r ∈ F with dist(x r
, y r ) → 0 as r → ∞. Then x r = (x ′r , x ′′r ) = (x ′r , 0) and y r = (y ′r , y ′′r ), hence ∥y ′′r ∥ 2 ≤ ∥x r -y r ∥ 2 → 0, and this implies q(y r ) = ∥y ′′r ∥ 2 → 0. Hence Q is a q-asymptote of F , because it now follows that ∥(x r , 0) -(y r , q(y r ))∥ → 0. □

This result shows that the notion of a q-asymptote is a natural extension of Klee's concept of f -asymptotes. We move from flat asymptotes to quadratic asymptotes. We are now ready to state the principal result of this section.

Theorem 3. A convex set F is Frank-and-Wolfe if and only if it is closed and has no q-asymptotes.

Proof: 1) Assume that there exists a quadratic function q : R n → R which is bounded below on F , but does not attain its infimum on F . We have to show that F has a qasymptote. Assume without loss that the infimum of q on F is 0. Since there exists x ∈ F with q(x) > 0 and y ̸ ∈ F with q(y) = 0, the set

Q = {x ∈ R n : q(x) = 0} is a quadric in R n .
Note that if F is not qFW, then by Theorem 1 the set F has an f -asymptote, and then has also a q-asymptote by Proposition 6. So we can assume that F is qFW, and by Proposition 1 we therefore know that orthogonal projections of F are closed.

We clearly have F ∩Q = ∅, so we have to show that dist({(x, q(x)) : x ∈ F }, Q×{0}) = 0. Since the statement we have to prove is invariant under an affine change of coordinates in R n , we may assume that the quadric Q is given by one of the following equations:

(4) Q : q(x) = p ∑ i=1 x 2 i - r ∑ i=p+1 x 2 i + γ = 0, (p < r ≤ n)
where γ ∈ {0, 1} if Q is a center quadric with 0 as its center, or

Q : q(x) = p ∑ i=1 x 2 i - r ∑ i=p+1 x 2 i + x r+1 = 0 (p ≤ r < n) if Q is a paraboloid. a) (5)
Let us first discuss the easier case of a paraboloid [START_REF] Frank | An algorithm for quadratic programming[END_REF]. Since q is a quadratic function, it satisfies a Łojasiewicz inequality at infinity. In other words, following [15, Theorem 2.1] there exist constants δ > 0, c > 0 and a Łojasiewicz exponent α > 0 at infinity such that for every x ∈ R n with |q(x)| < δ we have

|q(x)| ≥ c dist ( x, Q ) α ,
where

Q = Q ∪ Q 1 with Q = {x ∈ R n : q(x) = 0} , Q 1 = { x ∈ R n : ∂ ∂x r+1 q(x) = 0 } .
This result uses the fact that q is a monic polynomial of degree m = 1 in the variable x r+1 . Since ∂/∂x r+1 q(x) = 1, the set Q 1 is empty, hence we obtain

|q(x)| ≥ c dist (x, Q) α for |q(x)| < δ. Now choose a sequence x k ∈ F with q(x k ) → 0. Then from some k onward, q(x k ) ≥ c dist(x k , Q) α → 0, which proves dist({(x, q(x)) : x ∈ F }, Q × {0}) = 0.
This settles the case where Q is a paraboloid. b) Let us next consider the more complicated case where Q is a center quadric. Choose a sequence x k ∈ F such that q(x k ) → 0. We want to show dist(x k , Q) → 0, at least for a subsequence. Assume on the contrary that dist(x k , Q) > d > 0 for every k. Write x k = (ξ k 1 , . . . , ξ k n ), and note that ∥x k ∥ → ∞. We now have two principal cases. Case I is when (ξ k 1 , . . . , ξ k r ) → 0, while the part (ξ k r+1 , . . . , ξ k n ) on which q given by (4) does not depend satisfies ∥(ξ k r+1 , . . . , ξ r n )∥ → ∞. In this case we necessarily have r < n. Case II is when there exists i ∈ {1, . . . , r} such that ξ k i → ξ i ̸ = 0 for a subsequence k ∈ K, including the possibilities ξ i = ±∞.

We start by discussing case II. Suppose ξ i is finite and the signature of i is negative (i.e. i ∈ {p + 1, . . . , r}). Then there must also exist another index with positive signature j ∈ {1, . . . , p} say, for which ξ k j → ξ j ̸ = 0. (This is because in (4) the -ξ 2 i and ξ 2 j have to sum to γ ≥ 0. Therefore if there is a non-vanishing contribution from an index i with negative signature, there is necessarily also one from an index j with positive signature.) We may without loss assume that this contribution with positive signatures comes from missing is indeed an argument which allows to infer from f (x k ) → 0 for x k ∈ F that also dist(x k , Q) → 0, and for higher order polynomials such an argument may not exist.

Example 5.1. To illustrate Theorem 3 we consider the set F = {(x, y) ∈ R 2 : y ≥ x 2 } and claim that it is Frank-and-Wolfe. We check this by showing that F has no q-asymptotes. Suppose Q = {q = 0} is a q-asymptote of F . If Q is a hyperbola or consists of two lines, then F itself has lines as asymptotes, which is impossible, because F is a parabola. It is equally impossible that Q is an ellipse, so Q must be a parabola, too. But it is intuitively clear that no other parabola can be a q-asymptote of y = x 2 .

To prove this rigorously, suppose q(x, y) = ax 2 + bxy + cy 2 + dx + ey + f . By the definition of a q-asymptote there exist

(x k , y k ) ∈ Q and (x k , x 2 k ) ∈ F , such that ∥(x k , x 2 k )- (x k , y k )∥ → 0 and q(x k , x 2 k ) → 0.
Picking a subsequence, we may without loss assume

x k → +∞. Then q(x k , x 2 k ) = ax 2 k + bx 3 k + cx 4 k + dx k + ex 2 k + f → 0
implies successively c = 0, then b = 0, then a = -e, then d = 0 and f = 0, and finally a ̸ = 0. Hence Q = {(x, y) : a(x 2 -y) = 0}, but this is the boundary curve of F , which contradicts

F ∩ Q = ∅. □
Remark 7 suggests an equivalent geometric characterization of q-asymptotes, which we now develop. Let Q = {x ∈ R n : q(x) = 0} be a quadric and consider the associated one-parameter family Q = {Q α } α∈R of quadrics Q α = {x ∈ R n : q(x) -α = 0}. Note that Q is a geometric object, as an affine change of coordinates leads to the same family of sets. Informally, we intend to show that Q ∈ Q is a q-asymptote of the closed convex set F if and only if Q, F are asymptotic, and no other element Q ′ of the bundle Q can be squeezed in between F and Q.

Definition 5. Let F, Q be closed sets with F ∩ Q = ∅ and dist(F, Q) = 0. We say that the closed set Q ′ is squeezed in between F and Q if F ∩ Q ′ = ∅ = Q ∩ Q ′ and if every segment [x, y] with x ∈ F and y ∈ Q contains a point z ∈ Q ′ , i.e., [x, y] ∩ Q ′ ̸ = ∅. □
We now have the following Proposition 7. Let F be closed convex and let Q = {x ∈ R n : q(x) = 0} be a quadric. Then Q is a q-asymptote of F if and only if Q is asymptotic to F and no other member Q ′ of the bundle Q can be squeezed in between F and Q. In other words, Q is a tight quadric asymptote to F .

Proof: 1) Suppose Q is a q-asymptote of F . Then there exist

x k ∈ F , y k ∈ Q such that x k -y k → 0 and q(x k ) → 0. Clearly Q is asymptotic to F . Since F ∩ Q = ∅ and F is connected, we either have F ⊂ {x : q(x) > 0} or F ⊂ {x : q(x) < 0}. Assume without loss that F ⊂ {x : q(x) > 0}. Suppose Q ′ = {x : q(x) = α} can be squeezed in between Q and F . Since Q ∩ Q ′ = ∅, we have α ̸ = 0.
There are two cases to be discussed. Suppose first that α < 0. Then we find a point z k in the open segment (x k , y k ) such that q(z k ) = α < 0. But q(x k ) > 0, hence by the mean value theorem there exists another point v k in the open segment (x k , z k ) with q(v k ) = 0. Now we repeat the argument on [x k , v k ], which must also contain a point with value q = α. That leads to a contradiction, because we thereby find a third root of q -α on the segment [x k , y k ], which is impossible as q is quadratic. In consequence the squeezing value must be α > 0.

Suppose therefore that the quadric Q ′ which may be squeezed in between F and Q has α > 0. Then we have the following situation on the segment [x k , y k ]. There exists z k ∈ (x k , y k ) with q(z k ) = α > 0, while q(y k ) = 0 and q(x k ) → 0, 0 < q(x k ) ≪ α. Let L k be the line generated by [x k , y k ]. Since q is a quadratic function on L k , there exists a point v k ∈ L k preceding x k where q(v k ) = 0. Here preceding means that

x k ∈ [v k , y k ]. Since F ⊂ {q > 0}, we have v k ̸ ∈ F . In particular, F ∩ L k is contained in the segment [v k , x k ]. But v k ∈ Q, x k ∈ F , hence the segment [v k , x k ]
must also contain an element w k of Q ′ , i.e., with q(w k ) = α, and that is impossible because q is quadratic. Namely, the arrangement on the line

L k is now v k < w k < x k < z k < y k with q(y k ) = 0, q(z k ) = α > 0, q(x k ) ≪ α, q(w k ) = α, q(v k ) = 0. But q |L k
is concave, so this is impossible. This proves that Q ′ ∈ Q could not possibly be squeezed in between F and Q.

2) Conversely, suppose Q is asymptotic to F and is tight in the sense that no other member Q ′ of the bundle Q can be squeezed in between F and Q. Since F ∩ Q = ∅, we may assume F ⊂ {x : q(x) > 0}. Let γ := inf x∈F q(x). We claim that γ = 0. For suppose we had γ > 0 then on choosing 0 < α < γ we find that Q ′ = {x : q(x)-α = 0} is squeezed in between F and Q, which is impossible. Hence γ = 0. Now pick x k ∈ F with q(x k ) → 0 and y k ∈ Q. Using the argument of part 1) of the proof of Theorem 4, it follows that

y k -x k → 0. Hence (x k , q(x k )) -(y k , 0) → 0. That shows dist(Q × {0}, graph F (q)) = 0, hence Q is a q-asymptote of F . □ Remark 11.
In view of this new characterization of q-asymptotes we have the following description of Frank-and-Wolfe sets. Whenever Q is a quadric asymptote of a Frank and Wolfe set F , then there exists another quadric Q ′ in the bundle Q associated with Q that can be squeezed in between F and Q. We could say that Q ′ is a tighter asymptote than Q. As this argument can be repeated, the FW -set F has no tightest asymptote among the quadrics in Q.

Remark 12. It is instructive to give a direct argument for the fact that an f -asymptote in the sense of Klee is tight in the sense of the previous remark, hence is a q-asymptote.

To see this, suppose M = {x ∈ R n : Ax -b = 0} is an f -asymptote of F and represent M as the quadric M = {x : q(x) = ∥Ax -b∥ 2 = 0} = Q. Consider the associated bundle Q = {Q α } and suppose some Q α with α ̸ = 0 can be squeezed in between Q = M and F . Clearly this means α > 0, as the Q α ′ with α ′ < 0 are empty. But q is convex, hence F ⊂ {x : q(x) = ∥Ax -b∥ 2 > α}. Indeed, suppose there exists x ∈ F with q(x) < α.

Pick y ∈ M , then q(y) = 0. Since {z : q(z) < α} is a convex set containing x, y, it also contains the segment, [x, y] ⊂ {z : q(z) < α}. But since Q α is by assumption squeezed in between F and M , there exists z ∈ [x, y] with q(z) = α, a contradiction. So we have shown F ⊂ {x : q(x) > α}. But now we arrive at a contradiction with the fact that dist(M, F ) = 0, as the latter implies inf x∈F ∥Ax -b∥ = 0.

Generalized linear complementarity problem

Let F be a closed convex cone in R n , let A = A T ∈ R n×n , and b ∈ R n . Then we consider the following generalized linear complementarity problem on F with data (A, b): Proof: Let x 0 be a feasible solution, then (Ax 0 + b) T x ≥ 0 for every x ∈ F . Since F is a cone we have 2x 0 ∈ F , and (2Ax 0 + 2b) T x ≥ 0 for every x ∈ F . Due to feasibility the quadratic function q(x) = (Ax + 2b) T x is now bounded below by 2γ on F , and since F is Frank-and-Wolfe, there exists x * ∈ F such that (7) (Ax + 2b) T x ≥ (Ax * + 2b) T x * for every x ∈ F . For x ∈ F and 0 < t ≤ 1 we have x = x * + t(x -x * ) ∈ F , hence on substituting x in [START_REF] Hirsch | Fixed charge problems[END_REF] and expanding, we get t(Ax * + 2b) T (x -x * ) + t(x -x * ) T Ax * + t 2 (x -x * ) T A(x -x * ) ≥ 0.

Dividing by t and letting t → 0 gives 2(Ax * +b) T (x-x * ) ≥ 0, hence (Ax * +b) T (x-x * ) ≥ 0 for every x ∈ F . Putting x = 0 ∈ F we get (Ax * + b) T x * ≤ 0, while putting x = 2x * ∈ F gives (Ax * + b) T x * ≥ 0, so together we get complementarity Ax * + b ⊥ x * . From that follows (Ax * + b) T (x -x * ) = (Ax * + b) T x ≥ 0 for all x ∈ F , hence x * is a solution of (6). □

For sufficient conditions guaranteeing inf x∈F (Ax + b) T x > -∞ we refer to [START_REF] Gowda | Minimising quadratic functionals over closed convex cones[END_REF] and the references given there. Links with the linear complementarity problem can already be found in the original work [START_REF] Frank | An algorithm for quadratic programming[END_REF].

  Farkas-Minkowski-Weyl theorem (cf.[START_REF] Rockafellar | Convex Analysis[END_REF] Thm. 19.1] or[START_REF] Schrijver | Theory of Linear and Integer Programming[END_REF] Cor. 7.1a]) the polyhedral cone D is the linear image of the positive orthant of a space R p of appropriate dimension, i.e. D = {Zu : u ∈ R p , u ≥ 0}. This implies dom(f ) = {c : c T Zu ≥ 0 for every u ≥ 0 such that u T Z T GZu = 0}.

( 6 )Theorem 4 .

 64 Find x * ∈ F such that (Ax * + b) T x ≥ 0 for every x ∈ F , and Ax * + b ⊥ x * . Every x * ∈ F satisfying (6) is called a solution of the problem. We say that the generalized linear complementarity problem (6) is feasible if γ = inf x∈F (Ax + b) T x > -∞ and if there exists x 0 ∈ F such that (Ax 0 + b) T x ≥ 0 for every x ∈ F . Suppose problem (6) is feasible. If F is a Frank-and-Wolfe cone, then (6) has a solution x * .

Acknowledgments

J.E. Martínez-Legaz was partially supported by the MINECO of Spain, Grant MTM2014-59179-C2-2-P, the Severo Ochoa Programme for Centres of Excellence in R&D [SEV-2015-0563], and under the Australian Research Councils Discovery Projects funding scheme (project number DP140103213). He is affiliated with MOVE (Markets, Organizations and Votes in Economics). Wilfredo Sosa was partially supported by CNPq Grants 302074/2012-0 and 471168/2013-0. Part of this research was carried out during Wilfredo's visit to UAB (partially supported by the above mentioned MINECO grant) and during Juan Enrique's visit to UCB (as 'Professor visitante especial' in the program 'Ciência sem fronteiras'). D. Noll was supported by Fondation Mathématiques Jacques Hadamard (FMJH) under Programme Gaspard Monge pour l'Optimisation et la Recherche Opérationnelle (PGMO).

j = 1. A similar argument applies when ξ i = ±∞. We now have two subcases. Case II.1 is when ξ k 1 → ξ 1 ∈ (0, +∞], case II. 2 is when ξ k 1 → ξ 1 ∈ [-∞, 0). Let us discuss case II.1. Shrinking d if need be, we assume ξ 1 -d > 0, and then also ξ k 1 > d for all k large enough. (This works also for ξ 1 = +∞). Now define

Therefore the sequence g k converges to 0 in the space of quadratic polynomials in the variable t. But that implies its coefficients tend to 0, a contradiction with

, because the coefficient of t 2 is 1 and does not tend to 0. That is a contradiction in case II. 1, and therefore settles that case. Now consider case II. 2. Here we arrange ξ 1 + d < 0, and then also ξ k 1 + d < 0 for k sufficiently large, and that works also for ξ 1 = -∞. So here f k is positive and decreasing on I k . We use an analogous argument, and get a similar contradiction. That settles case II.

c) It remains to discuss case I. Note that here we must have γ = 0, so Q is a cone (in the sense of quadric theory). Suppose r > 0, then the sublevel set {x ∈ F : q(x) ≤ r} is nonempty and unbounded. Fix x in this set, then q(x) = q(x + td) for every d of the form d = (0, . . . , 0, ξ r+1 , . . . , ξ n ), because q does not depend on the coordinates ξ r+1 , . . . , ξ n . Now let P be the orthogonal projection on d ⊥ , then P (F ) is convex and, in addition, closed by what was observed at the beginning of the proof. But the infimum of q on P (F ) is still 0, and it is not attained. With regard to the form (4) we have therefore reduced the dimension n by 1, but the quadric is still of the form (4) with the same r. Continuing in this way, we end up with the case where r = n in (4). But then we are in case II, because remember that case I can only occur when r < n. That settles case I and therefore completes the first part of the proof.

2) Let us now prove that if F has a q-asymptote Q, then it is not Frank-and-Wolfe. From the definition of a q-asymptote we have F ∩ Q = ∅. We may therefore assume without loss that F ⊂ {x ∈ R n : q(x) > 0}, because F is connected and q is continuous. Now there exists a sequence x k ∈ F and a sequence y k ∈ Q such that dist((x k , q(x k )), (y k , 0)) → 0. That means 0 is the infimum of q on F , and it is not attained. □ Remark 9. One might be tempted to guess that F is Frank-and-Wolfe iff there is no quadratic Q which is asymptotic to F . The example of the positive orthant in remark 7 shows that this guess is incorrect. The corresponding condition is too strong.

Remark 10. The crucial point in the first part of the proof of Theorem 3 is that dist(Q, F ) = 0 is inferred from the fact that 0 is in the closure of the value set {q(x) : x ∈ F }. This is obviously strongly linked to the fact that q is quadratic, and we now indicate why there is little hope for an extension of the argument to higher degree polynomials. Consider for instance f (x, y, z) = (y 2 + (xy -1) 2 )z, then Q = {(x, y, z) : f (x, y, z) = 0} = {z = 0} is an affine manifold. We have f (k, 1 k , 1) → 0, but dist((k,

as k → ∞. Putting F = {(x, y, 1) ∈ R 3 : xy ≥ 1}, we see that f does not attain its infimum 0 on F , yet the affine manifold Q = {f = 0} is not asymptotic to F . What is