Hedy Attouch

Zaki Chbani
email: chbaniz@uca.ac.ma

Hassan Riahi
email: h-riahi@uca.ac.ma

FAST CONVEX OPTIMIZATION VIA TIME SCALING OF DAMPED INERTIAL GRADIENT DYNAMICS

Keywords: Convex optimization, inertial gradient systems, inertial proximal algorithms, Lyapunov analysis, Nesterov accelerated gradient method, time rescaling AMS subject classification. 37N40, 46N10, 49M30, 65K05, 65K10, 90B50, 90C25; Secondary: 37N40, 46N10, 49M30, 65K05, 65K10, 90B50, 90C25

In a Hilbert space setting, in order to develop fast first-order methods for convex optimization, we study the asymptotic convergence properties (t → +∞) of the trajectories of the inertial dynamics

The function Φ to minimize is supposed to be convex, continuously differentiable, γ(t) is a positive damping coefficient, and β(t) is a time scale coefficient. Convergence rates for the values Φ(x(t)) -min Φ and the velocities are obtained under conditions involving only β(t) and γ(t). In this general framework (Φ is only assumed to be convex with a non-empty solution set), the fast convergence property is closely related to the asymptotic vanishing property γ(t) → 0, and to the temporal scaling β(t) → +∞. We show the optimality of the convergence rates thus obtained, and study their stability under external perturbation of the system. The discrete time versions of the results provide convergence rates for a large class of inertial proximal algorithms. In doing so, we improve the classical results on the subject, including Güler's inertial proximal algorithm.

Introduction

Unless specified, throughout the paper we make the following standing assumptions

      
H is a real Hilbert space; Φ : H → R is a convex function of class C 1 , S = argmin Φ = ∅; t 0 > 0 is the origin of time; γ and β : [t 0 , +∞[→ R + are non-negative continuous functions.

We are interested in solving by (fast) first-order methods the convex optimization problem min x∈H Φ(x).

To deal with large-scale problems, first-order methods have become very popular in recent decades. Their direct link with gradient dynamics allows them to be approached with powerful tools from various fields such as mechanics, control theory, differential geometry and algebraic geometry. With this respect, the object of our study is the damped Inertial Gradient System (IGS) γ,β ẍ(t) + γ(t) ẋ(t) + β(t)∇Φ(x(t)) = 0, where ∇Φ : H → H is the gradient Φ. (IGS) γ,β involves two time-dependent parameters: γ(t) is a positive viscous damping coefficient, and β(t) is a time scale coefficient. The optimization property of this system comes from the viscous friction term γ(t) ẋ(t) which dissipates the global energy (potential + kinetic) of the system. Linking the tuning of the parameters γ(t) and β(t) to the corresponding rates of convergence of the values Φ(x(t)) -min H Φ is a subtle question. We will answer this question in a general context and obtain at the same time parallel results for the associated proximal algorithms. We will take advantage of the fact that proximal algorithms (obtained by implicit time discretization) usually inherit the properties of the continuous dynamics from which they come. Let us successively examine the role of the parameters γ(t) and β(t).

Date: May 23, 2019.

1.1. The damping parameter γ(t). Let's review some historical facts about the choice of γ(t) in (IGS) γ,1 ẍ(t) + γ(t) ẋ(t) + ∇Φ(x(t)) = 0, in relation to convex minimization. B. Polyak initiated the use of inertial dynamics to accelerate the gradient method. In [START_REF] Polyak | Some methods of speeding up the convergence of iteration methods[END_REF] (1964), he introduced the Heavy Ball with Friction system (HBF) ẍ(t) + γ ẋ(t) + ∇Φ(x(t)) = 0, which has a fixed viscous damping coefficient γ > 0 (the mass has been normalized equal to 1). In [START_REF] Álvarez | On the minimizing property of a second-order dissipative system in Hilbert spaces[END_REF] (2000), for a general convex function Φ, Alvarez proved that each trajectory of (HBF) converges weakly to a minimizer of Φ. For a strongly convex function Φ, (HBF) provides the convergence of Φ(x(t)) to min H Φ at an exponential rate by taking γ equal to 2 √ µ, where µ is the modulus of strong convexity of Φ. For general convex functions, the asymptotic convergence rate of (HBF) is O( 1 t ) (in the worst case). It's not better than the steepest descent. A decisive step was taken in 2014 by Su-Boyd-Candès [START_REF] Su | A differential equation for modeling Nesterov's accelerated gradient method: theory and insights[END_REF] who introduced the inertial system (AVD) α ẍ(t) + α t ẋ(t) + ∇Φ(x(t)) = 0.

For general convex functions, it provides a continuous version of the accelerated gradient method of Nesterov [START_REF] Nesterov | A method of solving a convex programming problem with convergence rate O(1/k2)[END_REF][START_REF] Nesterov | Introductory lectures on convex optimization: A basic course[END_REF] (described a little further). For α ≥ 3, each trajectory x(•) of (AVD) α satisfies the asymptotic convergence rate of the values Φ(x(t)) -min H Φ = O 1 t 2 . As a specific feature, the viscous damping coefficient α t vanishes (tends to zero) as time t goes to infinity, hence the terminology. Its close relationship with the Nesterov accelerated gradient method makes (AVD) α an interesting dynamic for convex optimization. As such, it has been the subject of many recent studies, see [START_REF] Apidopoulos | Convergence rate of inertial Forward-Backward algorithm beyond Nesterov's rule[END_REF], [START_REF] Attouch | Asymptotic stabilization of inertial gradient dynamics with time-dependent viscosity[END_REF][START_REF] Attouch | Convergence rates of inertial forward-backward algorithms[END_REF], [START_REF] Attouch | Rate of convergence of inertial gradient dynamics with time-dependent viscous damping coefficient[END_REF], [START_REF] Attouch | Fast convergence of inertial dynamics and algorithms with asymptotic vanishing viscosity[END_REF], [START_REF] Attouch | Rate of convergence of the Nesterov accelerated gradient method in the subcritical case α ≤ 3[END_REF], [START_REF] Attouch | The rate of convergence of Nesterov's accelerated forward-backward method is actually faster than 1/k 2[END_REF], [START_REF] Aujol | Stability of over-relaxations for the Forward-Backward algorithm, application to FISTA[END_REF], [START_REF] Aujol | Optimal rate of convergence of an ODE associated to the Fast Gradient Descent schemes for b > 0[END_REF], [START_REF] Bot | Second order forward-backward dynamical systems for monotone inclusion problems[END_REF], [START_REF] May | Asymptotic for a second-order evolution equation with convex potential and vanishing damping term[END_REF], [START_REF] Su | A differential equation for modeling Nesterov's accelerated gradient method: theory and insights[END_REF]. They allow to better understand Nesterov's method. Let's explain why α t is a clever tuning of the damping coefficient. i) In [START_REF] Cabot | On the long time behavior of second order differential equations with asymptotically small dissipation[END_REF], Cabot-Engler-Gaddat showed that the damping coefficient γ(•) in (IGS) γ,1 dissipates the energy, and hence makes the dynamic interesting for optimization, as long as +∞ t0 γ(t)dt = +∞. The damping coefficient can go to zero asymptotically but not too fast. The smallest which is admissible is of order 1 t . It is the one which enforces the most the inertial effect with respect to the friction effect.

ii) The tuning of the parameter α in front of 1 t comes naturally from the Lyapunov analysis and the optimality of the convergence rates obtained. Indeed, the case α = 3, which corresponds to Nesterov's historical algorithm, turns out to be critical. In the case α = 3, the question of the convergence of the trajectories remains an open problem (except in one dimension where convergence holds [START_REF] Attouch | Rate of convergence of the Nesterov accelerated gradient method in the subcritical case α ≤ 3[END_REF]). As a remarkable property, for α > 3, it has been shown by Attouch-Chbani-Peypouquet-Redont [START_REF] Attouch | Fast convergence of inertial dynamics and algorithms with asymptotic vanishing viscosity[END_REF] and May [START_REF] May | Asymptotic for a second-order evolution equation with convex potential and vanishing damping term[END_REF] that each trajectory converges weakly to a minimizer. Corresponding results for the algorithmic case have been obtained in [START_REF] Chambolle | On the convergence of the iterates of the Fast Iterative Shrinkage Thresholding Algorithm[END_REF] and [START_REF] Attouch | The rate of convergence of Nesterov's accelerated forward-backward method is actually faster than 1/k 2[END_REF]. Moreover for α > 3, it is shown in [START_REF] Attouch | The rate of convergence of Nesterov's accelerated forward-backward method is actually faster than 1/k 2[END_REF] and [START_REF] May | Asymptotic for a second-order evolution equation with convex potential and vanishing damping term[END_REF] that the asymptotic convergence rate of the values is o( 1 t 2 ). The subcritical case α < 3 has been examined by Apidopoulos-Aujol-Dossal [START_REF] Apidopoulos | Convergence rate of inertial Forward-Backward algorithm beyond Nesterov's rule[END_REF] and Attouch-Chbani-Riahi [START_REF] Attouch | Rate of convergence of the Nesterov accelerated gradient method in the subcritical case α ≤ 3[END_REF], with the convergence rate of the values f (x(t)) -min

H f = O 1 t 2α 3 .
A unifying view on the subject, dealing with the case of a general damping coefficient γ(t) in (IGS) γ,1 , has been developed by Attouch-Cabot [START_REF] Attouch | Asymptotic stabilization of inertial gradient dynamics with time-dependent viscosity[END_REF] and Attouch-Cabot-Chbani-Riahi [START_REF] Attouch | Rate of convergence of inertial gradient dynamics with time-dependent viscous damping coefficient[END_REF]. 1.2. The time scaling parameter β(t). Let's illustrate the role of β(t) in the following model situation. Start from the (AVD) α system with γ(t) = α t and α ≥ 3. Given a trajectory x(•) of (AVD) α , as explained above

Φ(x(t)) -min H Φ = O 1 t 2 as t → +∞.
Let's make the change of time variable in (AVD) α : t = s p , where p is a positive parameter. Set y(s) := x(s p ). By the derivation chain rule, we have For p > 1, we have α p > α, so the damping parameters for (1) are similar to those of (AVD) α . The only major difference is the coefficient s 2(p-1) in front of ∇Φ(y(s)), which explodes when s → +∞. From [START_REF] Álvarez | A second-order gradient-like dissipative dynamical system with Hessiandriven damping. Application to optimization and mechanics[END_REF] we observe that the convergence rate of values can be made arbitrarily fast (in the scale of powers of 1 s ) with p large. The physical intuition is clear. Fast optimization is associated with the fast parameterization of the trajectories of the (AVD) α system. One of our objective is to transpose these results to the proximal algorithms, taking advantage of the fact that implicit discretization usually preserves the properties of the continuous dynamics. The case γ(t) = α t , and β(t) general, i.e., (IGS) α t ,β

ẍ(t) + α t ẋ(t) + β(t)∇Φ(x(t)) = 0.
has been analyzed by Attouch-Chbani-Riahi [START_REF] Attouch | Fast proximal methods via time scaling of damped inertial dynamics[END_REF]. As explained above, the varying parameter t → β(t) comes naturally with the time reparametrization of the above dynamics, and plays a key role in the acceleration of its asymptotic convergence properties (the key idea is to take β(t) → +∞ as t → +∞ in a controlled way). The importance of the time scaling to accelerate algorithms was also stressed in [START_REF] Shi | Understanding the acceleration phenomenon via high-resolution differential equations[END_REF], [START_REF] Wibisono | A variational perspective on accelerated methods in optimization[END_REF].

1.3. Linking inertial dynamics with first-order algorithms. Let's first recall some classic results concerning the algorithms associated with the continuous steepest descent. Given λ > 0, the gradient descent method x k+1 = x k -λ∇Φ(x k ) and the proximal point method

x k+1 = prox λΦ (x k ) = (I + λ∂Φ) -1 (x k ) = argmin ξ∈H λΦ(ξ) + 1 2 x -ξ 2
are the basic blocks of the first-order methods for convex optimization. By interpreting λ as a fixed time step, they can be respectively obtained as the forward (explicit) discretization of the continuous steepest descent

(3) ẋ(t) + λ∇Φ(x(t)) = 0, and the backward (implicit) discretization of the differential inclusion (4) ẋ(t) + ∂Φ(x(t)) 0.

The gradient method goes back to Cauchy (1847). The proximal algorithm was first introduced by Martinet [START_REF] Martinet | Brève communication. Régularisation d'inéquations variationnelles par approximations successives[END_REF] (1970), and then developed by Rockafellar [START_REF] Rockafellar | Monotone operators and the proximal point algorithm[END_REF] who extended it to solve monotone inclusions. One can consult [START_REF] Bauschke | Convex Analysis and Monotone Operator Theory in Hilbert Spaces[END_REF], [START_REF] Parikh | Proximal algorithms[END_REF], [START_REF] Peypouquet | Convex optimization in normed spaces: theory, methods and examples[END_REF], [START_REF] Peypouquet | Evolution equations for maximal monotone operators: asymptotic analysis in continuous and discrete time[END_REF], [START_REF] Polyak | Introduction to optimization[END_REF], for a recent account on the proximal methods, that play a central role in nonsmooth optimization as a basic block of many splitting algorithms. Let's now come with second-order evolution equations, and illustrate their link with algorithms. Time discretization with a fixed time step h > 0 of (IGS) γ,β

ẍ(t) + γ(t) ẋ(t) + β(t)∇Φ(x(t)) = 0, gives (5) 1 h 2 (x k+1 -2x k + x k-1 ) + γ(kh) 1 h (x k -x k-1 ) + β(kh)∇Φ(ξ k ) = 0. Set α k = 1 -h 2 γ(kh) and β k = h 2 β(kh).
Following the choice of ξ k , we obtain one of the following algorithms:

• Implicit: ξ k = x k+1 gives the Inertial Proximal algorithm (Beck-Teboulle [START_REF] Beck | A fast iterative shrinkage-thresholding algorithm for linear inverse problems[END_REF], Güler [START_REF] Güler | New proximal point algorithms for convex minimization[END_REF]):

y k = x k + α k (x k -x k-1 ) x k+1 = prox β k Φ (y k ),
• Nesterov choice:

ξ k = x k + α k (x k -x k-1 )
gives the Inertial Gradient algorithm:

y k = x k + α k (x k -x k-1 ) x k+1 = y k -β k ∇Φ(y k ).
Nesterov's scheme corresponds to replacing the proximal step by a gradient step. This is illustrated below.

y k = x k + α k (x k -x k-1 ) • x k • x k-1 • x k+1 = y k -β k ∇Φ (y k ) S y k = x k + α k (x k -x k-1 ) • x k • x k-1 • x k+1 = prox β k Φ (y k ) S Figure 1. Inertial Gradient algorithm Inertial Proximal algorithm.
For additive structured optimization, the combination of the two algorithms yields FISTA type proximal-gradient algorithms. Their study was initiated by Beck-Teboulle [START_REF] Beck | A fast iterative shrinkage-thresholding algorithm for linear inverse problems[END_REF], see Attouch-Cabot [START_REF] Attouch | Convergence rates of inertial forward-backward algorithms[END_REF] for recent developments.

1.4. Presentation of the results. We are interested in the joint setting of the parameters γ(t) and β(t) providing a fast convergence of the values Φ(x(t)) -min Φ as t → +∞, and its algorithmic counterpart. In addition to Güler's accelerated proximal algorithm, our dynamic approach with general damping and scaling coefficients gives rise to a whole family of proximal algorithms with fast convergence properties. We start with the study of the continuous dynamic (sections 2 to 4), then consider the algorithmic aspects. In section 2, based on Lyapunov analysis, we study the asymptotic behaviour of the trajectories for (IGS) γ,β . According to [START_REF] Attouch | Asymptotic stabilization of inertial gradient dynamics with time-dependent viscosity[END_REF], the description of our result uses the auxiliary function Γ : [t 0 , +∞[→ R + defined by

Γ(t) = p(t) +∞ t du p(u)
where p(t) = e t t 0 γ(u) du .

In our main result, Theorem 2.1, we show that, under the following growth condition on β(•)

(H) γ,β Γ(t) β(t) ≤ β(t) (3 -2γ(t)Γ(t)) ,
we have the convergence rate of values Φ(x(t)) -min Γ(t) g(t) dt < +∞ (which means that it vanishes fast enough), we obtain similar convergence rates of the values (Theorem 4.1) and weak convergence of the trajectories (Theorem 4.2). In section 5, we make the link with the Güler inertial proximal algorithm, and provide its continuous interpretation. In section 6 (see Theorem 6.1), we study the fast convergence of values for a class of inexact inertial proximal algorithms, which takes into account a unified discretization of the damping term. We conclude with some Perspectives. In the Appendix section, we prove the existence and uniqueness of the global solution of the Cauchy problem associated with (IGS) γ,β,g , and complete with some technical lemmas.

H Φ = O 1 β(t)Γ(t)

2.

Convergence rates for the (IGS) system.

Based on the right tuning of the damping coefficient γ(t) and of the scaling coefficient β(t), we will analyze the convergence rate of the trajectories of the inertial dynamic (IGS) γ,β

(IGS) γ,β ẍ(t) + γ(t) ẋ(t) + β(t)∇Φ(x(t)) = 0.
We take for granted the existence of solutions to this system. The existence and uniqueness of a classical global solution for the corresponding Cauchy problem is detailed in the Appendix, Theorem 8.2. In the elementary case Φ ≡ 0, the direct integration of (IGS) γ,β makes appear the function ( 6)

p(t) = e t t 0 γ(u) du .
In the sequel, we will systematically assume that the following condition (H) 0 is satisfied

(H) 0 +∞ t0 du p(u) < +∞.
Under this assumption, we can define the function Γ :

[t 0 , +∞[→ R + (7) Γ(t) = p(t) +∞ t du p(u) .
Note that the definition of Γ does not depend on the choice of the origin of time t 0 . The asymptotic properties of the dynamic system (IGS) that will serve for the Lyapunov analysis. We have

E(t) = Γ(t) 2 W (t) + h(t) + Γ(t) ḣ(t) = Γ(t) 2 1 2 ẋ(t) 2 + β(t) Φ(x(t)) -min H Φ + 1 2 x(t) -z 2 + Γ(t) ẋ(t), x(t) -z ,
which gives ( 11)

E(t) = Γ(t) 2 β(t) Φ(x(t)) -min H Φ + 1 2 x(t) -z + Γ(t) ẋ(t) 2 .
Hence, E(•) is a non-negative function.

(H) γ,β Γ(t) β(t) ≤ β(t) (3 -2γ(t)Γ(t)) .
Then, for every solution trajectory x

: [t 0 , +∞[→ H of (IGS) γ,β ẍ(t) + γ(t) ẋ(t) + β(t)∇Φ(x(t)) = 0,
the following convergence rate of the values is satisfied: 2 as t → +∞.

(12) Φ(x(t)) -min H Φ = O 1 β(t)Γ(t)
Moreover, the trajectory x(•) is bounded on [t 0 , +∞[.

Proof. Set briefly m := min H Φ. Let's compute the time derivative of E(•), as formulated in [START_REF] Attouch | Fast proximal methods via time scaling of damped inertial dynamics[END_REF]. We first compute the derivative of its main ingredients, namely, W and h. The classical derivation chain rule and (IGS) γ,β give

Ẇ (t) = ẋ(t), ẍ(t) + β(t) ẋ(t), ∇Φ(x(t)) + β(t) (Φ(x(t)) -m) = ẋ(t), ẍ(t) + β(t)∇Φ(x(t)) + β(t) (Φ(x(t)) -m) = -γ(t) ẋ(t) 2 + β(t) (Φ(x(t)) -m) . ( 13
)
On the other hand, ḣ(t) = ẋ(t), x(t) -z and ḧ(t) = ẋ(t) 2 + ẍ(t), x(t) -z . It ensues that

ḧ(t) + γ(t) ḣ(t) = ẋ(t) 2 + ẍ(t) + γ(t) ẋ(t), x(t) -z = ẋ(t) 2 -β(t) ∇Φ(x(t)), x(t) -z ≤ ẋ(t) 2 -β(t)(Φ(x(t)) -m) ≤ ẋ(t) 2 , ( 14
)
where the above inequality follows from the convexity of Φ.

We have now all the ingredients to derivate E(•), as defined in [START_REF] Attouch | Fast proximal methods via time scaling of damped inertial dynamics[END_REF]. Collecting the above results we obtain

Ė(t) = Γ(t) 2 Ẇ (t) + 2Γ(t) Γ(t)W (t) + ḣ(t) + Γ(t) ḣ(t) + Γ(t) ḧ(t) = Γ(t) 2 [-γ(t) ẋ(t) 2 + β(t) (Φ(x(t)) -m)] + 2Γ(t) Γ(t) 1 2 ẋ(t) 2 + β(t) (Φ(x(t)) -m) + ḣ(t) + Γ(t) ḣ(t) + Γ(t) ḧ(t).
According to 1 + Γ(t) = γ(t)Γ(t) and ( 14), the above last line writes

ḣ(t) + Γ(t) ḣ(t) + Γ(t) ḧ(t) = Γ(t) ḧ(t) + γ(t) ḣ(t) ≤ Γ(t) ẋ(t) 2 -β(t)(Φ(x(t)) -m) .
Combining the above results, we obtain

Ė(t) ≤ Γ(t) 2 -γ(t) ẋ(t) 2 + β(t) (Φ(x(t)) -m) + 2Γ(t) Γ(t) 1 2 ẋ(t) 2 + β(t) (Φ(x(t)) -m) +Γ(t) ẋ(t) 2 -β(t)(Φ(x(t)) -m) ≤ Γ(t) ẋ(t) 2 1 + Γ(t) -γ(t)Γ(t) + (Φ(x(t)) -m) Γ(t) Γ(t) β(t) + 2 Γ(t)β(t) -β(t) .
Using again the relation 1 + Γ(t) = γ(t)Γ(t), we finally get

(15) Ė(t) ≤ (Φ(x(t)) -m) Γ(t) Γ(t) β(t) + 2 Γ(t)β(t) -β(t) .
According to 1 + Γ(t) = γ(t)Γ(t), this is equivalent to ( 16)

Ė(t) ≤ (Φ(x(t)) -m) Γ(t) Γ(t) β(t) + β(t)(2γ(t)Γ(t) -3) .
Therefore, by assumption (H) γ,β , we deduce that Ė(t) ≤ 0. Hence E(t) ≤ E(t 0 ) on [t 0 , +∞[. According to the formulation ( 11) of E(t), we deduce that, for all t ≥ t 0

(17) Φ(x(t)) -min H Φ ≤ E(t 0 ) β(t)Γ(t) 2 .
In addition, let us show that the trajectory remains bounded. According to [START_REF] Attouch | Convergence rates of inertial proximal algorithms with general extrapolation and proximal coefficients[END_REF], and E(•) decreasing, we have

x(t) -z + Γ(t) ẋ(t) 2 ≤ 2E(t) ≤ 2E(t 0 ).
After developing the above inequality, we obtain ( 18)

x(t) -z 2 + 2Γ(t) x(t) -z, ẋ(t) ≤ 2E(t 0 ). Set h(t) := 1 2 x(t) -z 2 , and q(t) := +∞ t ds p(s) . We have {q(t) : t ≥ t 0 } is bounded since +∞ t0 ds p(s) < +∞.
After dividing (18) by p(t), and noticing that Γ(t) p(t) = q(t), we obtain, with

C = E(t 0 ) 1 p(t) h(t) + q(t) ḣ(t) ≤ C p(t) , ∀t ∈ [t 0 , +∞[. Since q(t) = - 1 p(t)
, we equivalently have q(t) ḣ(t) -q(t)(h(t) -C) ≤ 0. After dividing by q(t) 2 , we obtain

d dt h(t) -C q(t) = 1 q(t) 2 q(t) ḣ(t) -q(t)(h(t) -C) ≤ 0.

Integration of this inequality gives

h(t) ≤ C 1 (1 + q(t)) for some C 1 > 0. Therefore, x(•) is bounded. Remark 1.
Let us analyze the condition (H) γ,β . We return to its formulation [START_REF] Bot | Second order forward-backward dynamical systems for monotone inclusion problems[END_REF]

) (19) Γ(t) β(t) + 2 Γ(t)β(t) -β(t) ≤ 0.
After multiplication by Γ(t), and by setting ξ(t) = β(t)Γ 2 (t) it writes equivalently as

(20) ξ(t) - 1 Γ(t) ξ(t) ≤ 0.
Integration of this first-order differential equation immediately gives

(21) ξ(t) ≤ ξ(t 0 ) exp t t0 1 Γ(s) ds on [t 0 , +∞[).
Therefore ( 22)

β(t) ≤ β(t 0 ) Γ 2 (t 0 ) Γ 2 (t) exp t t0 1 Γ(u)
du on [t 0 , +∞[. Thus, the condition (H) γ,β imposes a growth limitation for β(•), which depends on γ(•).

Remark 2.

The assumption argmin Φ = ∅ is crucial to guarantee that the trajectory remains bounded.

Otherwise when β ≡ 1, Φ is minorized and does not attain its infimum, i.e., argmin Φ = ∅, we may have lim t→+∞ Φ(x(t)) = inf H Φ (see [START_REF] Attouch | Fast convergence of inertial dynamics and algorithms with asymptotic vanishing viscosity[END_REF]). From this, we easily deduce that lim t→+∞ x(t) = +∞.

2.2. Rate of decay of the global energy. To obtain fast convergence of velocities to zero, we need to introduce the following slightly strengthened condition (H) + γ,β : there exist t 1 ≥ t 0 and ρ > 0 such that for t ≥ t 1 (H)

+ γ,β Γ(t) β(t) ≤ β(t) 3 -ρ -2γ(t)Γ(t) .
Note that (H) γ,β corresponds to the case ρ = 0 in (H) + γ,β . Let's first establish an integral estimate for the rescaled global energy function W . Proposition 1. Let Φ : H → R be a convex function of class C 1 such that argmin Φ = ∅. Suppose that the property (H) + γ,β is satisfied. Then, for any solution x(•) of (IGS) γ,β , the following integral energy estimate holds

(23) +∞ t0 Γ(t)W (t)dt < +∞,
where

W (t) := 1 2 ẋ(t) 2 + β(t) Φ(x(t)) -min H Φ . Equivalently +∞ t0 Γ(t) ẋ(t) 2 dt < +∞ and +∞ t0 Γ(t)β(t) Φ(x(t)) -min H Φ dt < +∞.
Proof. Start from the energy estimate ( 13), and set m := min H Φ. After multiplication by the positive scalar Γ(t) 2 , we get

Γ(t) 2 Ẇ (t) + γ(t)Γ(t) 2 ẋ(t) 2 = β(t)Γ(t) 2 (Φ(x(t)) -m) .
After integration by parts on (t 0 , t) we obtain

Γ(t) 2 W (t) -Γ(t 0 ) 2 W (t 0 ) -2 t t0 Γ(s) Γ(s)W (s)ds + t t0 γ(s)Γ(s) 2 ẋ(s) 2 ds = t t0 β(s)Γ(s) 2 (Φ(x(s)) -m) ds. Replace W (s) by its formulation W (s) := 1 2 ẋ(s) 2 + β(s) (Φ(x(s)) -min H Φ)
in the third term of the left member of the above formula. After simplification, we get

Γ(t) 2 W (t)+ t t0 Γ(s)[Γ(s)γ(s)-Γ(s)] ẋ(s) 2 ds = Γ(t 0 ) 2 W (t 0 )+ t t0 [2Γ(s) Γ(s)β(s)+ β(s)Γ(s) 2 ] (Φ(x(s)) -m) ds.
By (8), we have γ(s)Γ(s)γ(s) -Γ(s) = 1.

By (H) γ,β and (8), we have

d ds Γ(s) 2 β(s) = 2Γ(s) Γ(s)β(s) + β(s)Γ(s) 2 ≤ Γ(s)β(s).
Collecting the above results, we obtain

(24) Γ(t) 2 W (t) + t t0 Γ(s) ẋ(s) 2 ds ≤ t t0 Γ(s)β(s) (Φ(x(s)) -m) ds + Γ(t 0 ) 2 W (t 0 ).
In order to estimate the second member of (24), we return to ( 16)

Ė(t) ≤ Γ(t) Γ(t) β(t) + β(t)(2γ(t)Γ(t) -3) (Φ(x(t)) -m) .
According to the property (H) + γ,β , that we rewrite as follows

(25) Γ(t) β(t) + β(t) 2γ(t)Γ(t) -3 ≤ -ρβ(t), we get Ė(t) + ρΓ(t)β(t) (Φ(x(t)) -m) ≤ 0.
Integrating this inequality, and using that E(t) is non-negative, we obtain

(26) +∞ t0 Γ(s)β(s) (Φ(x(s)) -m) ds ≤ E(t 0 ) ρ < +∞.
Combining this inequality with [START_REF] Güler | New proximal point algorithms for convex minimization[END_REF], we obtain

(27) Γ(t) 2 W (t) + t t0 Γ(s) ẋ(s) 2 ds ≤ E(t0) ρ + Γ(t 0 ) 2 W (t 0 ). Hence ( 28
) +∞ t0 Γ(s) ẋ(s) 2 ds ≤ E(t0) ρ + Γ(t 0 ) 2 W (t 0 ).
By adding the inequalities ( 26) and ( 28), and using the definition of W , we finally get

+∞ t0 Γ(s)W (s)ds ≤ 3 2 E(t 0 ) ρ + 1 2 Γ(t 0 ) 2 W (t 0 ),
which gives (23).

2.3.

From O to o convergence rate of the values. According to Attouch-Peypouquet [START_REF] Attouch | The rate of convergence of Nesterov's accelerated forward-backward method is actually faster than 1/k 2[END_REF] and Attouch-Cabot [START_REF] Attouch | Asymptotic stabilization of inertial gradient dynamics with time-dependent viscosity[END_REF], let us improve the convergence rates obtained in the previous sections by passing from capital O to small o estimates.

Theorem 2.2. Let Φ : H → R be a convex function of class C 1 such that argmin Φ = ∅. Suppose that the property (H)

+ γ,β is satisfied. (i) If +∞ t0 1 Γ(t) dt = +∞, then Φ(x(t)) -min H Φ = o 1 β(t)Γ(t) 2 and ẋ(t) = o 1 Γ(t)
as t → +∞.

(ii) If +∞ t0 β(t)Γ(t)dt = +∞ and 2γ(t)β(t) + β(t) is nonnegative on [t 0 , +∞[, then Φ(x(t)) -min H Φ = o 1 t t0 β(s)Γ(s)ds and ẋ(t) 2 = o β(t) t t0 β(s)Γ(s)ds as t → +∞.
Proof. We examine successively the two cases: (i) Using the classical derivation chain rule, the definition of W , and the formula ( 13) for Ẇ , we obtain

d dt Γ(t) 2 W (t) = Γ(t) 2 Γ(t)W (t) + Γ(t) Ẇ (t) = Γ(t) 2 Γ(t)β(t) + Γ(t) β(t) (Φ(x(t)) -m) + Γ(t) Γ(t) -γ(t)Γ(t) ẋ(t) 2 .
According to Γ(t) -γ(t)Γ(t) = -1, and to the property (H) γ,β , i.e., d dt Γ(t) 2 β(t) ≤ Γ(t)β(t), we obtain

d dt Γ(t) 2 W (t) ≤ Γ(t)β(t) (Φ(x(t)) -m) .
According to Proposition 1, we have Using the classical derivation chain rule, the definition of W , and the formula [START_REF] Attouch | The rate of convergence of Nesterov's accelerated forward-backward method is actually faster than 1/k 2[END_REF] for Ẇ , we obtain

+∞ t0 Γ(t)β(t) (Φ(x(t)) -min H Φ) dt < +∞. The non-negative function k(t) := Γ(t) 2 W (t) verifies dk dt + ∈ L 1 (
dv dt (t) = - β(t) β(t) 2 W (t) + 1 β(t) Ẇ (t) = - β(t) β(t) 2 1 2 ẋ(t) 2 + β(t) (Φ(x(t)) -m) + 1 β(t) -γ(t) ẋ(t) 2 + β(t) (Φ(x(t)) -m) = - 2γ(t)β(t) + β(t) 2β 2 (t) ẋ(t) 2 .
By

β(s)Γ(s)ds = t t0 β(s)Γ(s)ds - r(t) t0 β(s)Γ(s)ds = (1 -δ) t t0 β(s)Γ(s)ds.
Recalling that v is non-increasing, we deduce that t r(t)

Γ(s)W (s)ds = t r(t) β(s)Γ(s)v(s)ds ≥ v(t) t r(t) β(s)Γ(s)ds = (1 -δ) W (t) β(t)
β(t) ≥ β(t 0 )e -2 t t 0 γ(s)ds = β(t 0 ) p(t) 2 .
Remark 4. The convergence of the trajectories generated by the dynamic system (IGS) γ,β will be analyzed in section 4, in the more general case where additional perturbations are taken into consideration.

Particular cases

3.1. Case β(t) = 1, γ(•) general.
The asymptotic convergence properties of the dynamical system (IGS) γ,1 are based on the behavior of Γ(t) as t → +∞. In this case, the condition (H) γ,β reduces to

(H) 1 γ(t)Γ(t) ≤ 3 2 ,
which is the condition introduced by Attouch-Cabot in [START_REF] Attouch | Asymptotic stabilization of inertial gradient dynamics with time-dependent viscosity[END_REF]. Thus, we recover Corollary 3.4. in [START_REF] Attouch | Asymptotic stabilization of inertial gradient dynamics with time-dependent viscosity[END_REF], that is, (H)

Corollary 1. Let Φ : H → R
+ 2 t β(t) ≤ β(t) (α -3 -ρ(α -1)) ⇐⇒ lim sup t→+∞ t β(t) β(t) < α -3.
Noticing that as t → +∞, see [START_REF] Attouch | Asymptotic stabilization of inertial gradient dynamics with time-dependent viscosity[END_REF], [START_REF] Attouch | Fast convergence of inertial dynamics and algorithms with asymptotic vanishing viscosity[END_REF], [START_REF] May | Asymptotic for a second-order evolution equation with convex potential and vanishing damping term[END_REF], [START_REF] Su | A differential equation for modeling Nesterov's accelerated gradient method: theory and insights[END_REF]. b) When β(t) = t p , we readily obtain that the condition (H) 2 is equivalent to α ≥ 3 + p.

+∞ t0 1 Γ(t) dt = +∞,
3.3. Case β(t) = β 0 t a ln(t) b , γ(t) = α t .
In this case, we readily obtain

t 2 β(t) = O t a+2 ln(t) b and t t0 sβ(s)ds = O t a+2 ln(t) b .
By specializing Corollary 4 and Corollary 5 to this situation, we obtain the following statement.

Corollary 6. Let x : [t 0 , +∞[→ H be a solution trajectory of (33) with β(t) = β 0 t a ln(t) b , and α > 1.

(i) If a ≤ α -3 and b ≤ 0, then Φ(x(t)) -min H Φ = O 1 t a+2 ln(t) b as t → +∞. (ii) If a < α -3, then Φ(x(t)) -min H Φ = o 1 t a+2 ln(t) b as t → +∞.
Proof. To show assertions (i) and (ii), we need to examine conditions (H) 2 and (H) + 2 , respectively. For t ≥ t 0 > 0, This will result from the following example showing that when γ(t) = α t , and β(t) = t δ , O(1/t 2+δ ) is the worst possible case for the convergence rate of values. The following example was obtained by rescaling the example of [START_REF] Attouch | Fast convergence of inertial dynamics and algorithms with asymptotic vanishing viscosity[END_REF]. Take H = R and Φ(x) = c|x| r , where c and r are positive parameters. We consider the evolution equation

t β(t) β(t) -α + 3 = a -α + 3 + b ln(t) . (i) If a ≤ α -3 = 0 and b ≤ 0, then t β(t) β(t) -α + 3 ≤ 0 for every t ≥ t 0 and (H) 2 is satisfied. (ii) If a < α -3 < 0, then have lim sup t→+∞ t β(t) β(t) -α + 3 < 0, and (H)
(IGS) α t ,δ ẍ(t) + α t ẋ(t) + t δ ∇Φ(x(t)) = 0,
and look for nonnegative solutions of the form x(t) = 1 t θ , with θ > 0. This corresponds to trajectories that are completely damped. We begin by determining the values of c, r,θ providing such solutions. On the one hand,

ẍ(t) + α t ẋ(t) = θ(θ + 1 -α) 1 t θ+2 .
On the other hand, ∇Φ(x) = cr|x| r-2 x, which gives

t δ ∇Φ(x(t)) = cr 1 t θ(r-1)-δ . Thus, x(t) = 1
t θ is solution of (IGS) α t ,δ if and only if, i) θ + 2 = θ(r -1) -δ, which is equivalent to r > 2 and θ = 2+δ r-2 ; and ii) cr = θ(α -θ -1), which is equivalent to α > r+δ r-2 and c = 2+δ r(r-2) (α -r+δ r-2 ). We have min Φ = 0 and

Φ(x(t)) = 2 + δ r(r -2) α - r + δ r -2 1 t r(2+δ) r-2
.

The convergence rate of Φ(x(t)) to 0 depends on the parameter r. When r goes to infinity, the exponent r(2+δ) r-2 > 2 + δ tends to 2 + δ. This limit situation is obtained by taking a function Φ which becomes very flat around the set of its minimizers. Therefore, without other geometrical hypotheses on Φ, we cannot expect a convergence rate better than O(1/t 2+δ ). It is precisely the convergence rate provided by Theorem 2.1, since t 2 β(t) = t 2+δ in this situation. Without rescaling, the optimality of the convergence rate of the values for the damped inertial gradient systems has been analyzed in a recent work of Apidopoulos-Aujol-Dossal-Rondepierre [START_REF] Apidopoulos | Convergence rates of an inertial gradient descent algorithm under growth and flatness condition[END_REF].

Numerical examples.

The following examples have been implemented with the Scilab opensource software version 5.5.2. In the context of convex optimization, we consider successively the strongly convex case with different conditionings, then the strictly convex case without strong convexity, and finally the case of a convex function with a continuum of solutions. The initial time t 0 has been taken equal to one.

Example 1. (Strongly convex).

We consider the strongly convex functions

Φ 1 (x 1 , x 2 ) := 5 • 10 -3 x 1 1 + x 2 2 (ill-conditioned) Φ 2 (x 1 , x 2 ) := 5x 1 1 + x 2 2
(well-condioned), which respectively illustrate the ill-conditioned case, and the well-conditioned case. We investigate numerically the convergence behaviour of x(t) -x 2 and |Φ i (x(t)) -Φ i (x)| (i = 1, 2) where x(•) is the solution of the dynamical system (IGS) γ,β , with the initial conditions (x 1 (1), x 2 (1)) = (1, 2) and ( ẋ1 (1), ẋ2 (1) = (0, 0). Three choices of γ and β are considered:

• γ(t) ≡ 2
√ µ i and β(t) ≡ 1, where µ i is the smallest eigenvalue for ∇ 2 Φ i . This choice is in accordance with the linear convergence results for the heavy ball method in the strongly convex case,

• γ(t) = 5 t and β(t) ≡ 1, • γ(t) = 5 t and β(t) = t 2 .
Figure 2. Graphical view of of x(t) -x 2 and |Φ i (x(t)) -Φ i (x)| for different values of γ(t) and β(t). The functions Φ i reach their infimum which is equal to zero uniquely at (0, 0). According to our theoretical results, we have the asymptotic convergence of values to zero. Compared results are shown in Figure 2. For the ill-conditioned function Φ 1 , the convergence of the values is faster for γ(t) = 5 t than for γ(t) constant.

3.5.2. Example 2. (Strictly convex, not strongly convex). Let's analyze the convergence properties, as t → +∞, of the trajectories of the dynamical system (IGS) γ,β in the case where Φ is defined on ]0, +∞[ 2 by Φ(x 1 , x 2 ) = x 1 + x 2 2 -ln(x 1 x 2 ). We can easily verify that Φ is a strictly convex (not strongly convex) function, and that x = (1, √ 2/2) is the unique global minimum point of Φ. We take γ(t) = α t and β(t) = t p so as to satisfy condition (H) 2 , which in this case is equivalent to α ≥ 3 + p. Specifically we compare the convergence rates for the solutions u 1 , u 2 , u 3 of the system (IGS) γ,β for (α 1 = 4, β 1 (t) = t 0.5 ), (α 2 = 5, β 2 (t) = t 1.5 ) and (α 3 = 6, β 3 (t) = t 2 ). This is illustrated in Figure 3, where the corresponding trajectories are displayed on the same screen with their optimal end point x. We observe that the numerical examples illustrated in Figure 3 are in agreement with the convergence rates predicted in Corollary 4. Precisely, in accordance with Φ(x(t)) -min H Φ = O

Example 3. (Non unique global solution)

. Take Φ : R 2 → R+ defined by Φ(x) = (x 1 + x 2 )

2 . The function Φ is convex but non strongly convex. Its solution set is the whole line argmin Φ = {(x 1 , x 2 ) ∈ R 2 : x 2 = -x 1 }. Depending on the initial data, we observe the convergence of the trajectories to different equilibria belonging to argmin Φ. In Figure 4 (left) are represented the trajectories of (IGS) γ,β corresponding to the initial position (x 1 (1), x 2 (1)) = (1, 2) and various initial velocity vectors ( ẋ1 (1), ẋ2 (1)). In Figure 4 (right) we observe a similar phenomenon when we change the initial point (x 1 (1), x 2 (1)) while keeping the same initial velocity vector ( ẋ1 (1), ẋ2 (1)) = (15, 0).

Existence and stability with respect to perturbations

Consider the perturbed version of the initial evolution system (IGS) γ,β

(IGS) γ,β,g ẍ(t) + γ(t) ẋ(t) + β(t)∇Φ(x(t)) = g(t),
where the second member g(•) can be interpreted as an external action on the system, a perturbation or a control term. For the existence and uniqueness of classical global solution of the Cauchy problem associated with the evolution system (IGS) γ,β,g , we refer to Theorem 8.2.

In this section, we show that the results of the previous sections remain satisfied if the perturbation g is sufficiently small asymptotically. Then, for any solution x : [t 0 , +∞[→ H of (IGS) γ,β,g , the following statements are satisfied:

(i) Under the condition (H) γ,β , x(t) and Γ(t) ẋ(t) are bounded on [t 0 , +∞[, and we have the convergence rate of the values:

(38) Φ(x(t)) -min H Φ = O 1 β(t)Γ(t) 2
as t → +∞.

(ii) Moreover, under condition (H) + γ,β , we have :

(39) +∞ t0 β(t)Γ(t) Φ(x(t)) -min H Φ dt < +∞.
Proof. The guiding idea of the proof is the same as in the Theorem 2.1. As a Lyapunov function, we use the same energy function E(•), which is defined for t ≥ t 0 by

E(t) = β(t)Γ(t) 2 Φ(x(t)) -min H Φ + 1 2 x(t) -z + Γ(t) ẋ(t) 2 .
The time derivative of E(•) is given by ( 40)

Ė(t) = d dt β(t)Γ(t) 2 (Φ(x(t)) -min H Φ) + β(t)Γ(t) 2 ∇Φ(x(t)), ẋ(t) + d dt (x(t) -z + Γ(t) ẋ(t)) , x(t) -z + Γ(t) ẋ(t) .
Then, according to [START_REF] Attouch | Fast convergence of inertial dynamics and algorithms with asymptotic vanishing viscosity[END_REF], note that the dynamical system (IGS) γ,β,g can be formulated as

(41) d dt (Γ(t) ẋ(t) + x(t) -z) = Γ(t)g(t) -β(t)Γ(t)∇Φ(x(t)).
From ( 40) and ( 41) we deduce that

Ė(t) = d dt β(t)Γ(t) 2 (Φ(x(t)) -min H Φ) + β(t)Γ(t) 2 ∇Φ(x(t)), ẋ(t) + Γ(t)g(t) -β(t)Γ(t)∇Φ(x(t)), x(t) -z + Γ(t) ẋ(t) = d dt β(t)Γ(t) 2 (Φ(x(t)) -min H Φ) + Γ(t)g(t), x(t) -z + Γ(t) ẋ(t) -β(t)Γ(t)∇Φ(x(t)), x(t) -z .
Using the Cauchy-Schwarz inequality and the convexity of Φ, that is,

∇Φ(x(t)), x(t) -z ≥ Φ(x(t)) -Φ(z), we obtain (42) Ė(t) ≤ d dt βΓ 2 (t) -(βΓ)(t) Φ(x(t)) -min H Φ + Γ(t) g(t) • x(t) -z + Γ(t) ẋ(t) .
Using the condition (H) γ,β and the definition of E, we deduce that

(43) Ė(t) ≤ Γ(t) g(t) • x(t) -z + Γ(t) ẋ(t) ≤ √ 2Γ(t) g(t) E(t),
By integrating the differential inequality (43) and using the assumption (H) g , we obtain

E(t) ≤ E(t 0 ) + 1 √ 2 +∞ t0 Γ(s) g(s) ds = Cte < +∞,
which gives the claim [START_REF] Shi | Understanding the acceleration phenomenon via high-resolution differential equations[END_REF]. In addition, we obtain that x(t) -z + Γ(t) ẋ(t) 2 is bounded, which gives

x(t) -z 2 + 2Γ(t) x(t) -z, ẋ(t) ≤ C. Set h(t) := 1 2 x(t) -z 2 .
The above inequality gives

h(t) + Γ(t) ḣ(t) ≤ 1 2 C.
By an argument similar to that of the unperturbed case, the integration of the above differential inequality gives that the trajectory x(

β(t)Γ(t) Φ(x(t)) -min H Φ dt ≤ 1 ρ E(t 0 ) + C +∞ t0 Γ(t) g(t) dt < +∞,
which completes the proof.

As in the unperturbed case, we can now pass from capital O estimates to small o estimates under the slightly stronger hypothesis (H) + γ,β . In addition, we obtain the convergence of trajectories. Theorem 4.2. Let Φ : H → R be a convex continuously differentiable function such that argmin Φ is nonempty. Suppose that the properties (H) + γ,β and (H) g are satisfied. Let x : [t 0 ; +∞[→ H be a global solution of (IGS) γ,β,g . Then x(t) converges weakly, as t → +∞ to a point in argmin Φ.

If moreover

+∞ t0 1 Γ(t) dt = +∞, we obtain Φ(x(t)) -min H Φ = o 1 β(t)Γ(t) 2 and ẋ(t) = o 1 Γ(t)
as t → +∞.

Proof. For the weak convergence, the proof is based on Opial's Lemma 8.3. By elementary calculus, convexity of Φ, and equation (IGS) γ,β,g , one can first establish, as in [START_REF] Attouch | Fast convex minimization via inertial dynamics with Hessian driven damping[END_REF], that for any z ∈ argmin Φ, the function

h z (t) := 1 2 x(t) -z 2 satisfies ḧz (t) + γ(t) ḣz (t) ≤ ẋ(t) 2 + g(t), x(t) -z .
Taking the norm of each member and using the boundedness of x(t), see Theorem

+∞ t0 d dt Γ(t) 2 W (s) dt ≤ +∞ t0 Γ(t)β(t) (Φ(x(s)) -m) dt + +∞ t0 Γ(t) 2 g(t) • ẋ(t) dt ≤ +∞ t0 Γ(t)β(t) (Φ(x(s)) -m) dt + sup t≥t0 Γ(t) ẋ(t) +∞ t0 Γ(t) g(t) dt < +∞.
From ( 39) and ( 46) we have

+∞ t0 Γ(t)W (s)dt < +∞.
In addition, by using

+∞ t0 dt Γ(t)
= +∞, we obtain that lim t→+∞ Γ(t) 2 W (t) = 0, which ends the proof.

Continuous modeling of Güler's Inertial Proximal Point Algorithm

In [START_REF] Güler | On the convergence of the proximal point algorithm for convex optimization[END_REF], Güler first studied the convergence rate of the proximal point algorithm without inertia. Then, in [START_REF] Güler | New proximal point algorithms for convex minimization[END_REF], he introduced the so-called Inertial Proximal Point Algorithm, which combines the ideas of Nesterov and Martinet as follows:

• Initialization of ν 0 and A 0 > 0.

• Step k :

• Choose β k > 0, and calculate g k > 0 by solving

• g 2 k + g k A k β k -A k β k = 0. • y k = (1 -g k )x k + g k ν k ; • x k+1 = prox β k Φ (y k ); • ν k+1 = ν k + 1 g k (x k+1 -y k ); • A k+1 = (1 -g k )A k . ( 47
)
Let us show that Güler's proximal algorithm (47) can be written as:

(IP) α k ,β k y k = x k + (1 -α k )(x k -x k-1 ) x k+1 = prox β k Φ (y k ).
This result was first obtained by the authors in [START_REF] Attouch | Fast proximal methods via time scaling of damped inertial dynamics[END_REF]. We reproduce it here briefly for the convenience of the reader. First verify that, for all k ≥ 1 (48)

ν k = x k-1 + 1 g k-1 (x k -x k-1 ) .
For this, we use an induction argument. Suppose ( 48) is satisfied at step k, then show that it will be at step k + 1. Using successively (47) and (48), we obtain

ν k+1 = ν k + 1 g k (x k+1 -y k ) = x k-1 + 1 g k-1 (x k -x k-1 ) + 1 g k (x k+1 -y k ) = 1 g k x k+1 + x k-1 + 1 g k-1 (x k -x k-1 ) - 1 g k ((1 -g k )x k + g k ν k ) = 1 g k x k+1 - 1 -g k g k x k = x k + 1 g k (x k+1 -x k ) ,
which shows that ( 48) is satisfied at step k + 1. Then, combining onesmore (47) with (48), we obtain

y k = (1 -g k )x k + g k ν k = (1 -g k )x k + g k x k-1 + 1 g k-1 (x k -x k-1 ) = x k + g k g k-1 -g k (x k -x k-1 ) .
Hence, Güler's proximal algorithm can be written as (IP) α k ,β k with (49)

α k = g k 1 g k-1 -1 .
By construction of g k , we have

g k = 1 2 -A k β k + (A k β k ) 2 + 4A k β k ,
which, by elementary calculation, gives 0 ≤ g k < 1. According to (49), we deduce that α k > 0. As a result, this makes Güler's algorithm (47) as an inertial proximal algorithm (IP) α k ,β k . From (47), we also get:

(50) A k = A 0 k-1 j=0 (1 -g j ) and g 2 k = A k β k (1 -g k ) = β k A k+1 ;
and then, we obtain the following relation between β k and g k :

(51)

β k = g 2 k A 0 k j=0 (1 -g j )
.

According to (49)-(51), we have obtained that all the parameters entering into Güler's algorithm can be expressed according to the single parameter g k . Let's come with the dynamic interpretation of Güler's algorithm, as formulated in (IP) α k ,β k . According to the formulation (49) of α k we get

x k+1 + β k ∂Φ(x k+1 ) y k = x k + g k 1 g k-1 -1 (x k -x k-1 ).
Equivalently, (52)

x k+1 -2x k + x k-1 + (g k - g k -g k-1 g k-1 )(x k -x k-1 ) + β k ∂Φ(x k+1 ) = 0.
This can be interpreted as a backward time discretization of the second-order evolution equation (when Φ is smooth)

(53) ẍ(t) + g(t) - ġ(t) g(t) ẋ(t) + β(t)∇Φ(x(t)) = 0.
So, modeling the Güler's accelerated Backward algorithm (IP) α k ,β k , we derive a second-order ordinary differential equation

(IGS) γ,β ẍ(t) + γ(t) ẋ(t) + β(t)∇Φ(x(t)) = 0.
where the damping coefficient is expressed as

(54) γ(t) = g(t) - ġ(t) g(t) = g(t) 1 + d dt 1 g (t) .
The algorithm (IP) α k ,β k features a new optimal convergence rate than Nesterov's method and also can be applied for non-smooth convex function Φ. The terminology (IGS) γ,β refers to the Inertial Gradient System with damping coefficient γ(t) and time scale coefficient β(t). Indeed, the parameter β(•) comes naturally with the time scaling of these dynamics.

Taking w := 1 g , then (54) is equivalent to solve the non-autonomous linear differential equation ẇ(t) -γ(t)w(t) = -1.

Set p(t) := exp t t0 γ(τ )dτ for t ≥ t 0 , then, following [5, Proposition 2.1] and assuming that +∞ t0 ds p(s) < +∞, we obtain Γ(t) = p(t) ∞ t 1 p(s) ds as the unique solution satisfying the limit condition lim t→+∞

Γ(t) p(t) = 0. Hence the general solution of (54) is g(t) = 1 Cp(t)+Γ(t) with 1 C = lim t→+∞ p(t)g(t).
When β(t) ≡ β > 0 is fixed, fast convergence of the values is obtained in [START_REF] Attouch | Asymptotic stabilization of inertial gradient dynamics with time-dependent viscosity[END_REF]Corollary 3.4] under the condition γ(t)Γ(t) ≤ 3 2 . According to (54), in terms of g(t), this condition takes the equivalent form

g(t) 1 + d dt 1 g (t) 1 g(t) = 1 + d dt 1 g (t) ≤ 3 2 . So, the condition γ(t)Γ(t) ≤ 3 2 becomes d dt 1 g (t) ≤ 1 2
, with the corresponding convergence rate of the values

(55) Φ(x(t)) -min H Φ = O g(t) 2 .
This shows the obvious interest in formulating the damping coefficient in the form g(t)

1 + d dt 1 g (t)
. There is no loss of generality, and the conditions for obtaining rapid convergence results can be formulated directly on the data g. For example, let's start with g(t) = α-1 t . From (54), we immediately obtain γ(t) = α t . Then, (55) shows that the well-known condition α ≥ 3 provides the O 1 t 2 convergence rate of values (see [START_REF] Attouch | Asymptotic stabilization of inertial gradient dynamics with time-dependent viscosity[END_REF][START_REF] Su | A differential equation for modeling Nesterov's accelerated gradient method: theory and insights[END_REF]). Following [6, Theorem 1], when β k ≡ β > 0 is fixed, we obtain a fast convergence of the values under the condition ∀k ≥ 1, t 2 k+1 -t 2 k ≤ t k+1 where t k := 1 + +∞ i=k i j=k α j . This condition can be equivalently formulated in terms of g k as (56)

g k ≥ 1 - g k g k-1 2 ,
with the corresponding convergence rate of the values

(57) Φ(x k ) -min H Φ = O g 2 k .
Starting from g k = α-1 k , (54) immediately gives γ k = α k . Then (56)-(57) shows that the well-known condition α ≥ 3 provides the O 1 k 2 convergence rate of values, a classical result (see [START_REF] Attouch | Fast convergence of inertial dynamics and algorithms with asymptotic vanishing viscosity[END_REF][START_REF] Attouch | The rate of convergence of Nesterov's accelerated forward-backward method is actually faster than 1/k 2[END_REF][START_REF] Chambolle | On the convergence of the iterates of the Fast Iterative Shrinkage Thresholding Algorithm[END_REF][START_REF] Su | A differential equation for modeling Nesterov's accelerated gradient method: theory and insights[END_REF]).

A class of Inexact inertial proximal algorithms

Motivated by the results above, we consider the proximal algorithms that can be obtained (when Φ is smooth) by various temporal discretizations of the second-order evolution equation

(58) ẍ(t) + g(t) 1 + d dt 1 g (t) ẋ(t) + λ(t)∇Φ(x(t)) = 0.
We'll see that the convergence analysis for the inertial proximal algorithm can be developed within this setting. As a major advantage, the convergence results can be expressed directly on the parameters describing the algorithm.

6.1.

A parametrized family of proximal inertial algorithms. Let us start from the second-order evolution equation ( 58) and introduce various temporal discretizations. When considering the implicit discretization for the potential term, which gives proximal algorithms, we can take a general convex lower semicontinuous proper function Φ. As novelty, we introduce a parameter θ ∈ [0, 1] which takes into account different discretizations of the damping term: for k ≥ 1,

(x k+1 -2x k + x k-1 ) + g k (1 -θ) 1 + 1 g k+1 - 1 g k (x k+1 -x k ) +g k θ 1 + 1 g k - 1 g k-1 (x k -x k-1 ) + λ k ∂Φ(x k+1 ) 0.
After dividing by g k , we obtain

1 g k + (1 -θ) 1 + 1 g k+1 - 1 g k (x k+1 -x k ) - 1 g k + θ 1 g k-1 - 1 g k -1 (x k -x k-1 ) + λ k g k ∂Φ(x k+1 ) 0. Set, for k ≥ 1, θ k := 1 g k + θ 1 g k-1 -1 g k -1 , then 1 g k + (1 -θ) 1 + 1 g k+1 - 1 g k = 1 + θ k+1 .
So, we can reformulate the above algorithm in the condensed form (59)

x k+1 + λ k g k (1 + θ k+1 ) ∂Φ(x k+1 ) x k + θ k 1 + θ k+1 (x k -x k-1 ).
For numerical reasons, it is important to allow approximate computation of the proximal mappings, which are not always available in close form. The following formulation of the algorithm (59) combines additive errors with the use of -subgradients. It extends the framework of the Inertial Proximal algorithm studied in [START_REF] Attouch | Convergence rates of inertial proximal algorithms with general extrapolation and proximal coefficients[END_REF]. One can consult [START_REF] Schmidt | Convergence rates of inexact proximal-gradient methods for convex optimization[END_REF] and [START_REF] Villa | Accelerated and inexact forward-backward[END_REF] for related results concerning the introduction of errors in proximal based algorithms.

Inexact Inertial Proximal algorithm.

(IP) α k ,β k , k y k = x k + α k (x k -x k-1 ) x k+1 ≈ prox k β k Φ (y k -r k ), α k := θ k 1+θ k+1 ; r k := - 1 g k (1+θ k+1 ) e k ; β k := λ k g k (1+θ k+1 ) ; θ k := 1 g k + θ 1 g k-1 -1 g k -1 . (60)
By definition of the inexact proximal operator, the iteration at step k of (IP) α k ,β k , k can be written as

1 β k (y k -x k+1 -r k ) ∈ ∂ k Φ(x k+1 ),
where ∂ k Φ(x) := {u ∈ H : Φ(x) ≤ Φ(y) -u, y -x + k , ∀y ∈ H}.

6.2. Convergence rates. The objective of this section is to study the rapid convergence of values for sequences generated by the algorithm (IP) α k ,β k , k .

Theorem 6.1. Consider the algorithm (IP) α k ,β k , k and suppose that 0 < g k ≤ 1, 0 ≤ θ ≤ 1, and the parameters (g k ), (λ k ) and θ satisfy the growth condition: there exists k 1 ∈ N such that for all k ≥ k 1

(K g k ,λ k ,θ ) λ k+1 ≤ g k+1 g k θ k+1 + 1 θ k+2 λ k .
Suppose that the sequences (r k ) ⊂ H, ( k ) ⊂ R + satisfy the summability properties

(61) (1 + θ k+1 ) r k < +∞ and (1 + θ k+1 ) λ k k g k < +∞.
Then, for any sequence (x k ) generated by the algorithm (IP) α k ,β k ,θ , we have

             (i) Φ(x k ) -min H Φ = O g k-1 λ k-1 (1 + θ k ) , as k → +∞; (ii) k≥1 β k,θ (Φ(x k ) -min H Φ) < +∞ where β k,θ := λ k-1 g k-1 (1 + θ k ) -λ k g k θ k+1 is non-negative by (K g k ,λ k ,θ ).
Proof. To make the presentation simpler, without loss of generality, we take k 1 = 1. By definition of the proximal operator, the iteration at step k of the algorithm (IP) α k ,β k ,θ writes

1 β k (y k -x k+1 -r k ) ∈ ∂ k Φ(x k+1 ).
Equivalently, we have the following subdifferential inequalities: for any x ∈ H

(62) Φ(x) + k ≥ Φ(x k+1 ) + 1 β k ( x -x k+1 , y k -x k+1 -x -x k+1 , r k ) .
Let us write successively inequality (62) at x = x k and x = x * ∈ argmin Φ. We obtain the two inequalities

Φ(x k ) + k ≥ Φ(x k+1 ) + 1 β k ( x k -x k+1 , y k -x k+1 -x k -x k+1 , r k ), (63) Φ(x * ) + k ≥ Φ(x k+1 ) + 1 β k ( x * -x k+1 , y k -x k+1 -x * -x k+1 , r k ). (64) Using x k -x k+1 = x k -y k + y k -x k+1 in (63) and x * -x k+1 = x * -y k + y k -x k+1 in (64) we obtain Φ(x k ) + k ≥ Φ(x k+1 ) + 1 β k ( x k -y k , y k -x k+1 -x k -x k+1 , r k + y k -x k+1 2 ), (65) Φ(x * ) + k ≥ Φ(x k+1 ) + 1 β k ( x * -y k , y k -x k+1 -x * -x k+1 , r k + y k -x k+1 2
). ( 66) Multiplying (65) by θ k α k -1 ≥ 0, then adding (66), we derive that

k θ k α k + θ k α k -1 (Φ(x k ) -Φ(x * )) ≥ θ k α k (Φ(x k+1 ) -Φ(x * )) + θ k α k β k y k -x k+1 2 + 1 β k x k+1 -y k , θ k α k -1 (y k -x k ) + y k -x * + 1 β k θ k α k -1 (x k+1 -x k ) + x k+1 -x * , r k . ( 67
)
By definition of y k we have

θ k α k -1 (y k -x k ) + y k = θ k α k -1 α k (x k -x k-1 ) + x k + α k (x k -x k-1 ) = x k + θ k (x k -x k-1 ) = z k where z k := x k + θ k (x k -x k-1 ). Moreover θ k α k -1 (x k+1 -x k ) + x k+1 = θ k+1 (x k+1 -x k ) + x k+1 = z k+1 .
We then deduce from (67) that

k θ k α k + θ k α k -1 (Φ(x k ) -Φ(x * )) ≥ θ k α k (Φ(x k+1 ) -Φ(x * )) + 1 β k x k+1 -y k , z k -x * + 1 β k r k , z k+1 -x * + θ k α k β k y k -x k+1 2 .
Equivalently, after multiplication by

β k k θ k β k α k + β k θ k α k -1 (Φ(x k ) -Φ(x * )) ≥ β k θ k α k (Φ(x k+1 ) -Φ(x * )) + x k+1 -y k , z k -x * + r k , z k+1 -x * + θ k α k y k -x k+1 2 . ( 68
)
To write (68) in a recursive form, we use z k+1 -

z k = (1 + θ k+1 ) (x k+1 -y k ) . It ensues that z k+1 -x * 2 = z k -x * 2 + 2(1 + θ k+1 ) x k+1 -y k , z k -x * + (1 + θ k+1 ) 2 x k+1 -y k 2 ,
which gives

x k+1 -y k , z k -x * = 1 2(1 + θ k+1 ) z k+1 -x * 2 -z k -x * 2 - (1 + θ k+1 ) 2 x k+1 -y k 2 .
Using this equality in (68), we obtain

k θ k β k α k + β k θ k α k -1 (Φ(x k ) -Φ(x * )) ≥ β k θ k α k (Φ(x k+1 ) -Φ(x * )) + r k , z k+1 -x * + 1 2(1 + θ k+1 ) z k+1 -x * 2 -z k -x * 2 + (1 + θ k+1 ) 2 x k+1 -y k 2 ,
where we have used

θ k α k -(1+θ k+1 ) 2 = (1+θ k+1 ) 2
(a consequence of the definition of α k ). After multiplication by

(1 + θ k+1 ), and neglecting the non-negative term (1+θ k+1 ) 2

x k+1 -y k 2 , we obtain

k θ k β k α k (1 + θ k+1 ) + β k (1 + θ k+1 ) θ k α k -1 (Φ(x k ) -Φ(x * )) + 1 2 z k -x * 2 ≥ β k (1 + θ k+1 )θ k α k (Φ(x k+1 ) -Φ(x * )) + 1 2 z k+1 -x * 2 + (1 + θ k+1 ) r k , z k+1 -x * . (69) According to β k (1 + θ k+1 ) = λ k g k , and θ k α k -1 = θ k+1 we have β k (1 + θ k+1 ) θ k α k -1 = λ k g k θ k+1 .
Hence, (69) can be equivalently written as

λ k g k (1 + θ k+1 ) k + λ k g k θ k+1 (Φ(x k ) -Φ(x * )) + 1 2 z k -x * 2 ≥ λ k g k (1 + θ k+1 )(Φ(x k+1 ) -Φ(x * )) + 1 2 z k+1 -x * 2 + (1 + θ k+1 ) r k , z k+1 -x * .
This naturally leads us to introduce the sequence (E k ) (70)

E k = λ k-1 g k-1 (1 + θ k )(Φ(x k ) -Φ(x * )) + 1 2 z k -x * 2 .
Thus, we have obtained the following inequality

E k + λ k g k (1 + θ k+1 ) k ≥ E k+1 + λ k-1 g k-1 (1 + θ k ) - λ k g k θ k+1 (Φ(x k ) -min H Φ) +(1 + θ k+1 ) r k , z k+1 -x * . (71) Under condition (K g k ,λ k ,θ ) we have λ k-1 g k-1 (1 + θ k ) -λ k g k θ k+1 ≥ 0. Hence, (72)
E k+1 ≤ E k + (1 + θ k+1 ) r k • z k+1 -x * + λ k g k (1 + θ k+1 ) k .
By summing inequalities (72) from j = 1 to k -1, we obtain (73)

E k ≤ E 1 + k-1 j=1 (1 + θ j+1 ) r j • z j+1 -x * + k-1 j=1 (1 + θ j+1 ) λ j j g j . Since E k ≥ 1 2 z k -x * 2 and A := (1 + θ j+1 ) λj j gj < +∞, we deduce that (74) z k -x * 2 ≤ 2E 1 + 2A + k j=1 2(1 + θ j ) r j-1 • z j -x * .
Let us apply the Gronwall Lemma 8.4 with a j = z j -x * , b j = 2(1+θ j ) r j-1 , and c = 2(E 1 + A). According to assumption (61), we obtain

z k -x * ≤ c + ∞ j=1 2(1 + θ j ) r j-1 < +∞.
Returning to (73), we deduce from the convergence of the series

B := (1 + θ j+1 ) r j that (75) E k ≤ C := E 1 + B (c + B) + A < +∞. By definition of E k , we obtain, for all k ≥ k 1 λ k-1 g k-1 (1 + θ k )(Φ(x k ) -Φ(x * )) ≤ E k ≤ C,
which gives the claim. The last item follows directly by summing (71).

Depending on the choice of the parameter θ, we obtain a specific algorithm, with its convergence rate. Let's consider the following cases of particular interest: a) Case θ = 1 corresponds to the explicit discretization of the damping term:

(76) (x k+1 -2x k + x k-1 ) + g k 1 + 1 g k - 1 g k-1 (x k -x k-1 ) + β k ∂ k Φ(x k+1 ) - g k g k-1 1 -g k-1 e k ,
since, for θ = 1 in the formula giving the parameters, we have

θ k := 1 g k + 1 g k-1 - 1 g k -1 = 1 g k-1 -1, θ k+1 + 1 = 1 g k and β k = λ k .
So, the formula (K g k ,λ k ,θ ) in Theorem 6.1 becomes (77)

β k+1 ≤ g k+1 g k 1 g k 1 g k+1 -1 β k = g k+1 g 2 k ( 1 g k+1 -1) β k .
So we recover the same growth condition as in [START_REF] Attouch | Convergence rates of inertial proximal algorithms with general extrapolation and proximal coefficients[END_REF]Theorem 4]. Let us now compare the convergence rates. Theorem 6.1 gives

Φ(x k ) -min H Φ = O g k-1 λ k-1 (1 + θ k ) . From θ k + 1 = 1 g k-1 we get g k-1 λ k-1 (1+θ k ) = g 2 k-1 λ k-1 = 1 t 2 k β k-1
, and then we recover the same convergence rate as in [START_REF] Attouch | Convergence rates of inertial proximal algorithms with general extrapolation and proximal coefficients[END_REF]Theorem 4]. b) Case θ = 0 corresponds to the implicit discretization of the damping term

(78) (x k+1 -2x k + x k-1 ) + g k 1 + 1 g k+1 - 1 α k (x k+1 -x k ) + β k ∂Φ(x k+1 ) - g k g k-1 1 -g k-1 e k .
We have

θ k = 1 g k , α k = g k+1 g k (1+g k+1 ) and β k = λ k g k+1 g k (1+g k+1 )
, which by Theorem 6.1 give that, under the condition

(K g k ,λ k ,θ ), i.e., λ k+1 ≤ (1+g k+1 )g k+2 g k λ k , we have Φ(x k ) -min H Φ = O g k g k-1 λ k-1 (1 + g k ) , as k → +∞.
Consider the case g k = α-1 k-1 . This gives α k = k-1 k+α-1 , which corresponds to a variant of the Nesterov acceleration scheme considered by several authors (see [START_REF] Attouch | Fast convergence of inertial dynamics and algorithms with asymptotic vanishing viscosity[END_REF], [START_REF] Chambolle | On the convergence of the iterates of the Fast Iterative Shrinkage Thresholding Algorithm[END_REF], [START_REF] Su | A differential equation for modeling Nesterov's accelerated gradient method: theory and insights[END_REF]). An elementary calculation shows that the growth condition above and the corresponding convergence rate give results comparable to those of the explicit case.

Perspectives

In general, the presence of oscillations is not a desirable property for optimization problems. In this respect, in order to improve the inertial methods, various strategies have recently been developed. It would be interesting to combine them with time scaling. Let us list some of them.

• To overcome the presence of wild oscillations (which may occur for ill-conditioned minimization problems), one has to consider a damping which takes into account the geometry of Φ. In this direction, the Hessian-driven damping (79) ẍ(t) + α t ẋ(t) + ∇ 2 Φ(x(t) ẋ(t) + ∇Φ(x(t)) = 0, combines the Nesterov acceleration with the Newton method, see [START_REF] Álvarez | A second-order gradient-like dissipative dynamical system with Hessiandriven damping. Application to optimization and mechanics[END_REF], [START_REF] Attouch | Fast convex minimization via inertial dynamics with Hessian driven damping[END_REF]. At first glance, the presence of the Hessian may seem like a numerical difficulty. The crucial point is that the Hessian comes in the form ∇ 2 Φ(x(t)) ẋ(t), which is equal to the derivative of ∇Φ(x(t)). As a consequence, the temporal discretization of this dynamics provides first-order algorithms. Time scaling in this context would lead to consider the Hessian-driven damping dynamics ẍ(t) + γ(t) ẋ(t) + β(t)∇ 2 Φ(x(t) ẋ(t) + b(t)∇Φ(x(t)) = 0.

• To avoid oscillations, while retaining the advantage of the inertial effect, the restarting method considers the damping coefficient as a control variable. The strategy is to maintain a high speed along the orbit by stopping the dynamic when the speed begins to decrease. After stopping, restart with zero speed, see Su-Boyd-Candès (2016). As well, time scaling in this framework is an interesting subject to study.

• In [START_REF] Ghisi | The remarkable effectiveness of time-dependent damping terms for second order evolution equations[END_REF] Ghisi-Gobbino-Haraux consider t → γ(t) as a pulsating function that alternates big and small values in a suitable way. They prove the effectiveness of the method for quadratic minimization, and apply it to ordinary differential equations and partial differential equations of hyperbolic type.

• In the above approaches, the damping is considered as a control variable. In this respect, it would be interesting to consider the damping as a closed-loop control, as opposed to the open-loop approach developed in most of the papers devoted to inertial methods in optimization.

• In our study of the continuous dynamic, the potential function Φ to minimize has been assumed to be continuously differentiable. Indeed, the entire study can be performed to deal with non-smooth functions, simply replacing Φ with its Moreau envelopes. This operation preserves the optimal value and optimal set, and leads to relaxed proximal algorithms whose numerical complexity is the same. Without time scaling, this approach has been recently developed by Attouch-Peypouquet [START_REF] Attouch | Convergence rate of proximal inertial algorithms associated with Moreau envelopes of convex functions[END_REF].

Appendix

In what follows, we prove the existence and the uniqueness of a global solution to the Cauchy problem associated with the evolution system (IGS) γ,β,g . We will use the following lemma.

Lemma 8.1. ([25, Prop. 6.2.1]) Let F : I × X → X where I = [t 0 , +∞[ and X is a Banach space. Assume that (i) for every x ∈ X , F (•, x) ∈ L 1 loc (I, X ); (ii) for a.e. t ∈ I, for every x, y ∈ X , F (t, x) -F (t, y) ≤ K(t, x + y ) x -y , where K(•, r) ∈ L 1 loc (I), ∀r ∈ R + ; (iii) for a.e. t ∈ I, for every x ∈ X , F (t, x) ≤ P (t)(1 + x ), where P ∈ L 1 loc (I). Then, for every s ∈ I, x ∈ X , there exists a unique solution u s,x ∈ W 1,1 loc (I, X ) of the Cauchy problem: us,x (t) = F (t, u s,x (t)) for a.e. t ∈ I, and u s,x (s) = x.

For simplicity, we give a short proof in the case where the gradient of Φ is Lipschitz continuous. Theorem 8.2. Suppose that Φ : H → R is convex, C 1 , with Lipschitz continuous gradient ∇Φ. Assume that β, γ : [t 0 , +∞[→ R * + and g : [t 0 , +∞[→ H are locally integrable. Then, the evolution system (IGS) γ,β,g , with initial condition (x(t 0 ), ẋ(t 0 )) = (x 0 , ẋ0 ) ∈ H × H, admits a unique global solution x : [t 0 , +∞[→ H. Proof. To prove the existence and uniqueness for (IGS) γ,β,g with initial condition (x(t 0 ), ẋ(t 0 )) = (x 0 , ẋ0 ), we formulate it in the phase space. Set I = [t 0 , +∞[, and define F : I × H × H → H by F (t, x, y) = (y, g(t) -β(t)∇Φ(x) -γ(t)y).

Set u(t) = (x(t), y(t)). The Cauchy problem for (IGS) γ,β,g can be equivalently formulated as ≤ (1 + γ(t)) y + Lβ(t) x -x 0 + β(t) ∇Φ(x 0 ) + g(t)

≤ max (1 + γ(t), Lβ(t), β(L x 0 + ∇Φ(x 0 ) + g(t)

) (1 + x + y ) = r(t)(1 + u ),
where r(t) = max (1 + γ(t), Lβ(t), β(L x 0 + ∇Φ(x 0 ) + g(t) ). Since r(•) ∈ L 1 loc (I, R), we conclude that all the conditions of Lemma 8.1 are satisfied. So, there exists a unique global solution of (IGS) γ,β,g satisfying the initial condition (x(t 0 ), ẋ(t 0 )) = (x 0 , ẋ0 ). Lemma 8.3. ( [START_REF]Weak convergence of the sequence of successive approximations for nonexpansive mappings[END_REF]) Let S be a nonempty subset of H and let x : [t 0 , +∞[→ H. Assume that (i) for every z ∈ S, lim t→∞ x(t) -z exists; (ii) every weak sequential cluster point of x(t), as t → ∞, belongs to S. Then x(t) converges weakly as t → ∞ to a point in S.

  ) + p 2 s 2(p-1) ∇Φ(y(s)) = 0, where α p = 1 + (α -1)p. The convergence rate of values becomes[START_REF] Álvarez | A second-order gradient-like dissipative dynamical system with Hessiandriven damping. Application to optimization and mechanics[END_REF] Φ(y(s)) -minH Φ = O 1 s 2pas s → +∞.

  )Γ(t)dt = +∞.

+ 2 is satisfied. 3 . 4 .

 34 Optimality of the results. Let us show the optimality of the convergence rate obtained in Theorem 2.1.

1 t 2

 12 β(t) = O 1 t 2+pas t → +∞, we have, with δ i ≥ 2 + p i , max i=1,2,3 max 1≤t≤2000 t δi (|Φ(u i (t)) -min Φ|) ≤ 0.5.

Figure 3 .

 3 Figure 3.

Figure 4 .

 4 Figure 4.

Theorem 4 . 1 .

 41 Let Φ : H → R be a C 1 convex function with argmin Φ = ∅. Take β : [t 0 , +∞[→ R * + a continuous function. Suppose that γ : [t 0 , +∞[→ R + is a continuous function that satisfies (H) 0 . Suppose that the function g : [t 0 , +∞[→ H is locally integrable and verifies (H) g +∞ t0 Γ(t) g(t) dt < +∞.

  ) = F (t, u(t)) for a.e. t ∈ I, u(t 0 ) = (x 0 , ẋ0 ).Let us verify the three conditions of Lemma 8.1. (i) For each (x, y) ∈ H × H, F (•, x, y) ∈ L 1 loc (I, H), since the functions g, β and γ are so. (ii) Denote by L the Lipschitz constant of ∇Φ. For every u = (x, y), u = (x , y ) ∈ H × H and a.e.t ∈ I F (t, u) -F (t, u ) = y -y + β(t)(∇Φ(x) -∇Φ(x )) + γ(t)(y -y ) ≤ (1 + Lβ(t) + γ(t)) ( x -x + y -y ) = (1 + Lβ(t) + γ(t)) (x, y) -(x , y )and then the second condition is verified, since the real function t → 1 + Lβ(t) + γ(t) belongs to L 1 loc (I, R). (iii) For every u = (x, y) ∈ H × H and a.e. t ∈ I F (t, u) = y + β(t)(∇Φ(x) -∇Φ(x 0 )) + β(t)∇Φ(x 0 ) + γ(t)y + g(t)

Lemma 8 . 4 ([ 8 ,

 848 Lemma 5.14]). Let (a k ) be a sequence of nonnegative numbers such that, for all k ∈ N,a 2 k ≤ c 2 + k j=1 b j a j ,where (b j ) is a summable sequence of nonnegative numbers, and c ≥ 0. Then, for all k ∈ N, a k ≤ c + ∞ j=1 b j .

  where z ∈ argmin Φ is given. They are the basic constitutive blocks of the function E : [t 0 , +∞[→ R +

	and the anchor function
				h(t) :=	1 2	x(t) -z 2 ,
	(10)	E(t) := Γ(t) 2 W (t) + h(t) + Γ(t) ḣ(t)
	By differentiating (7), we immediately obtain the relation
	(8)			Γ(t) = γ(t)Γ(t) -1,
	which plays a central role in Lyapunov's analysis. Let us define the rescaled global energy function
	(9)	W (t) :=	1 2	ẋ(t) 2 + β(t) Φ(x(t)) -min H	Φ ,

γ,β are based on the behavior of the functions t → Γ(t) and t → β(t) as t → +∞.

  Let Φ : H → R be a convex function of class C 1 such that argmin Φ = ∅. Let us assume that β : [t 0 , +∞[→ R + is a continuous function which is positive. Suppose γ : [t 0 , +∞[→ R + is a continuous function that satisfies (H) 0 . Assume that the following growth condition (H) γ,β linking α(t) with β(t) is satisfied:

	2.1. O-rate of convergence for the values. Based on the decreasing property of E(•), we are going to prove
	the following theorem.
	Theorem 2.1.

  Therefore, the function u is an increasing bijection from [t 0 , +∞[ onto [0, +∞[. Set r(t) := u -1 (δu(t)) where δ ∈]0, 1[ is fixed and t ∈ [t 0 , +∞[. In view of the increasing property of u, we have r(t) ≤ t for all t ∈ [t 0 , +∞[. By definition of u and of r(t), we have

	assumption, 2γ(t)β(t) + β(t) is non-negative on [t 0 , +∞[. Therefore v(•) is non-increasing on [t 0 , +∞[.	dv dt	(t) ≤ 0, which gives that the function
		+∞
	By assumption,	t0	β(s)Γ(s)ds = +∞. r(t)	t
			β(s)Γ(s)ds = u(r(t)) = δu(t) = δ		β(s)Γ(sds.
			t0	t0
	Therefore,
		t
		r(t)

  be a convex function of class C 1 such that argmin Φ = ∅. Let us assume that γ : [t 0 , +∞[→ R + is a continuous function satisfying (H) 0 and (H) 1 . Then, every solution trajectory x : [t 0 , +∞[→ H of (IGS) γ,β satisfies the following convergence rate of the values: Suppose that the conditions of Corollary 1 are satisfied, together with

	(30)	Φ(x(t)) -min H	Φ = O	1 Γ(t) 2	as t → +∞.
	The condition (H)
	+ 1			lim sup t→+∞	γ(t)Γ(t) <	3 2	.
	This condition is equivalent to existence of m < 3 2 such that γ(t)Γ(t) ≤ m on [t 1 , +∞[. Therefore, the assertion
	(ii) of Theorem 2.2 gives
	Corollary 2. ([5, Theorem 3.6]) +∞ t0 Γ(t)dt = +∞ and (H) + 1 . Then, as t → +∞
	(31)	Φ(x(t)) -min H	Φ = o	1 t0 Γ(s)ds t	, and	ẋ(t) 2 = o	1 t0 Γ(s)ds t
									+∞ t0	1 Γ(t) dt = +∞ and
	(H) + 1 . Then, as t → +∞
	(32)	Φ(x(t)) -min H	Φ = o		1 Γ(t) 2	and	ẋ(t) = o	1 Γ(t)	.
	(34)	Φ(x(t)) -min H	Φ = O	1 t 2 β(t)	as t → +∞,
	Similar computation gives the following formulation of condition (H)

+

γ,β , which means ∃t 1 ≥ t 0 , ∃ρ > 0 such that 2γ(t)Γ(t) -3 + ρ ≤ 0 for every t ≥ t 1 , becomes (H) .

The assertion (i) of Theorem 2.2, in the case β(t) = 1, is new and can be formulated as follows.

Corollary 3. Suppose that the conditions of Corollary 1 are satisfied together with

3.2. Case β(•) general, γ(t) = α t , α > 1.

In this case, the inertial dynamic (IGS) γ,β writes

(33) ẍ(t)

+ α t ẋ(t) + β(t)∇Φ(x(t)) = 0. Elementary computation gives p(t) = t t0 α . Hence +∞ t0 du p(u) < +∞ for α > 1, and condition (H) 0 is satisfied. From this, we readily obtain Γ(t) = t α-1 . Condition (H) γ,β reduces to (H) 2 t β(t) ≤ (α -3)β(t) for t ≥ t 0 . As a consequence of Theorem 2.1 we recover the convergence rate of values of [10, Theorem 8.1], which can be formulated as follows: Corollary 4. Let Φ : H → R be a convex function of class C 1 such that argmin Φ = ∅. Let us assume that β : [t 0 , +∞[→ R + is a continuous function, and that condition (H) 2 is satisfied. Then every solution trajectory x : [t 0 , +∞[→ H of (33) satisfies the following convergence rate of the values: + γ,β :

  a direct application of Theorem 2.2 gives the following result: which can be seend as a continuous version of the Nesterov method. In this case, the condition (H) 2 simply writes α ≥ 3. As a result, for α ≥ 3, we get Φ(x(t)) -min H Φ = O 1 t 2 , and for α > 3 Φ(x(t)) -min H Φ = o 1

	Remark 5. a) When β(t) ≡ 1 we get the classical results for the evolution equation
		ẍ(t) +	α t	ẋ(t) + ∇Φ(x(t)) = 0,
	t 2
	Corollary 5. (i) Under the condition (H) + 2 , every solution trajectory x : [t 0 , +∞[→ H of (33) satisfies the
	following convergence rate of the values:
	(35)	Φ(x(t)) -min H	Φ = o	1 t 2 β(t)	as t → +∞.
	(ii) Suppose moreover that
	(36)

+∞ t0 tβ(t)dt = +∞ and 2αβ(t) + t β(t) ≥ 0, ∀t ∈ [t 0 , +∞[, then (37) Φ(x(t)) -min H Φ = o 1 t t0 sβ(s)ds and ẋ(t) 2 = o β(t) t t0 sβ(s)ds as t → +∞.


Mounir Mokhtari

Romain Endelin

Hamdi Aloulou
email: hamdi.aloulou@mines-telecom.fr

Thibaut Tiberghien

{mounir Mokhtari

Cathy Bodine

Sumi Helal

Tao Gu

Measur ing the Impact of ICTs on the Quality of Life of Ageing People with Mild Dementia

Keywords:

Abstr act

Intr oduction

Today, providing assistive services for frail and dependant people could be done following two ways: the first option consists in looking for a reliable industrial solution to be deployed in the user's living space, which may not fit exactly with the users' specific requirements and usually imposes to modify extensively the living environment itself. The second option is to design a specific application, which meets exactly the needs of the end-users, but developers will be confronted with a wide spread of technologies and APIs which impose a huge amount of human-efforts and associated cost. This second option may require a long time (several years) to build a reliable running prototype. Consequently, in both options, the development cost is high, and this is also the reason why most smart home projects are still at a laboratory prototype level worldwide [START_REF] Orwat | Towards pervasive computing in health care-A literature review[END_REF]. There are efforts trying to transfer these prototypes into homes or hospitals [START_REF] Bardram | Experiences from real-world deployment of context-aware technologies in a hospital environment[END_REF], but migrating systems from a well controlled research lab environment into a much more complex real-world environment introduces tremendous challenges in terms of correctness, reliability and fault-tolerance of the system.

Ageing is a highly individualized, irreversible and inevitable process by which a person becomes more vulnerable and dependent on others [START_REF] Routhier | Aging health and aging need[END_REF]. It proceeds at different rates and within different functions depending on people. Changes, that can occur in cognitive, physiological and social conditions, are not necessarily related to a disease since they are, in a certain magnitude, a normal part of the ageing process.

Cognitive changes related to normal ageing span across several aspects of the mind [START_REF] Kausler | Learning and memory in normal aging[END_REF]. Sensory memory is the ability for each of the five senses to hold a large amount of sensory information for a very short period of time, and is independent of the attention to the stimulus. It suffers no major influence of ageing. Short term memory is a 20 to 30 seconds memory used to hold information for processing. It is a working memory that can hold 7 elements for direct manipulation. It is highly involved in Activities of Daily Living (ADL) as it enables multi-tasking and manipulation of information. The capacity to hold 7 elements is not affected by age, but the manipulation of this memory becomes difficult as elders have a limited capacity to divide their attention between two related tasks or inhibit unimportant information. Long term memory is a series of memory modules each responsible for holding different sorts of information. It is subject to three mechanisms (encoding, storage, retrieval) that are not affected by age in the same way. Encoding is usually subject to a less spontaneous organization of information, so elders might need support on this.

Conducting a survey that involves aged people and their associated caregivers will help to understand the needs of this portion of the society and to provide solutions that Respond to their requirements. Our ultimate goal is to maintain the quality of life of aged people in their own home as long as possible by the integration of ICTs in an acceptable way for both the end-users and the stakeholders [START_REF] Mokhtari | New trends to support independence in persons with mild dementia -a mini-review[END_REF]. This means that economical and social impacts are key issues in our approach.

Methods

To properly define and address this issue, our methodology consisted on performing 2 parallel activities, in one hand a survey investigating the need of 123 frail aged people living in their own homes, and on the other hand designing and developing a suitable system for the provision of assistive services.

Sur vey -The lifestyle of the elder s

In order to observe the lifestyle of the elders within their home, we have sent a questionnaire to elderly people, in collaboration with a national association of retired people1

In the Fig. 1 below, we observe the causes of insecurity, as felt by the elders as well as their caregivers. We observe that, many of the elders have developed a fear of other people (20%). But only 14% are concerned by their own condition and the risk of having an accident. On the other hand, from the caregivers' point of view, 23% have fear because the house is not suited for the patient (e.g. presence of stairs). In total, 50% are worried for the health of the patient, dispatched between an unsafe house, cognitive problems, autonomy problem, and the risk of accident.

. The same questions have been asked to both the elders and their family caregivers, so that we can observe the bias in their perception. We have gathered 246 questionnaires, making up a total of 35,178 questions answered. After investigating the population of the elders and the caregivers, this survey gives insights on the elders' activities at home and outdoor, as well as the healthcare services they receive. From this survey, we aim to determine the most critical needs of elderly people in their daily lives.

Fig. 1. Causes of patient's insecurity within the house

From this study, we observe a strong divergence in opinion between the patient and the caregiver. The caregivers tend to be more worried about the elders' condition, whereas the elders themselves may deny it, and develop social complications. Fig. 2 shows the caregivers' opinions about healthcare services for elderly people. As we can see, 1% of them are concerned about health monitoring, while 29% want to support the elderly people for living independently, and 29% mention the improvements in the patient's quality of life. 18% of the caregivers acknowledge that these services maintain a social connection for the elders (Fig. 1 has shown the importance of it, as patients may have social inabilities). Finally, 21% of the family caregivers say that assistive services are also a support for the family, giving a glimpse of the difficult condition of the family caregivers who support the patients in their daily life.

Fig. 2. Families' expectations about assistive services

As we can see from Fig. 2, the caregivers tend to be far more concerned by the quality of life and the autonomy of the elderly people, rather than by their medical condition.

System design har dwar e & softwar e

Our main approach for an easy adoption of an ambient assistive living solution, is to limit both the number of devices to be deployed, and the types of sensors to be used. The goal is to provide a smart home in the box approach that is easy to install on site and maintain remotely in order to allow a large-scale deployment in several homes while leveraging a server based software architecture able to instantiate each home and analyze the data coming from its deployed sensors independently. Fig. 3 illustrates a typical home environment and its associated sensors. The idea is to deploy low-cost and non-intrusive sensors (motion sensors, contact sensors) with an internet connected gateway in each home.

Fig. 3. Smart home is a box setup for activities recognition

On the server side, a framework is in charge of processing events sent by each home gateway and providing assistive services [START_REF] Aloulou | An Adaptable and Flexible Framework for Assistive Living of Cognitively Impaired People[END_REF]. Our software framework is based on the OSGi (Open Service Gateway initiative) modular principle. This allows decomposing the platform into several modules, having each its specific functionality, and all communicating with each others. These modules are responsible of sensors' events reception, reasoning, taking decisions about users' situations, and providing the assistive services. A semantic model is used by the platform to represent the environment and the end-user profile.

The platform's OSGi modules are identical for each deployed house and the abstract semantic model is instantiated with information concerning each new house. Therefore, on the server side, a new instance of our framework is created and launched after the deployment of our system (sensors and gateway) in a new house and the semantic model is instantiated with the new user profile (name, address, disease, etc), as well as the description of the house and the deployed sensors characteristics (type, code, possible events, localization, etc). Fig. 4 represents the architecture of the framework developed on the server side. ) of each house sends events received from the sensors deployed to the central server (hosted by CNRS LIRMM Lab. in Montpellier city in France). The CGI (Common Gateway Interface) on the server receives these events and routes them to the appropriate OSGi instance. After reasoning and making decisions, assistive services are provided to the end-users (patients or caregivers) on different interaction devices using the D3.js (Data-Driven Documents) library.

The goal of this experimentation was to confirm the feasibility of reasoning in an AAL solution based on a stripped-down hardware deployment. This means that we must demonstrate that valuable knowledge can be extracted about an elderly person's lifestyle from coarse sensor data. Moreover, we must verify that the scalability of the incurred processing remains within an economical level, where an economy of scale is possible with one server providing enough processing power to cater for hundreds of houses.

Multiple assistive services can be provided for the end-users. These services are based on the patients' locations and activities inferred by our platform deployed on the centralized server. Fig. 5 represents a Real-time visualization of a patient's activities during the day.

Fig. 5. Real time visualization of ADL

Relying on our conducted survey, we have identified several assistive services which are requested by the end-users and that can be provided based on the inferred information from our platform. Table 1 illustrates some of the assistive services that can be provided.

Discussion

Several technical issues have emerged from deploying technological systems in real living spaces. To perform the validation, these issues had to be dealt with, which is often considered as a "waste of time" by researchers and engineers developing systems in their lab. However, this experience allowed us to learn a lot about the targeted users and stakeholders in general, and it provides us with essential and extremely valuable knowledge related to bringing value out of our research work and making an impact in society. This knowledge, collected through an extensive survey, is mainly related to the feedback received from the stakeholders, the acceptance of the solutions, their ease of deployment and maintenance, usage issues, etc. Even though such deployments felt like a burden at some point in time, we can only recommend to researchers in our field to get out of the lab, deploy their solutions, and include stakeholders early in the research work.

After deploying in four bedrooms in a nursing home in Singapore [START_REF] Aloulou | Deployment of assistive living technology in a nursing home environment: methods and lessons learned[END_REF] and three individual homes in France [START_REF] Tiberghien | Strategies for Context Reasoning in Assistive Livings for the Elderly[END_REF], we have improved the system along two aspects, on one hand, to simplify the hardware architecture by limiting drastically the number of devices; and on the other hand, to design a server-based framework able to instantiate each deployed site independently in a scalable manner. Experimentation within living labs in Grenoble (France) and Starhome (Singapore) provided promising results. Our next target is to deploy 10 individual homes and one nursing home in France within the Quality of Life Chair.

Fig. 4 .

 4 Fig. 4. Server based Software architecture The gateway (BeagleBone Black 2

Table 1 .

 1 Assistive services deduced from the conducted survey

	End users expectations	Assistive services
	Improve Quality of Life	Serious games for memory stimulation
		Wandering at night alert
	Promoting Independent Living	Meals time reminders
		Medication agenda
	Support for the Family	Elderly ADL visualization
		Emergency calls in critical situations
	Maintaining Social Connexions	Video conferences with families and friends

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Acknowledgements

This project is funded under Quality of Life Chair of Institut Mines Telecom and supported by Mutuelle Generale and REUNICA.

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