Hedy Attouch
email: hedy.attouch@umontpellier.fr

Zaki Chbani
email: chbaniz@uca.ac.ma

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

FAST PROXIMAL METHODS VIA TIME SCALING OF DAMPED INERTIAL DYNAMICS

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

In a Hilbert setting, we consider a class of inertial proximal algorithms for nonsmooth convex optimization, with fast convergence properties. They can be obtained by time discretization of inertial gradient dynamics which have been rescaled in time. We will rely specifically on the recent developement linking Nesterov's accelerated method with vanishing damping inertial dynamics. Doing so, we somehow improve and obtain a dynamical interpretation of the seminal papers of Güler on the convergence rate of the proximal methods for convex optimization.

Introduction

Throughout the paper, H is a real Hilbert space with scalar product •, • and norm • , and Φ : H → R ∪ {+∞} is a convex lower-semicontinuous and proper function such that argmin Φ = ∅. Our study falls within the general setting of the Inertial Proximal Algorithm, (IPA) α k ,λ k for short

(IPA) α k ,λ k y k = x k + α k (x k -x k-1 ) x k+1 = prox λ k Φ (y k ),
where (α k ) is a sequence of positive extrapolation parameters, and (λ k ) is a sequence of positive proximal parameters.

On the basis of an appropriate tuning of α k and λ k , we will show that for any sequence (x k ) generated by (IPA) α k ,λ k , the convergence of values Φ(x k ) → min H Φ can be done arbitrarily fast. Recall that, for λ > 0, the proximal mapping prox λΦ : H → H is defined by prox λΦ (x) = argmin ξ∈H Φ(ξ) + 1 2λ

x -ξ 2 .

Equivalently, prox λΦ (x) + λ∂Φ (prox λΦ (x))

x, that is, prox λΦ = (I + λ∂Φ) -1 is the resolvent of index λ of the maximally monotone operator ∂Φ. The proximal mapping enters as a basic block of many splitting methods for nonsmooth structured optimization. A rich literature has been devoted to proximal-based algorithms. One can consult [START_REF] Attouch | Convergence rates of inertial forward-backward algorithms[END_REF], [START_REF] Bauschke | Convex Analysis and Monotone Operator Theory in Hilbert spaces[END_REF], [START_REF] Beck | A fast iterative shrinkage-thresholding algorithm for linear inverse problems[END_REF], [START_REF] Combettes | Signal recovery by proximal forward-backward splitting[END_REF], [START_REF] Parikh | Proximal algorithms[END_REF], [START_REF] Peypouquet | Convex optimization in normed spaces: theory, methods and examples[END_REF] for some recent contributions to the subject in the convex optimization setting.

As a guideline of our approach, we consider proximal algorithms corresponding (when Φ is smooth) to various time discretizations of the second-order evolution equation (AVD) α,β ẍ(t) + α t ẋ(t) + β(t)∇Φ(x(t)) = 0.

The case β(t) ≡ 1 corresponds to the dynamic introduced by Su-Boyd-Candès [START_REF] Su | A differential equation for modeling Nesterov's accelerated gradient method: theory and insights[END_REF] as a continuous version of the Nesterov accelerated gradient method, see also [START_REF] Attouch | Convergence rates of inertial forward-backward algorithms[END_REF], [START_REF] Attouch | Fast convergence of inertial dynamics and algorithms with asymptotic vanishing viscosity[END_REF]. The terminology (AVD) refers to Asymptotic Vanishing Damping, a specific characteristic of this dynamic in which the damping coefficient α t vanishes in a controlled manner (neither too fast nor too slowly), as t goes to infinity. The introduction of the varying parameter t → β(t) comes naturally with the time reparametrization of this dynamic, 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). Doing so, we obtain a dynamic interpretation of Güler's founding articles [START_REF] Güler | On the convergence of the proximal point algorithm for convex optimization[END_REF][START_REF] Güler | New proximal point algorithms for convex minimization[END_REF] on the convergence rate of the proximal methods for convex optimization. Our work is part of the study of the link between continuous dynamics and algorithms in optimization. It is a living subject, and particularly delicate in the non-autonomous case, here are some recent references on the subject [START_REF] Álvarez | A second-order gradient-like dissipative dynamical system with Hessian-driven damping. Application to optimization and mechanics[END_REF], [START_REF] Attouch | Fast convergence of inertial dynamics and algorithms with asymptotic vanishing viscosity[END_REF], [START_REF] Attouch | A dynamical approach to an inertial forward-backward algorithm for convex minimization[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] Bach | From Averaging to Acceleration, There is Only a Step-size[END_REF], [START_REF] Bot | A second order dynamical approach with variable damping to nonconvex smooth minimization[END_REF], [START_REF] Ghisi | The remarkable effectiveness of time-dependent damping terms for second order evolution equations[END_REF], [START_REF] Peypouquet | Evolution equations for maximal monotone operators: asymptotic analysis in continuous and discrete time[END_REF], [START_REF] Shi | Understanding the acceleration phenomenon via high-resolution differential equations[END_REF], [START_REF] Su | A differential equation for modeling Nesterov's accelerated gradient method: theory and insights[END_REF].

As a model example of our results, consider the algorithm (IPA) α k ,λ k associated with the following discretization of (AVD) α,β [START_REF] Álvarez | On the minimizing property of a second-order dissipative system in Hilbert spaces[END_REF] (

x k+1 -2x k + x k-1 ) + α -1 k (x k+1 -x k ) + 1 k (x k -x k-1 ) + β k ∇Φ(x k+1 ) = 0.
The parameter β k is the discrete version of β(t). Along with β(t) → +∞ as t → +∞, we will pay special attention to the case β k → +∞ as k → +∞. Taking β k = k δ (it corresponds to β(t) = t δ in (AVD) α,β ) gives the parameters Date: November 29, 2018.

α k = 1 - α k + α -1
, and

λ k = k δ+1 k + α -1
. Assuming that α > 3, and 0 < δ < α -3, we will show that for any sequence (x k ) generated by the algorithm (IPA)

α k ,λ k , Φ(x k ) -min Φ = o 1 k 2+δ .
This result provides with a much simpler algorithm the convergence rate obtained by Güler in [START_REF] Güler | New proximal point algorithms for convex minimization[END_REF]. As a result, by taking the parameter α large enough, we can take a large parameter δ, and thus obtain an arbitrarily fast convergence rate of values (in the scale of powers of 1 k ). In doing so, α k is close to one (following Nesterov's acceleration), and λ k is large (this is the large step proximal method). In addition, we obtain convergence rates to zero for speed and acceleration, and we show that the sequence (x k ) converges weakly to some x ∞ belonging to the solution set argmin Φ. Our study also opens new perspectives on the acceleration of proximal methods for inclusions governed by maximally monotone operators. This is an active research subject (link with ADMM algorithm) where proximal methods with large steps play an important role, see the recent studies [START_REF] Attouch | Convergence of damped inertial dynamics governed by regularized maximally monotone operators[END_REF], [START_REF] Attouch | Convergence of a relaxed inertial proximal algorithm for maximally monotone operators[END_REF], [START_REF] Attouch | Convergence of a relaxed inertial forward-backward algorithm for structured monotone inclusions[END_REF], [START_REF] Attouch | Convergence of inertial dynamics and proximal algorithms governed by maximal monotone operators[END_REF].

The paper is organized as follows: In section 2, we introduce the accelerated proximal algorithms via an implicit discretization of the rescaled dynamic (AVD) α,β . In section 3, we show that a proper tuning of the parameters provides fast convergent algorithms. In section 4, we show the convergence of the iterates to optimal solutions. In section 5, we compare our results with those of Güler. In section 6, we study the stability of the algorithms with respect to perturbations and errors. Finally, in section 7 we analyze the fast convergence properties of a general class of inertial proximal algorithms that extend the situation studied in the previous sections. The Appendix contains a brief analysis of the convergence properties of the associated dynamics, as well as some useful technical lemmas.

Accelerated proximal algorithms via time rescaling of inertial dynamics

In this section, we aim to introduce the algorithms and their fast convergence properties from a dynamic point of view. To simplify the presentation and consideration of inertial dynamics, just for this section we assume that Φ is convex continuously differentiable.

2.1. Inertial dynamics for convex optimization. We will rely on the recent developments linking Nesterov accelerated method for convex optimization with inertial gradient dynamics. As a main originality of our approach, we will show that time rescaling of these dynamics leads to proximal algorithms that converge arbitrarily fast.

Precisely, (IPA) α k ,λ k bears close connection with the Inertial Gradient System

(2) (IGS) γ ẍ(t) + γ(t) ẋ(t) + ∇Φ(x(t)) = 0, which is a non-autonomous second-order differential equation where γ(•) is a positive viscous damping parameter.

As pointed out by Su-Boyd-Candès in [START_REF] Su | A differential equation for modeling Nesterov's accelerated gradient method: theory and insights[END_REF], the (IGS) γ system with γ(t) = 3 t can be seen as a continuous version of the accelerated gradient method of Nesterov (see [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]). This method has been developed to deal with large scale structured convex minimization problems, such as the FISTA algorithm of Beck-Teboulle [START_REF] Beck | A fast iterative shrinkage-thresholding algorithm for linear inverse problems[END_REF]. These methods guarantee (in the worst case) the convergence rate Φ

(x k ) -min H Φ = O 1 k 2 ,
where k is the number of iterations. Convergence of the sequences generated by FISTA, has not been established so far (except in the one dimensional case, see [START_REF] Attouch | Rate of convergence of the Nesterov accelerated gradient method in the subcritical case α ≤ 3[END_REF]). This is a puzzling question in the study of numerical optimization methods. By making a slight change in the coefficient of the damping parameter, one can overcome this difficulty. Recently, 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] showed convergence of the trajectories of the (IGS) γ system with γ(t) = α t and α > 3

(3) (AVD) α ẍ(t) + α t ẋ(t) + ∇Φ(x(t)) = 0.
They also obtained the improved convergence rate Φ(x(t)) -min H Φ = o( 1 t 2 ) as t → +∞. Corresponding results for the algorithmic case have been obtained by Chambolle-Dossal [START_REF] Chambolle | On the convergence of the iterates of the Fast Iterative Shrinkage Thresholding Algorithm[END_REF], and by 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].

2.2. Time rescaling: implicit versus explicit time discretization. Let us show that, by time rescaling, we can make converge the trajectories of (AVD) α arbitrarily fast to the infimal value of Φ. Suppose that α ≥ 3. Given a trajectory x(•) of (AVD) α , we know that (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] Su | A differential equation for modeling Nesterov's accelerated gradient method: theory and insights[END_REF])

(4) Φ(x(t)) -min H Φ = O 1 t 2 .
Let's make the change of time variable t = τ (s) in (AVD) α , where τ (•) is an increasing function from R to R, which satisfies lim s→+∞ τ (s) = +∞. We have

(5) ẍ(τ (s)) + α τ (s) ẋ(τ (s)) + ∇Φ(x(τ (s))) = 0. Set y(s) := x(τ (s))
. By the derivation chain rule, we have

ẏ(s) = τ (s) ẋ(τ (s)), ÿ(s) = τ (s) ẋ(τ (s)) + τ (s) 2 ẍ(τ (s)).
Reformulating [START_REF] Attouch | Convergence rates of inertial forward-backward algorithms[END_REF] in terms of y(•) and its derivatives, we obtain

1 τ (s) 2 ÿ(s) - τ (s) τ (s) ẏ(s) + α τ (s) 1 τ (s) ẏ(s) + ∇Φ(y(s)) = 0.
Hence, y(•) is solution of the rescaled equation

(6) ÿ(s) + α τ (s) τ (s) - τ (s) τ (s) ẏ(s) + τ (s) 2 ∇Φ(y(s)) = 0.
The inequality (4) becomes ( 7) Φ(y(s)) -min

H Φ = O 1 τ (s) 2 .
Hence, by making a fast time reparametrization, we can obtain arbitrarily fast convergence property of the values. The damping coefficient of ( 6) is equal to

γ(s) = α τ (s) τ (s) - τ (s) τ (s) = α τ (s) 2 -τ (s)τ (s) τ (s) τ (s) .
As a model example, take τ (s) = s p , where p is a positive parameter. Then γ(s) = αp s , where α p = 1 + (α -1)p, and ( 6) writes [START_REF] Attouch | Convergence of a relaxed inertial forward-backward algorithm for structured monotone inclusions[END_REF] ÿ(s) + α p s ẏ(s) + p 2 s 2(p-1) ∇Φ(y(s)) = 0.

From [START_REF] Attouch | Convergence of a relaxed inertial proximal algorithm for maximally monotone operators[END_REF] we have ( 9) Φ(y(s)) -min

H Φ = O 1 s 2p .
For p > 1, we have α p > α, so the same damping features as for (AVD) α . The only major difference is the coefficient 1) in front of ∇Φ(y(s)) which explodes when s → +∞. As a general rule, implicit discretization preserves the convergence properties of the continuous dynamics. Precisely, we are going to show that the implicit discretization of (8) provides proximal algorithms whose convergence rate can be made arbitrarily fast with p large. The physical intuition is clear. Fast convergence just corresponds to fast parametrization of the trajectories of the (AVD) α system.

s 2(p-
The situation is completely different when we consider the gradient algorithms obtained by the explicit dicretization of [START_REF] Attouch | Convergence of a relaxed inertial forward-backward algorithm for structured monotone inclusions[END_REF]. Indeed, the fast convergence rate (9) cannot be transposed to the gradient methods: As a general rule, when passing from continuous dynamics to explicit discretized versions, in order to preserve the optimization properties, a step size smaller than the inverse of the Lipschitz constant of the gradient of the potential function must be chosen. Since the Lipschitz constant of s 2(p-1) ∇f tends to +∞ as s → +∞, this is not compatible with taking a fixed positive step size for the time discretization. Indeed, we know that the optimal convergence rate of the values (best possible in the worst case) for first-order gradient methods is O 1 k 2 , see [START_REF] Nesterov | Introductory lectures on convex optimization: A basic course[END_REF]Theorem 2.1.7]. 2.3. Introducting the scaled proximal inertial algorithm from a dynamic perspective. Motivated by the fast convergence properties of the trajectories of (8), we consider the second-order differential equation [START_REF] Attouch | Accelerated forward-backward algorithms with perturbations. Application to Tikhonov regularization[END_REF] (AVD) α,β

ẍ(t) + α t ẋ(t) + β(t)∇Φ(x(t)) = 0,
where the positive damping parameter α satisfies α ≥ 1, and β(•) is a positive time dependent scaling coefficient. From our perspective, the most interesting case is when β(t) → +∞ as t → +∞. We will then specialize our result in the important case β(t) = t p considered above. Let us consider the following implicit discretization of (AVD) α,β where for simplicity, the time step size has been normalized equal to one: for k ≥ 1, [START_REF] Attouch | Fast convergence of inertial dynamics and algorithms with asymptotic vanishing viscosity[END_REF] (

x k+1 -2x k + x k-1 ) + α -1 k (x k+1 -x k ) + 1 k (x k -x k-1 ) + β k ∇Φ(x k+1 ) = 0.
Note the special form of the discretization for the damping term α t ẋ(t), which was used above. This proves to be practical for our study. In section 7, we will study other types of discretization of the damping term, for which similar convergence properties hold. But for the moment, for the sake of simplicity, we will study this specific case as a model example. Equivalently, [START_REF] Attouch | Fast convergence of inertial dynamics and algorithms with asymptotic vanishing viscosity[END_REF] writes as follows

(1 + α -1 k )(x k+1 -x k ) + β k ∇Φ(x k+1 ) = (1 - 1 k )(x k -x k-1 ).
Setting

α k = k -1 k + α -1 and λ k = kβ k k + α -1
, we obtain the inertial proximal algorithm

(IPA) α k ,λ k y k = x k + α k (x k -x k-1 ) x k+1 = prox λ k Φ (y k ).
The algorithm (IPA) α k ,λ k still makes sense for a general convex lower-semicontinuous proper function Φ : H → R ∪ {+∞}. In this case, equality [START_REF] Attouch | Fast convergence of inertial dynamics and algorithms with asymptotic vanishing viscosity[END_REF] is replaced by the inclusion

(12) (x k+1 -2x k + x k-1 ) + α -1 k (x k+1 -x k ) + 1 k (x k -x k-1 ) + β k ∂Φ(x k+1 ) 0.
Remark 2.1. It is interesting to note that similar proximal inertial algorithms can be obtained by discretizing (AVD) α (i.e., with β ≡ 1) with a variable step size h k . Then β k = h 2 k , and so taking h k large corresponds to taking β k large. In [START_REF] Attouch | Convergence rates of inertial forward-backward algorithms[END_REF] Attouch-Cabot consider the case of a general extrapolation coefficient α k , but their study is limited to the case of a fixed step size, h k ≡ h > 0, which therefore does not cover our situation.

Fast convergence results

We now return to the general situation where Φ : H → R ∪ {+∞} is a convex lower-semicontinuous proper function such that argmin Φ = ∅. We will analyze the convergence rate of the values for the sequences (x k ) generated by the algorithm (IPA) α k ,λ k . Let's recall the basic result concerning the case α k = 1 -α k , λ k ≡ µ > 0, which is directly related to the Nesterov accelerated method (see [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] Beck | A fast iterative shrinkage-thresholding algorithm for linear inverse problems[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]). When α ≥ 3, we have Φ(x k ) -min Φ = O 1 k 2 . Indeed, we are going to show that the introduction of the scaling factor β k into the algorithm allows us to improve the convergence rate, and so obtain, for any sequence (x k ) generated by the algorithm (IPA)

α k ,λ k Φ(x k ) -min Φ = O 1 k 2 β k . 3.1. Convergence of the values. Theorem 3.1. Suppose α ≥ 1. Take α k = k -1 k + α -1 , λ k = kβ k k + α -1
. Suppose that the sequence (β k ) satisfies the growth condition: there exists

k 1 ∈ N such that for all k ≥ k 1 (H β ) β k+1 ≤ k(k + α -1) (k + 1) 2 β k .
Then, for any sequence (x k ) generated by the algorithm (IPA) α k ,λ k , we have

                   (i) Φ(x k ) -min H Φ = O 1 k 2 β k , (ii) k≥1 k 2 β 2 k ξ k 2 < +∞, with ξ k ∈ ∂Φ(x k+1 ), (iii) k≥1 Γ k (Φ(x k+1 ) -min H Φ) < +∞ where Γ k := k(k + α -1)β k -(k + 1) 2 β k+1 is non-negative by (H β ).
Proof. Let us denote briefly m := min H Φ. Fix z ∈ argmin Φ, that is Φ(z) = min H Φ = m, and consider, for k ≥ 1, the energy function:

E k := k 2 β k (Φ(x k ) -m) + 1 2 v k 2 , with v k := (α -1)(x k -z) + (k -1)(x k -x k-1 ).
Let's look for conditions on β k so that the sequence (E k ) k is non-increasing. To this end, we evaluate the term

E k+1 -E k . ( 13
) E k+1 -E k = (k + 1) 2 β k+1 (Φ(x k+1 ) -m) -k 2 β k (Φ(x k ) -m) + 1 2 v k+1 2 -1 2 v k 2 = (k + 1) 2 (β k+1 -β k ) (Φ(x k+1 ) -m) + (k + 1) 2 β k (Φ(x k+1 ) -m) -k 2 β k (Φ(x k ) -m) + 1 2 v k+1 2 -1 2 v k 2 = (k + 1) 2 (β k+1 -β k ) + (2k + 1)β k (Φ(x k+1 ) -m) + k 2 β k (Φ(x k+1 ) -Φ(x k )) + 1 2 v k+1 2 -1 2 v k 2 .
On the other hand,

v k+1 -v k = (α -1)(x k+1 -x k ) + k(x k+1 -x k ) -(k -1)(x k -x k-1 ) = (α -1)(x k+1 -x k ) + (x k -x k-1 ) + k(x k+1 -2x k + x k-1 ) = -kβ k ξ k , with ξ k ∈ ∂Φ(x k+1
), where the last equality comes from [START_REF] Attouch | Rate of convergence of the Nesterov accelerated gradient method in the subcritical case α ≤ 3[END_REF]. Combining the above formula with the definition of v k , we obtain

v k+1 -v k , v k+1 = (α -1)(x k+1 -z) + k(x k+1 -x k ), -kβ k ξ k = (α -1)kβ k ξ k , z -x k+1 + k 2 β k ξ k , x k -x k+1 ≤ (α -1)kβ k (Φ(z) -Φ(x k+1 )) + k 2 β k (Φ(x k ) -Φ(x k+1 ))
, where the last inequality follows from α ≥ 1, the convexity of Φ, and ξ k ∈ ∂Φ(x k+1 ). Using the elementary algebraic equality

(14) 1 2 v k+1 2 - 1 2 v k 2 = v k+1 -v k , v k+1 - 1 2 v k+1 -v k 2 ,
we obtain

1 2 v k+1 2 - 1 2 v k 2 ≤ (α -1)kβ k (Φ(z) -Φ(x k+1 )) + k 2 β k (Φ(x k ) -Φ(x k+1 )) - 1 2 kβ k ξ k 2 .
Combining the above inequality with [START_REF] Attouch | The rate of convergence of Nesterov's accelerated forward-backward method is actually faster than 1/k 2[END_REF], and after simplification, we obtain

E k+1 -E k + 1 2 k 2 β 2 k ξ k 2 ≤ (k + 1) 2 (β k+1 -β k ) + (2k + 1)β k -(α -1)kβ k (Φ(x k+1 ) -Φ(z)) ≤ (k + 1) 2 β k+1 -kβ k (k + α -1) (Φ(x k+1 ) -Φ(z))
.

Hence

E k+1 -E k + 1 2 k 2 β 2 k ξ k 2 + Γ k (Φ(x k+1 ) -Φ(z)) ≤ 0, (15)
where

Γ k := k(k + α -1)β k -(k + 1) 2 β k+1 . By assumption (H β ), for all k ≥ k 1 we have Γ k ≥ 0, and hence E k+1 ≤ E k . The sequence (E k ) k≥k1 is non-increasing and minorized by zero. Consequently, it is convergent. By definition of E k , we obtain, for all k ≥ k 1 k 2 β k (Φ(x k ) -min Φ) ≤ E k ≤ E k1 , which gives item (i), Φ(x k ) -min Φ = O 1 k 2 β k .
Moreover, from inequality [START_REF] Attouch | A dynamical approach to an inertial forward-backward algorithm for convex minimization[END_REF] and Γ k ≤ 0 for k ≥ k 1 , we obtain, for all i ≥ k 1

E i+1 -E i + 1 2 i 2 β 2 i ξ i 2 ≤ 0.

Summing the above inequalities from

i = k 1 to k ≥ k 1 , we get 1 2 k i=k1 i 2 β 2 i ξ i 2 ≤ E k1 -E k+1 ≤ E k1 , and hence k≥1 k 2 β 2 k ξ k 2 < +∞,
which gives item (ii).

For item (iii), we go back to [START_REF] Attouch | A dynamical approach to an inertial forward-backward algorithm for convex minimization[END_REF]. By summing the corresponding inequalities for k ≥ k 1 , we obtain

0 ≤ ∞ k=k1 Γ k (Φ(x k+1 ) -Φ(z)) ≤ E k1 < +∞,
which gives the claim.

3.2.

Convergence rate to zero of the velocities and the accelerations. To obtain fast convergence of velocities to zero, we need to introduce the following slightly strengthened version of (H β ).

Definition 3.2. We say that the sequence (β k ) satisfies the growth condition

(H + β ) if there exists k 1 ∈ N and ρ > 0 such that for all k ≥ k 1 (H + β ) β k+1 ≤ k(k + (α -1)(1 -ρ)) (k + 1) 2 β k .
Note that (H β ) corresponds to the case ρ = 0. Let's give an equivalent form of (H + β ) convenient for calculation. From (H + β ) we immediately get

(k + 1) 2 β k+1 -k 2 β k -(α -1)(1 -ρ)kβ k ≤ 0. Hence (16) ρ(α -1))kβ k ≤ -(k + 1) 2 β k+1 + k 2 β k + (α -1))kβ k = Γ k .
We can now establish the following rate of convergence for the velocities, and the acceleration. Note that the quantity

x k+1 + 2x k -x k-1 = (x k+1 -x k ) -(x k -x k-1
) is a discrete form of the norm of the acceleration.

Proposition 3.3. Suppose that α > 3 2 . Under condition (H β ) + we have +∞ k=1 k x k -x k-1 2 < +∞. and ∞ k=1 k 2 x k+1 -2x k + x k-1 2 < +∞. Moreover ∞ k=1 kβ k Φ(x k+1 ) -min H Φ < +∞.
Proof. Consider, for k ≥ 1, the global energy function:

W k := β k (Φ(x k ) -m) + 1 2 w k 2 , with w k := x k -x k-1 . Let's evaluate the term (k + 1) 2 W k+1 -k 2 W k . ( 17
) (k + 1) 2 W k+1 -k 2 W k = (k + 1) 2 β k+1 (Φ(x k+1 ) -m) -k 2 β k (Φ(x k ) -m) + (k+1) 2 2 w k+1 2 -k 2 2 w k 2 = (k + 1) 2 (β k+1 -β k ) (Φ(x k+1 ) -m) + (k + 1) 2 β k (Φ(x k+1 ) -m) -k 2 β k (Φ(x k ) -m) + (k+1) 2 2 w k+1 2 -k 2 2 w k 2 = (k + 1) 2 (β k+1 -β k ) + (2k + 1)β k (Φ(x k+1 ) -m) + k 2 β k (Φ(x k+1 ) -Φ(x k )) + k 2 2 w k+1 2 -w k 2 + 2k+1 2 w k+1 2 ≤ (α -1)kβ k (Φ(x k+1 ) -m) + k 2 β k (Φ(x k+1 ) -Φ(x k )) + k 2 2 w k+1 2 -w k 2 + 2k+1 2 w k+1 2
where the last inequality comes from assumption (H β ).

On the other hand,

1 2 w k+1 2 -1 2 w k 2 = -1 2 w k+1 -w k 2 + w k+1 -w k , w k+1 = -1 2 x k+1 -2x k + x k-1 2 + x k+1 -2x k + x k-1 , x k+1 -x k = -1 2 x k+1 -2x k + x k-1 2 -α-1 k (x k+1 -x k ) + 1 k (x k -x k-1 ) + β k ξ k , x k+1 -x k with ξ k ∈ ∂Φ(x k+1
), where the last equality comes from [START_REF] Attouch | Rate of convergence of the Nesterov accelerated gradient method in the subcritical case α ≤ 3[END_REF]. After multiplying by k 2 , we obtain

k 2 2 ( w k+1 2 -w k 2 ) = -k 2 2 x k+1 -2x k + x k-1 2 -(α -1)(x k+1 -x k ) + (x k -x k-1 ) + kβ k ξ k , k(x k+1 -x k ) ≤ -k 2 2 x k+1 -2x k + x k-1 2 -(α -1)k x k+1 -x k 2 -k x k+1 -x k , x k -x k-1 -k 2 β k (Φ(x k+1 ) -Φ(x k )) ,
where the last inequality follows from the convexity of Φ, and ξ k ∈ ∂Φ(x k+1 ).

Combining the above inequality with [START_REF] Aujol | Optimal rate of convergence of an ODE associated to the Fast Gradient Descent schemes for b > 0[END_REF], and after simplification, we obtain

(k + 1) 2 W k+1 -k 2 W k + k 2 2 x k+1 -2x k + x k-1 2 ≤ (α -1)kβ k (Φ(x k+1 ) -m) -(α -1)k x k+1 -x k 2 -k x k+1 -x k , x k -x k-1 + 2k + 1 2 x k+1 -x k 2 . Equivalently (k + 1) 2 W k+1 -k 2 W k + k 2 2 w k+1 -w k 2 + (α -1)k w k+1 2 + k w k+1 , w k - 2k + 1 2 w k+1 2 ≤ (α -1)kβ k (Φ(x k+1 ) -m) .
By elementary algebraic operations

k 2 2 w k+1 -w k 2 + (α -1)k w k+1 2 + k w k+1 , w k - 2k + 1 2 w k+1 2 = k 2 2 w k+1 -w k 2 + (α -1)k w k+1 2 + k 2 w k+1 2 + k 2 w k 2 - k 2 w k+1 -w k 2 - 2k + 1 2 w k+1 2 = k(k -1) 2 w k+1 -w k 2 + (α - 3 2 )k - 1 2 w k+1 2 + k 2 w k 2 .
For α > 3 2 , and k sufficiently large, all the above quantities are non-negative. Hence

(k + 1) 2 W k+1 -k 2 W k + k 2 x k -x k-1 2 + k(k -1) 2 x k+1 -2x k + x k-1 2 ≤ (α -1)kβ k (Φ(x k+1 ) -m) .
By condition (H β ) + , as formulated in ( 16), we have ρ(α -1)kβ k ≤ Γ k for some ρ > 0, and k sufficiently large. Hence

(18) (k + 1) 2 W k+1 -k 2 W k + k 2 x k -x k-1 2 + k(k -1) 2 x k+1 -2x k + x k-1 2 ≤ 1 ρ Γ k (Φ(x k+1 ) -m) .
Let's sum the above inequalities for k ≥ k 1 . According to the estimation k≥1 Γ k (Φ(x k+1 ) -min H Φ) < +∞ (see Theorem 3.1 (iii)), we obtain

∞ k=1 k x k -x k-1 2 < +∞.
and

∞ k=1 k 2 x k+1 -2x k + x k-1 2 < +∞,
which gives the claim.

Remark 3.4. In Proposition 3.3 above we proved that, under condition (

H β ) + , ∞ k=1 kβ k (Φ(x k+1 ) -min H Φ) < +∞.
Let's show that the following estimates holds too:

(19) ∞ k=1 kβ k Φ(x k ) -min H Φ < +∞.
This results from the following elementary majorizations. From (H β ),

(k + 1) 2 β k+1 ≤ k(k + α -1)β k ≤ 2k(k + 1)β k
where the last inequality is valid for k ≥ α -2. After simplification we get (k + 1)

β k+1 ≤ 2kβ k . Hence ∞ k=1 (k + 1)β k+1 Φ(x k+1 ) -min H Φ ≤ 2 ∞ k=1 kβ k Φ(x k+1 ) -min H Φ < +∞,
which gives the result, after reindexation.

3.3.

From O to o estimates. We rely on the following result from 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]. Suppose that α > 3. Given a trajectory x(•) of (AVD) α , the following rate of convergence of the values holds: [START_REF] Beck | A fast iterative shrinkage-thresholding algorithm for linear inverse problems[END_REF] Φ(x(t)) -min

H Φ = o 1 t 2 .
Hence, for the corresponding time rescaled dynamic (6), we have

(21) Φ(x(t)) -min H Φ = o 1 τ (s) 2 .
Based on the dynamical approach to the algorithm (IPA) α k ,λ k , we can expect improving the rates of convergence in Theorem 3.1, replacing O by o estimates. Precisely, we are going to prove the following result.

Theorem 3.5. Suppose α > 3 2 . Take α k = k -1 k + α -1 , λ k = kβ k k + α -1
. Suppose that the sequence (β k ) satisfies the growth condition (H + β ). Then, for any sequence (x k ) generated by the algorithm (IPA) α k ,λ k , we have

Φ(x k ) -min H Φ = o 1 k 2
Proof. Let's consider the sequence of global energies (W k ) introduced in the proof of Proposition 3.3

W k := β k (Φ(x k ) -m) + 1 2 x k -x k-1 2 .
By Proposition 3.3, we have

+∞ k=1 k x k -x k-1 2 < +∞ and ∞ k=1 kβ k (Φ(x k ) -min H Φ) < +∞, see Remark 3.4 formula (19). Hence ∞ k=1 kW k < +∞.
On the other hand, returning to [START_REF] Bach | From Averaging to Acceleration, There is Only a Step-size[END_REF] we have

(k + 1) 2 W k+1 -k 2 W k ≤ 1 ρ Γ k (Φ(x k+1 ) -m) .
The nonnegative sequence (a k ) with a k = k 2 W k satifies the relation

a k+1 -a k ≤ ω k with ω k = 1 ρ Γ k (Φ(x k+1 ) -m). According to k≥1 Γ k (Φ(x k+1 ) -min H Φ) < +∞ (see Theorem 3.1 (iii)), we have (w k ) ∈ l 1 (N)
. By a standard argument, we deduce that the limit of the sequence (a k ) exists, that is , we need to take β k → +∞ to get an improved convergence rate compared to the classical situation. Let's calculate the best convergence rate we can expect on the sequence (β k ), which is supposed to satisfy the growth condition (H β ). For simplicity of the presentation, we take k 1 = 1, the extension to a general k 1 is straightforward. Hence, for j = 1, 2, ..., k

β j ≤ (j -1)(j + α -2) (j) 2 β j-1 .
By taking the product of the above inequalities when j varies from 2 to k, we obtain

β k ≤ β 1 k j=2 (j -1)(j + α -2) j 2 .
Equivalently, for any k ≥ 2

β k ≤ β 1 k j=2 1 - 1 j 1 + α -2 j .
Taking the logarithm, we obtain the equivalent inequality

ln β k ≤ ln β 1 + k j=2 ln 1 - 1 j + ln 1 + α -2 j .
According to the inequality ln(1 + x) ≤ x for any x > -1, we deduce that

ln β k ≤ ln β 1 + (α -3) k j=2 1 j .
By a classical comparison argument between series and integral, we have k j=2

1 j ≤ k 1 1 t dt = ln k. Hence ln β k ≤ ln β 1 + (α -3) ln k, which gives β k ≤ β 1 k α-3 .
Let us show that the above majorization is sharp and that, for

β k = k δ with δ < α -3, the condition (H β ) is satisfied. Indeed, for β k = k δ we have (H β ) ⇐⇒ (k + 1) δ ≤ k(k + α -1) (k + 1) 2 k δ ⇐⇒ (k + 1) δ+2 ≤ k δ+1 (k + α -1) ⇐⇒ (1 + 1 k ) δ+2 ≤ 1 + α -1 k . ( 22
)
For k large, 1 k is close to zero. Then, the left member of the above inequality is equivalent to 1 + δ+2 k . So inequality ( 22) is satisfied for k sufficiently large if δ + 2 < α -1, that is δ < α -3. Thus, if α > 3, we can take β k = k δ for any δ < α -3. In addition, we have

Γ k = k(k + α -1)β k -(k + 1) 2 β k+1 = k δ+1 (k + α -1) -(k + 1) δ+2 = (α -3 -δ)k δ+1 + • k δ+1
Since we argue with strict inequalities, it is immediate to verify that (H + β ) is also satisfied under the assumption α > 3. Note that the condition δ < α -3 allows us to take δ < 0, which corresponds to the case β k → 0. But for our purpose of getting a fast convergent algorithm, the most interesting case is δ > 0, which corresponds to β k → +∞.

Let's summarize the above results in the following statement.

Corollary 3.6. Take α > 3,

α k = 1 - α k + α -1 , λ k = k δ+1 k + α -1 with 0 < δ < α -3.
Then, for any sequence (x k ) generated by the algorithm (IPA) α k ,λ k , we have

                 Φ(x k ) -min Φ = o 1 k 2+δ ; +∞ k=1 k 2(1+δ) ξ k 2 < +∞ with ξ k ∈ ∂Φ(x k+1 ); +∞ k=1 k δ+1 (Φ(x k+1 ) -min H Φ) < +∞; +∞ k=1 k x k -x k-1 2 < +∞.
3.5. Back to the dynamical interpretation. Let us show that the above results are consistent with the dynamic interpretation of the algorithm, via temporal rescaling. For the rescaled inertial dynamic

(23) ẍ(t) + α p t ẋ(t) + p 2 t 2(p-1) ∇Φ(x(t)) = 0,
we showed that, for α ≥ 3 and p > 1

(24) Φ(x(t)) -min H Φ = O 1 t 2p .
By passing to the implicit discretized version, we expect to maintain the same convergence rate and thus obtain

(25) Φ(x k ) -min H Φ = O 1 k 2p .
Let's verify that this is the case. When β(t) = p 2 t 2(p-1) , we have 1) . By Theorem 3.1 and Corollary 3.6, for the corresponding algorithm (IPA) α k ,λ k , by taking

β k = p 2 k 2(p-
β k = k δ with δ = 2p -2, we have 2 + δ = 2p, so (26)
Φ(x k ) -min Φ = O 1 k 2+δ = O 1 k 2p .
Thus, the continuous approach to the algorithm and its direct independent study by a Lyapunov argument are consistent, and give the same convergence rates.

Convergence of the iterates

Let us now fix x * ∈ H, and define the sequence (h k ) by

h k = 1 2 x k -x * 2 .
The next result will be useful for establishing the convergence of the iterates of (IPA) α k ,λ k . The proof follows the line of [5, Proposition 4.1]. Proposition 4.1. We have

(27) h k+1 -h k -α k (h k -h k-1 ) = 1 2 (α 2 k + α k ) x k -x k-1 2 -y k -prox λ k Φ (y k ), y k -x * + 1 2 y k -prox λ k Φ (y k ) 2 .
If moreover x * ∈ argmin Φ, then

h k+1 -h k -α k (h k -h k-1 ) ≤ 1 2 (α 2 k + α k ) x k -x k-1 2 -λ k (Φ(x k+1 ) -min H Φ) - 1 2 y k -prox λ k Φ (y k ) 2 .
Proof. Observe that

y k -x * 2 = x k + α k (x k -x k-1 ) -x * 2 = x k -x * 2 + α 2 k x k -x k-1 2 + 2α k x k -x * , x k -x k-1 = x k -x * 2 + α 2 k x k -x k-1 2 + α k x k -x * 2 + α k x k -x k-1 2 -α k x k-1 -x * 2 = x k -x * 2 + α k ( x k -x * 2 -x k-1 -x * 2 ) + (α 2 k + α k ) x k -x k-1 2 = 2[h k + α k (h k -h k-1 )] + (α 2 k + α k ) x k -x k-1 2 .
Setting briefly

A k = h k+1 -h k -α k (h k -h k-1
), we deduce that

A k = 1 2 x k+1 -x * 2 - 1 2 y k -x * 2 + 1 2 (α 2 k + α k ) x k -x k-1 2 = x k+1 -y k , 1 2 (x k+1 + y k ) -x * + 1 2 (α 2 k + α k ) x k -x k-1 2 = x k+1 -y k , y k -x * + 1 2 x k+1 -y k 2 + 1 2 (α 2 k + α k ) x k -x k-1 2 .
Using the equality x k+1 = prox λ k Φ (y k ), we obtain [START_REF] Drori | Performance of first-order methods for smooth convex minimization: a novel approach[END_REF].

Let us now assume that x * ∈ argmin Φ. By definition of x k+1 = prox λ k Φ (y k ), we have

1 λ k (y k -x k+1 ) ∈ ∂Φ(x k+1 ). Hence, by convexity of Φ Φ(x * ) ≥ Φ(x k+1 ) + 1 λ k y k -x k+1 , x * -x k+1 . Equivalently Φ(x * ) ≥ Φ(x k+1 ) + 1 λ k y k -x k+1 , x * -y k + 1 λ k y k -x k+1 2 .
Returning to [START_REF] Drori | Performance of first-order methods for smooth convex minimization: a novel approach[END_REF], by using the above inequality, we obtain

h k+1 -h k -α k (h k -h k-1 ) ≤ 1 2 (α 2 k + α k ) x k -x k-1 2 -λ k (Φ(x k+1 ) -Φ(x * )) - 1 2 y k -prox λ k Φ (y k ) 2 ,
which completes the proof of Proposition 4.1.

Theorem 4.2. Assume (H β ) + . Then, any sequence (x k ) generated by algorithm (IPA) α k ,λ k converges weakly, and its limit belongs to argmin Φ.

Proof. We apply the Opial lemma, see Lemma 8.3.

(i) By Theorem 3.5 we have Φ(x k ) -min H Φ = o 1 k 2 , and hence lim k→+∞ Φ(x k ) = min H Φ. Assume that there exist x ∈ H and a sequence (k n ) such that k n → +∞, and x kn

x weakly as n → +∞. Since the convex function Φ is lower semicontinuous, it is lower semicontinuous for the weak topology, hence satisfies

Φ(x) ≤ lim inf n→+∞ Φ(x kn ) = lim k→+∞ Φ(x k ) = min H Φ.
It ensues that x ∈ argmin Φ, which shows the first point.

(ii) Let us now fix x * ∈ argmin Φ, and show that lim k→+∞ x k -x * exists. For that purpose, let us set

h k = 1 2 x k -x * 2
. From Proposition 4.1, the sequence (h k ) satisfies the following inequalities

h k+1 -h k -α k (h k -h k-1 ) ≤ 1 2 (α 2 k + α k ) x k -x k-1 2 ≤ x k -x k-1 2 since α k ∈ [0, 1].
Taking the positive part, we find

(h k+1 -h k ) + ≤ α k (h k -h k-1 ) + + x k -x k-1 2 .
From Proposition 3.3, we have

+∞ k=1 k x k -x k-1 2 < +∞.
By applying Lemma 8.4 (given in the appendix) with

a k = (h k -h k-1 ) + and ω k = x k -x k-1 2 , we obtain +∞ k=1 (h k -h k-1 ) + < +∞.
Since (h k ) is nonnegative, this classically implies that lim k→+∞ h k exists. The second point of the Opial lemma is shown, which ends the proof.

Comparaison with Güler's results

In a founding work for the study of proximal algorithms, based on the Nesterov accelerated scheme for convex optimization, Güler, see [30, Theorem 2.2], introduced algorithms that accelerate the classical proximal point algorithm. He obtained the convergence rate of values

f (x k ) -min H f = O 1 ( k i=1 √ λ i ) 2
, where (λ i ) is the sequence of proximal parameters. Our dynamic approach to accelerating proximal algorithms and Güler's proximal algorithms find their roots in the Nesterov acceleration gradient method. So, they provide comparable but, as we will see, significantly different results. We will list below some advantages of our approach. Recall first Güler's proximal algorithm, where we slightly modify the notations of his seminal paper [START_REF] Güler | New proximal point algorithms for convex minimization[END_REF] to fit our framework.

Güler's proximal algorithm: a) Initialization of ν 0 and A 0 . b)

Step k:

• Choose λ k > 0, and calculate γ k > 0 by solving the second-order algebraic equation ( 28)

γ 2 k + γ k A k λ k -A k λ k = 0.
• Define

y k = (1 -γ k )x k + γ k ν k ; (29) x k+1 = prox λ k Φ (y k ); (30) ν k+1 = ν k + 1 γ k (x k+1 -y k ); (31) A k+1 = (1 -γ k )A k . (32)
Let us show that the above Güler's proximal algorithm can be written as an inertial proximal algorithm (IPA) α k ,λ k . First prove that, for all k ≥ 1 (33)

ν k = x k-1 + 1 γ k-1 (x k -x k-1 ) .
For this, we use an induction argument. Suppose ( 33) is satisfied at step k, and then show that it will be at step k + 1.

Using successively ( 31), ( 33), [START_REF] Güler | On the convergence of the proximal point algorithm for convex optimization[END_REF], and (33) again, we obtain

ν k+1 = ν k + 1 γ k (x k+1 -y k ) = x k-1 + 1 γ k-1 (x k -x k-1 ) + 1 γ k (x k+1 -y k ) = 1 γ k x k+1 + x k-1 + 1 γ k-1 (x k -x k-1 ) - 1 γ k ((1 -γ k )x k + γ k ν k ) = 1 γ k x k+1 + x k-1 + 1 γ k-1 (x k -x k-1 ) - 1 -γ k γ k x k -x k-1 - 1 γ k-1 (x k -x k-1 ) = 1 γ k x k+1 - 1 -γ k γ k x k = x k + 1 γ k (x k+1 -x k ) ,
which shows that ( 33) is satisfied at step k + 1. Then, combining ( 29) with ( 33) we obtain

y k = (1 -γ k )x k + γ k ν k = (1 -γ k )x k + γ k x k-1 + 1 γ k-1 (x k -x k-1 ) = x k + γ k γ k-1 -γ k (x k -x k-1 ) .
Hence, Güler's proximal algorithm can be written as the algorithm (IPA) α k ,λ k (34)

y k = x k + α k (x k -x k-1 ) x k+1 = prox λ k Φ (y k ), where (35)
α k = γ k 1 γ k-1 -1 .
By construction of the γ k , we have 0 ≤ γ k ≤ 1, which gives α k ≥ 0. From ( 28) and ( 32), we have

γ 2 k = A k λ k (1 -γ k ) = λ k A k+1
, which gives the following relation between λ k and γ k :

(36) λ k = γ 2 k A 0 k j=0 (1 -γ j )
.

Let's come to the comparison of the convergence rates obtained by the two methods. If (λ k ) k is nondecreasing, we have (

k i=1 √ λ i ) 2 ≤ k 2 λ k . In our construction, λ k ∼ β k .
As a result, in the setting of Theorem 3.1, our convergence rates are at least as good as those obtained by Güler. In the setting of Theorem 3.5 they are better. The comparison in the general case is a non-trivial question, which requires further studies. Some advantages of our approach are listed below.

• Based on the dynamic approach of the Nesterov method recently discovered by Su-Boyd-Candès [START_REF] Su | A differential equation for modeling Nesterov's accelerated gradient method: theory and insights[END_REF], the time rescaling technique developed in this paper gives much simpler results. It also provides a valuable guide for the proofs, which result from standard Lyapunov analysis.

• The convergence of iterates is obtained (see section 4), which is not known by either the Nesterov method or the Güler algorithm. We rely on the recent progress of Chambolle-Dossal [START_REF] Chambolle | On the convergence of the iterates of the Fast Iterative Shrinkage Thresholding Algorithm[END_REF] on this subject. Based on the related results concerning the o rate of convergence results of 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], in Theorem 3.5 we obtain the convergence rate o 1 k 2 β k , which slightly improves the convergence rates, as mentioned above.

Note that Güler result, which is in line with the seminal Nesterov method, is based on taking γ k equal to the positive root of the second order equation [START_REF] Ghisi | The remarkable effectiveness of time-dependent damping terms for second order evolution equations[END_REF]. Indeed, the above mentioned progress simply relies on the fact that one can argue with an inequality instead of the equality in [START_REF] Ghisi | The remarkable effectiveness of time-dependent damping terms for second order evolution equations[END_REF]. • The flexibility of our approach allows us to provide a large family of inertial proximal algorithms with similar convergence rates (see section 7).

Stability with respect to perturbations, errors

Consider the perturbed version of the evolution equation (AVD) α,β

(37) ẍ(t) + α t ẋ(t) + β(t)∇Φ(x(t)) = g(t),
where the second member of (37), denoted by g(•), can be interpreted as an external action on the system, a perturbation, or a control term. By following a parallel approach to the time discretization procedure described in section 2.3, we obtain

(38) (x k+1 -2x k + x k-1 ) + α -1 k (x k+1 -x k ) + 1 k (x k -x k-1 ) + β k ∂Φ(x k+1 ) g k .
From the algorithmic point of view, the sequence (g k ) of elements of H takes into account the presence of perturbations, approximations, or errors. Setting

α k = k -1 k + α -1 , λ k = kβ k k + α -1 , e k = k k + α -1 g k , we obtain the inertial proximal algorithm (IPA) α k ,λ k ,e k y k = x k + α k (x k -x k-1 ) x k+1 = prox λ k Φ (y k + e k ).
Note that g k and e k are asymptotically equivalent, which makes them play a similar role as perturbation variables.

The following result extends Theorem 3.1 to the perturbed case.

Theorem 6.1. Suppose α ≥ 1. Take

α k = k -1 k + α -1 , λ k = kβ k k + α -1
, and assume that the sequence (β k ) satisfies the growth condition (H β ). Suppose that the sequence (e k ) satisfies the summability property

k≥1 k e k < ∞.
Then, for any sequence (x k ) generated by the algorithm (IPA) α k ,λ k ,e k , we have

(39) Φ(x k ) -min H Φ = O 1 k 2 β k and k≥1 Γ k Φ(x k+1 ) -min H Φ < +∞,
where

Γ k := k(k + α -1)β k -(k + 1) 2 β k+1 is non-negative by (H β ).
Proof. We use the same energy function as in the unperturbed case, namely

E k := k 2 β k (Φ(x k ) -m) + 1 2 v k 2 ,
where v k is defined by

v k := (α -1)(x k -z) + (k -1)(x k -x k-1 ).
A computation similar to that of the proof of Theorem 3.1 gives

(40) E k+1 -E k = (k + 1) 2 (β k+1 -β k ) + (2k + 1)β k (Φ(x k+1 ) -m) + k 2 β k (Φ(x k+1 ) -Φ(x k )) + 1 2 v k+1 2 -1 2 v k 2 .
Let's majorize the last above expression 1 2 v k+1 2 -1 2 v k 2 with the help of the convex inequality

1 2 v k+1 2 - 1 2 v k 2 ≤ v k+1 -v k , v k+1 .
According to the formulation (38) of the algorithm, we have

v k+1 -v k = (α -1)(x k+1 -x k ) + (x k -x k-1 ) + k(x k+1 -2x k + x k-1 ) = -kβ k ξ k + kg k . Hence v k+1 -v k , v k+1 = (α -1)kβ k ξ k , z -x k+1 + k 2 β k ξ k , x k -x k+1 + kg k , v k+1 ≤ (α -1)kβ k (Φ(z) -Φ(x k+1 )) + k 2 β k (Φ(x k ) -Φ(x k+1 )) + kg k , v k+1 ,
where the last inequality follows from α ≥ 1, the convexity of Φ, and ξ k ∈ ∂Φ(x k+1 ). As a consequence,

1 2 v k+1 2 - 1 2 v k 2 ≤ (α -1)kβ k (Φ(z) -Φ(x k+1 )) + k 2 β k (Φ(x k ) -Φ(x k+1 )) + kg k , v k+1 .
Combining the above inequality with [START_REF] Polyak | Some methods of speeding up the convergence of iteration methods[END_REF], and after simplification, we obtain

E k+1 -E k ≤ (k + 1) 2 β k+1 -kβ k (k + α -1) (Φ(x k+1 ) -Φ(z)) + kg k , v k+1 .
Hence

E k+1 -E k + Γ k (Φ(x k+1 ) -Φ(z)) ≤ kg k v k+1 . (41)
By assumption (H β ), Γ k is non-negative. Hence

E k+1 -E k ≤ kg k v k+1 .
Summing up the above inequalities obtained for j = 1, ..., k -1, and after reindexing, we obtain

E k ≤ E 1 + k j=2 (j -1)g j-1 v j . ( 42
)
By definition of E k , we have 1 2 v k 2 ≤ E k . Therefore, according to (42), we deduce that

v k 2 ≤ 2E 1 + 2 k j=2 (j -1)g j-1 v j . (43)
Let's apply the Gronwall Lemma 8.5 with

a k = v k and b k = (k -1) g k-1 . We obtain v k ≤ C := 2E 1 + 2 ∞ j=1 jg j

From the condition

k k e k < +∞, and e k = k k + α -1 g k , we have k k g k < +∞, and hence C is finite. Returning to [START_REF] Schmidt | Convergence rates of inexact proximal-gradient methods for convex optimization[END_REF], we obtain

E k ≤ E 1 + C ∞ j=1 jg j < +∞.
Hence, (E k ) is bounded from above, which gives the claim. Precisely,

Φ(x k ) -min H Φ ≤ E 1 + ( √ 2E 1 + 2 ∞ j=1 jg j ) ∞ j=1 jg j k 2 β k .
By arguing as in Theorem 3.1, we complete the proof of (39).

A general class of proximal algorithms with fast convergence properties

One can of course wonder if the fast convergence results obtained in the previous sections are specifically based on the type of discretization chosen in the section 2.3. We will show that there is some flexibility, and will present a whole family of proximal algorithms (IPA) α k ,λ k for which similar results are valid. They can be obtained by time discretization of (AVD) α,β , implicit with respect to the potential term, and semi-implicit with respect to the damping term according to a real parameter θ. Precisely, consider the following discretization of (AVD) α,β where we take directly a general convex lower semicontinuous proper function Φ: for k ≥ 1,

(44) (x k+1 -2x k + x k-1 ) + α -θ k (x k+1 -x k ) + θ k (x k -x k-1 ) + β k ∂Φ(x k+1 ) 0.
Equivalently,

(1 + α -θ k )(x k+1 -x k ) + β k ∂Φ(x k+1 ) (1 - θ k )(x k -x k-1 ),
which gives

x k+1 + kβ k k + α -θ ∂Φ(x k+1 x k + k -θ k + α -θ (x k -x k-1 ).
Setting

α k = k-θ k+α-θ and λ k = kβ k k + α -θ
, we end up with the inertial proximal algorithm

(IPA) α k ,λ k y k = x k + α k (x k -x k-1 ) x k+1 = prox λ k Φ (y k ).
Note that when θ = 1 we recover the previous scheme where the coefficients were taken equal to α k = k -1 k + α -1 and

λ k = kβ k k + α -1
. But, for a general θ, we must make an independent study of the algorithm.

7.1. Rate of convergence of the values. We will use the following equivalent formulation of the algorithm:

(45) k(x k+1 -2x k + x k-1 ) + (α -θ)(x k+1 -x k ) + θ(x k -x k-1 ) + kβ k ξ k = 0, with ξ k ∈ ∂Φ(x k+1 ). Theorem 7.1. Suppose α ≥ 1. Take α k = k -θ k + α -θ and λ k = kβ k k + α -θ
. Suppose that the sequence (β k ) satisfies the growth condition: there exists

k 1 ∈ N such that for all k ≥ k 1 (H β,θ ) β k+1 ≤ k(k + α -θ) (k + 1)(k + 2 -θ) β k .
Then, for any sequence (x k ) generated by the algorithm (IPA) α k ,λ k , we have

                   (i) Φ(x k ) -min H Φ = O 1 k 2 β k , (ii) k≥1 k 2 β 2 k ξ k 2 < +∞, with ξ k ∈ ∂Φ(x k+1 ), (iii) k≥1 Γ k,θ (Φ(x k+1 ) -min H Φ) < +∞ where Γ k,θ := k(k + α -θ)β k -(k + 1)(k + 2 -θ)β k+1 is non-negative by (H β,θ ).
Proof. Let us denote briefly m := min H Φ. Fix z ∈ argmin Φ, that is Φ(z) = min H Φ = m, and consider, for k ≥ 1, the energy function:

E k,θ := k(k + 1 -θ)β k (Φ(x k ) -m) + 1 2 v k 2 , with v k,θ := (α -1)(x k -z) + (k -θ)(x k -x k-1
). Let's look for conditions on (β k ) k so that the sequence (E k,θ ) k is non-increasing. To this end, we evaluate the term E k+1,θ -E k,θ . Unambiguously, we write v k for v k,θ in the following computation, but note that v k,θ is slightly different from the v k used in Theorem 3.1. By a similar computation as in Theorem 3.1, we have (46)

E k+1,θ -E k,θ = (k + 1)(k + 2 -θ)β k+1 (Φ(x k+1 ) -m) -k(k + 1 -θ)β k (Φ(x k ) -m) + 1 2 v k+1 2 -1 2 v k 2 = (k + 1)(k + 2 -θ)(β k+1 -β k ) (Φ(x k+1 ) -m) + (k + 1)(k + 2 -θ)β k (Φ(x k+1 ) -m) -k(k + 1 -θ)β k (Φ(x k ) -m) + 1 2 v k+1 2 -1 2 v k 2 = [(k + 1)(k + 2 -θ)(β k+1 -β k ) + (2k + 2 -θ)β k ] (Φ(x k+1 ) -m) +k(k + 1 -θ)β k (Φ(x k+1 ) -Φ(x k )) + 1 2 v k+1 2 -1 2 v k 2
where, to obtain the last relation, we used the relation

(k + 1)(k + 2 -θ) = k(k + 1 -θ) + (2k + 2 -θ).
Let's evaluate the last term of the equality above 1 2 v k+1 2 -1 2 v k 2 using the elementary algebraic equality

1 2 v k+1 2 - 1 2 v k 2 = v k+1 -v k , v k+1 - 1 2 v k+1 -v k 2 .
We have

v k+1 -v k = (α -1)(x k+1 -x k ) + (k + 1 -θ)(x k+1 -x k ) -(k -θ)(x k -x k-1 ) = (α -1)(x k+1 -x k ) + k(x k+1 -2x k + x k-1 ) + (1 -θ)(x k+1 -x k ) + θ(x k -x k-1 ) = k(x k+1 -2x k + x k-1 ) + (α -θ)(x k+1 -x k ) + θ(x k -x k-1 ) = -kβ k ξ k ,
with ξ k ∈ ∂Φ(x k+1 ), where the last equality comes from [START_REF] Su | A differential equation for modeling Nesterov's accelerated gradient method: theory and insights[END_REF]. Combining the above formula with the definition of v k , we obtain

v k+1 -v k , v k+1 = -kβ k ξ k , (α -1)(x k+1 -z) + (k + 1 -θ)(x k+1 -x k ) = (α -1)kβ k ξ k , z -x k+1 + k(k + 1 -θ)β k ξ k , x k -x k+1 ≤ (α -1)kβ k (Φ(z) -Φ(x k+1 )) + k(k + 1 -θ)β k (Φ(x k ) -Φ(x k+1 )) ,
where the last inequality follows from the convexity of Φ, and ξ k ∈ ∂Φ(x k+1 ). As a consequence,

1 2 v k+1 2 - 1 2 v k 2 ≤ (α -1)kβ k (Φ(z) -Φ(x k+1 )) + k(k + 1 -θ)β k (Φ(x k ) -Φ(x k+1 )) .
Combining the above inequality with (46), and after simplification, we obtain

E k+1,θ -E k,θ ≤ [(k + 1)(k + 2 -θ)(β k+1 -β k ) + (2k + 2 -θ)β k -(α -1)kβ k ] (Φ(x k+1 ) -Φ(z)) ≤ [(k + 1)(k + 2 -θ)β k+1 -kβ k (k + α -θ)] (Φ(x k+1 ) -Φ(z)) . Hence E k+1,θ -E k,θ + Γ k,θ (Φ(x k+1 ) -Φ(z)) ≤ 0, ( 47
) where Γ k,θ := k(k + α -θ)β k -(k + 1)(k + 2 -θ)β k+1 .
By assumption (H β,θ ), we have Γ k,θ ≥ 0 for all k ≥ k 1 , and hence E k+1,θ ≤ E k,θ . The sequence (E k,θ ) k≥k1 is non-increasing and minorized by zero. Consequently, it is convergent. By definition of E k,θ , we obtain, for all k ≥ k 1

k(k + 1 -θ)β k Φ(x k ) -min H Φ ≤ E k,θ ≤ E k1,θ . Consequently, Φ(x k ) -min H Φ = O 1 k 2 β k ,
that's item i). The end of the proof is similar to Theorem 3.1.

7.2. Rate of convergence of the velocities. To obtain fast convergence of velocities to zero, we need to introduce the following slightly strengthened version of (H β ).

Definition 7.2. We say that the sequence (β k ) satisfies the growth condition (H + β,θ ) if there exists k 1 ∈ N and ρ > 0 such that for all k ≥ k 1

(H + β,θ ) β k+1 ≤ k(k + α -θ -ρ(α -1)) (k + 1)(k + 2 -θ) β k .
Note that (H β,θ ) corresponds to the case ρ = 0. Let's give an equivalent form of (H + β,θ ) convenient for calculation: (48) ρ(α -1))kβ k ≤ Γ k,θ .

We can now establish the following rate of convergence for the velocities, and the acceleration.

Proposition 7.3. Suppose that α > 1 + θ 2 . Under condition (H β,θ ) + we have

+∞ k=1 k x k -x k-1 2 < +∞ and ∞ k=1 k 2 x k+1 + 2x k -x k-1 2 < +∞. Moreover ∞ k=1 kβ k Φ(x k+1 ) -min H Φ < +∞.
Proof. Consider, for k ≥ 1, the global energy function

W k := β k (Φ(x k ) -m) + 1 2 w k 2 , with m = inf H Φ and w k := x k -x k-1 . Let's evaluate the term (k + 1)(k + 2 -θ)W k+1 -k(k + 1 -θ)W k .
A similar computation as in Theorem 7.1 gives (49)

(k + 1)(k + 2 -θ)W k+1 -k(k + 1 -θ)W k = (k + 1)(k + 2 -θ)β k+1 (Φ(x k+1 ) -m) -k(k + 1 -θ)β k (Φ(x k ) -m) + (k+1)(k+2-θ) 2 w k+1 2 -k(k+1-θ) 2 w k 2 = [(k + 1)(k + 2 -θ)(β k+1 -β k ) + (2k + 2 -θ)β k ] (Φ(x k+1 ) -m) + k(k + 1 -θ)β k (Φ(x k+1 ) -Φ(x k )) + k(k+1-θ) 2 w k+1 2 -w k 2 + 2k+2-θ 2 w k+1 2 ≤ k(α -1)β k (Φ(x k+1 ) -m) + k(k + 1 -θ)β k (Φ(x k+1 ) -Φ(x k )) + k(k+1-θ) 2 w k+1 2 -w k 2 + 2k+2-θ 2 w k+1 2
where, to obtain the last relation, we used the hypothesis (H β,θ ). On the other hand,

1 2 w k+1 2 -1 2 w k 2 = -1 2 w k+1 -w k 2 + w k+1 -w k , w k+1 = -1 2 x k+1 -2x k + x k-1 2 + x k+1 -2x k + x k-1 , x k+1 -x k = -1 2 x k+1 -2x k + x k-1 2 -α-θ k (x k+1 -x k ) + θ k (x k -x k-1 ) + β k ξ k , x k+1 -x k with ξ k ∈ ∂Φ(x k+1
), where the last equality comes from [START_REF] Su | A differential equation for modeling Nesterov's accelerated gradient method: theory and insights[END_REF]. After multiplying by k(k + 1 -θ), we obtain

k(k+1-θ) 2 ( w k+1 2 -w k 2 ) = -k(k+1-θ) 2 x k+1 -2x k + x k-1 2 -(α -θ)(x k+1 -x k ) + θ(x k -x k-1 ) + kβ k ξ k , (k + 1 -θ)(x k+1 -x k ) ≤ -k(k+1-θ) 2 x k+1 -2x k + x k-1 2 -(α -θ)(k + 1 -θ) x k+1 -x k 2 -θ(k + 1 -θ) x k+1 -x k , x k -x k-1 -k(k + 1 -θ)β k (Φ(x k+1 ) -Φ(x k )) ,
where the last inequality follows from the convexity of Φ, and ξ k ∈ ∂Φ(x k+1 ).

Combining the above inequality with (49), and after simplification, we obtain

(k + 1)(k + 2 -θ)W k+1 -k(k + 1 -θ)W k + k(k + 1 -θ) 2 x k+1 -2x k + x k-1 2 ≤ (α -1)kβ k (Φ(x k+1 ) -m) -(α -θ)(k + 1 -θ) x k+1 -x k 2 -θ(k + 1 -θ) x k+1 -x k , x k -x k-1 + 2k + 2 -θ 2 x k+1 -x k 2 . Equivalently (k + 1)(k + 2 -θ)W k+1 -k(k + 1 -θ)W k + A k ≤ (α -1)kβ k (Φ(x k+1 ) -m) ,
where

A k := k(k + 1 -θ) 2 w k+1 -w k 2 + (α -θ)(k + 1 -θ) w k+1 2 + θ(k + 1 -θ) w k+1 , w k - 2k + 2 -θ 2 w k+1 2 .

By elementary algebraic operations

A k = k(k + 1 -θ) 2 w k+1 -w k 2 + (α -θ)(k + 1 -θ) w k+1 2 + 1 2 θ(k + 1 -θ) w k+1 2 + 1 2 θ(k + 1 -θ) w k 2 - 1 2 θ(k + 1 -θ) w k+1 -w k 2 - 2k + 2 -θ 2 w k+1 2 = (k + 1 -θ)(k -θ) 2 w k+1 -w k 2 + (α -1 - θ 2 )k + α -θα + θ 2 2 -1 w k+1 2 + 1 2 θ(k + 1 -θ) w k 2 .
For α > 1 + θ 2 , and k sufficiently large, all the above quantities are non-negative. Hence

(k + 1)(k + 2 -θ)W k+1 -k(k + 1 -θ)W k + (α -1 -θ 2 )k + α -θα + θ 2 2 -1 x k+1 -x k 2 + (k+1-θ)(k-θ) 2 x k+1 + 2x k -x k-1 2 ≤ (α -1)kβ k (Φ(x k+1 ) -m) .
By condition (H β,θ ) + , as formulated in (48), we have ρ(α -1)kβ k ≤ Γ k,θ for some ρ > 0, and k sufficiently large. Hence

(k + 1)(k + 2 -θ)W k+1 -k(k + 1 -θ)W k + (α -1 -θ 2 )k + α -θα + θ 2 2 -1 x k+1 -x k 2 + (k+1-θ)(k-θ) 2 x k+1 + 2x k -x k-1 2 ≤ 1 ρ Γ k,θ (Φ(x k+1 ) -m) . (50)
Let's sum the above inequalities for k ≥ k 1 . According to the estimation k≥1 Γ k,θ (Φ(x k+1 ) -min H Φ) < +∞ (see Theorem 7.1 (iii)), we obtain

∞ k=1 k x k+1 -x k 2 < +∞. and ∞ k=1 k 2 x k+1 + 2x k -x k-1 2 < +∞,
which gives the claim. This results from the following elementary majorizations. From (H β,θ ),

(k + 1)β k+1 ≤ kβ k k + α -θ k + 2 -θ ≤ 2kβ k
where the last inequality is valid for k

≥ α -4 + θ. Hence ∞ k=1 (k + 1)β k+1 Φ(x k+1 ) -min H Φ ≤ 2 ∞ k=1 kβ k Φ(x k+1 ) -min H Φ < +∞,
which gives the result, after reindexation.

7.3.

From O to o estimates. In a parallel way to Theorem 3.5, we are going to prove the following result.

Theorem 7.5. Take

α k = k -θ k + α -θ and λ k = kβ k k + α -θ with α > 1 + θ 2 , θ ∈ R.
Suppose that the sequence (β k ) satisfies the growth condition (H + β,θ ). Then, for any sequence (x k ) generated by the algorithm (IPA) α k ,λ k , we have

Φ(x k ) -min H Φ = o 1 k 2 .
Proof. Let's consider the sequence of global energies (W k )

W k := β k (Φ(x k ) -m) + 1 2 x k -x k-1 2
.

By Proposition 7.3, we have

+∞ k=1 k x k -x k-1 2 < +∞ and ∞ k=1 kβ k (Φ(x k ) -min H Φ) < +∞, see Remark 7.4 formula (51). Hence ∞ k=1 kW k < +∞.
On the other hand, returning to (50) we have

(k + 1)(k + 2 -θ)W k+1 -k(k + 1 -θ)W k ≤ 1 ρ Γ k,θ (Φ(x k+1 ) -m) .
The nonnegative sequence (a k ) with

a k = k(k + 1 -θ)W k satifies the relation a k+1 -a k ≤ ω k with ω k = 1 ρ Γ k,θ (Φ(x k+1 ) -m). According to k≥1 Γ k,θ (Φ(x k+1 ) -min H Φ) < +∞ (see Theorem 7.1 (iii)), we have (w k ) ∈ l 1 (N)
. By a standard argument, we deduce that the limit of the sequence (a k ) exists, that is

lim k→+∞ k(k + 1 -θ)W k = lim k→+∞ k 2 W k exists.
(52) h k+1 -h k -α k (h k -h k-1 ) ≤ 1 2 (α 2 k + α k ) x k -x k-1 2 -λ k (Φ(x k+1 ) -min H Φ) - 1 2 y k -prox λ k Φ (y k ) 2 ,
is valid for any algorithm (IPA) α k ,λ k , and hence it is valid in our setting,

α k = k -θ k + α -θ and λ k = kβ k k + α -θ . From
this result, in parallel to Theorem 4.2, we will deduce the convergence of the iterates.

Theorem 7.6. Take

α k = k -θ k + α -θ and λ k = kβ k k + α -θ with α > 1 + θ 2 , θ ∈ R.
Assume (H β,θ ) + . Then, any sequence (x k ) generated by algorithm (IPA) α k ,λ k converges weakly, and its limit belongs to argmin Φ.

Proof. We apply the Opial lemma, see Lemma 8.3. (ii) Let us now fix x * ∈ argmin Φ, and show that lim k→+∞ x k -x * exists. For that purpose, let us set h k = 1 2 x k -x * 2 . From (52), the sequence (h k ) satisfies the following inequalities

h k+1 -h k -α k (h k -h k-1 ) ≤ 1 2 (α 2 k + α k ) x k -x k-1 2 ≤ x k -x k-1 2 since α k ∈ [0, 1].
Taking the positive part, we find

(h k+1 -h k ) + ≤ α k (h k -h k-1 ) + + x k -x k-1 2 .
From Proposition 3.3, we have +∞ k=1 k x k -x k-1 2 < +∞. By applying Lemma 8.4 (given in the appendix) with a k = (h k -h k-1 ) + and ω k = x k -x k-1 2 , we obtain +∞ k=1 (h k -h k-1 ) + < +∞.

Since (h k ) is nonnegative, this classically implies that lim k→+∞ h k exists. The second point of the Opial lemma is shown, which ends the proof. , we need to take β k → +∞ to get an improved convergence rate compared to the classical situation. For simplicity of the presentation, we take k 1 = 1, the extension to a general k 1 is straightforward. As a model situation, take β k = µk δ with µ > 0. Then,

(H β,θ ) ⇐⇒ (k + 1) δ ≤ k(k + α -θ) (k + 1)(k + 2 -θ) k δ ⇐⇒ (k + 1) δ+1 (k + 2 -θ) ≤ k δ+1 (k + α -θ) ⇐⇒ (1 + 1 k ) δ+1 (1 + 2 -θ k ) ≤ 1 + α -θ k . ( 53
)
For k large, 1 k is close to zero. Then, the left member of the above inequality is equivalent to 1 + δ+3-θ k . So inequality (53) is satisfied for k sufficiently large if δ + 3 -θ < α -θ, that is δ < α -3. As a striking prpoerty, note that the condition is independent of θ. It is the same as the one obtained in the case θ = 1. Thus, if α > 3, we can take β k = k δ for any 0 ≤ δ < α -3. In addition, we have

Γ k,θ = k(k + α -θ)β k -(k + 1)(k + 2 -θ)β k+1 = µk δ+1 (k + α -θ) -µ(k + 1) δ+1 (k + 2 -θ) = µ(α -3 -δ)k δ+1 + • k δ+1 .
Once more we can observe that the result is indepent of θ. Thus, with δ < α -3, the condition (H β,θ ) is satisfied, and we have the following results: x k+1 = prox β k Φ (y k ), As a particular case, take β k ≡ µ > 0. This corresponds to δ = 0 in the above model example, which fits the condition 0 ≤ δ < α -3, since α has been supposed strictly greater than 3. Doing so, we recover the classical results concerning the proximal method based on Nesterov's accelerated scheme, see [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] Beck | A fast iterative shrinkage-thresholding algorithm for linear inverse problems[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]. In particular, when α > 3, we

lim k→+∞ k 2 W 3 . 4 .

 234 k exists. Let c := lim k→+∞ k 2 W k . Hence kW k ∼ c k . According to ∞ k=1 kW k < +∞, we must have c = 0. Hence, lim k→+∞ k 2 W k = 0, which gives the claim. On the condition (H) β and (H) + β . According to the formula Φ(x k+1 ) -min Φ = O 1 k 2 β k

Remark 7 . 4 .

 74 In Proposition 7.3 we proved that, under condition (Hβ,θ ) + , ∞ k=1 kβ k (Φ(x k+1 ) -min H Φ) < +∞.Let's show that the following estimates holds too:(51) ∞ k=1 kβ k Φ(x k ) -min H Φ < +∞.

7 . 4 .

 74 Let c := lim k→+∞ k 2 W k . Hence kW k ∼ c k . According to ∞ k=1 kW k < +∞, we must have c = 0. Hence, lim k→+∞ k 2 W k = 0, which gives the claim. Convergence of iterates. Fix x * ∈ argmin Φ, and define the sequence (h k ) by h k = 1 2 x k -x * 2 . The result of Proposition 4.1

  (i) By Theorem 7.5 we have Φ(x k ) -min H Φ = o 1 k 2 and hence lim k→+∞ Φ(x k ) = min H Φ. Assume that there exist x ∈ H and a sequence (k n ) such that k n → +∞, and x knx weakly as n → +∞. Since the convex function Φ is lower semicontinuous, it is lower semicontinuous for the weak topology, hence satisfiesΦ(x) ≤ lim inf n→+∞ Φ(x kn ) = lim k→+∞ Φ(x k ) = min Φ.It ensues that x ∈ argmin Φ, which shows the first point.

7. 5 .

 5 The caseβ k = µk λ . According to the formula Φ(x k+1 ) -min Φ = O 1 k 2 β k

Corollary 7 . 7 . 7 . 6 .

 7776 Let θ ∈ R, and µ > 0 arbitrarily chosen. Given α > 3, takeα k = 1 -α k + α -θ , λ k = µ k δ+1 k + α -θ with 0 ≤ δ < α -3. Then, for any sequence (x k ) generated by the algorithm (IPA) α k ,λ k , we have k ) -min Φ = o 1 k 2+δ ; +∞ k=1 k 2(1+δ) ξ k 2 < +∞ with ξ k ∈ ∂Φ(x k+1 ); +∞ k=1 k δ+1 (Φ(x k+1 ) -min H Φ) < +∞.; +∞ k=1 k x k -x k-1 2 < +∞. Some examples.Depending on the choice of θ, we obtain a specific algorithm. It is worth noticing that the corresponding convergence rates do not depend on θ, and therefore of the type of discretization chosen for the damping term. This is a new result compared to the classical situation (considered below) where the explicit discretization of the damping term is used. Let's consider the following cases of particular interest: a) Case θ = α: it corresponds to the classical explicit discretization of the damping term(54) (x k+1 -2x k + x k-1 ) + α k (x k -x k-1 ) + β k ∂Φ(x k+1 ) 0, which gives the algorithm (IPA) α k ,λ k with α k = 1 -α k and λ k = β k :    y k = x k + (1 -α k )(x k -x k-1 )

  have Φ(x k ) -min Φ = o 1 k 2 .b) Case θ = 1: it corresponds to the semi-implicit discretization of the damping term(55) (x k+1 -2x k + x k-1 ) + α -1 k (x k+1 -x k ) + 1 k (x k -x k-1 ) + β k ∇Φ(x k+1 ) = 0.It provides the algorithm studied in the previous sections withα k = 1 -α k + α -1 and λ k = kβ k k + α -1 .

c) Case θ = 0: it corresponds to the implicit discretization of the damping term

This gives the algorithm (IPA) α k ,λ k with α k = 1 -α k + α and λ k = kβ k k + α :

8. Auxiliary results

8.1. Continuous dynamics. Recall the continuous evolution system (AVD) α,β defined in [START_REF] Attouch | Accelerated forward-backward algorithms with perturbations. Application to Tikhonov regularization[END_REF], that served as a guide for the introduction of the inertial proximal algorithms (IPA) α k ,λ k :

In the following theorem, we specify the hypotheses on the parameters α and β that guarantee the existence and uniqueness of global trajectories for the Cauchy problem associated with (AVD)

Then, for any x 0 and v 0 in H, the (AVD) α,β system has a unique twice continuously differentiable global solution

Moreover, the trajectory is bounded and satisfies the convergence rate: as t → +∞,

Proof. First write (AVD) α,β as a first-order system, for example (58) ẋ(t) = y(t)

Local existence and uniqueness follows classically from the Cauchy-Lipschitz theorem. Then, passing from local to global existence will result from global estimates on the trajectory. Just like for the algorithm, a key point is to prove that the trajectory remains bounded. We follow a parallel argument to the algorithmic case. Given z ∈ argmin Φ, we introduce the energy function

that will serve as a Lyapunov function. By classical differential calculus, using equation (AVD) α,β , and a convex differential inequality, we obtain

By assumption (H β ), we have Γ(t) ≥ 0, which implies that E(•) is non-increasing on [t 0 , +∞[. Therefore, it is bounded from above, which gives (57). In addition, (α -1)(x(t) -z) + t ẋ(t) 2 is bounded above by a constant which, after development, gives

Integration of this inequality immediately gives that h(•), and hence the trajectory x(•), is bounded. Then the solution does not blow up in any finite time interval. By a standard argument we deduce that (58), and hence (AVD) α,β , has a unique maximal solution on [t 0 , +∞[ verifying the Cauchy data

Remark 8.2. Recall the growth condition on the sequence (β k ) that has been used in Theorem 3.1

It can be equivalently written as

2 β k , which can be viewed as a discretized version of the condition used in the continuous evolution system

This justifies the use of the same terminology (H β ) for continuous and discrete cases.

8.2. Discrete case. Let us state the discrete version of Opial's lemma.

Lemma 8.3. Let S be a non empty subset of H, and (x k ) a sequence of elements of H. Assume that (i) every weak sequential cluster point of (x k ), as k → ∞, belongs to S.

(ii) for every z ∈ S, lim k→+∞ x k -z exists. Then x k converges weakly as k → ∞ to a point in S.

The following result allows us to establish the summability of a nonnegative sequence (a k ) satisfying some suitable inequality. Let's recall that in our setting

Let (a k ) and (ω k ) be two sequences of nonnegative numbers such that, for all k ≥ 0, (60)


Paul Stief
email: paul.stief@ensam.eu

Jean-Yves Dantan

Alain Etienne

Ali Siadat

Uzair Khaleeq Uz Zaman
email: uzair-khaleeq-uz.zaman@ensam.eu

Mickael Rivette

Ahmed Baqai

A new methodology to analyze the functional and physical architecture of existing products for an assembly oriented product family identification

Keywords: Assembly, Design method, Family identification Additive manufacturing, Decision making, Integrated product-process design Additive manufacturing, Decision making, Integrated product-process design

In today's business environment, the trend towards more product variety and customization is unbroken. Due to this development, the need of agile and reconfigurable production systems emerged to cope with various products and product families. To design and optimize production systems as well as to choose the optimal product matches, product analysis methods are needed. Indeed, most of the known methods aim to analyze a product or one product family on the physical level. Different product families, however, may differ largely in terms of the number and nature of components. This fact impedes an efficient comparison and choice of appropriate product family combinations for the production system. A new methodology is proposed to analyze existing products in view of their functional and physical architecture. The aim is to cluster these products in new assembly oriented product families for the optimization of existing assembly lines and the creation of future reconfigurable assembly systems. Based on Datum Flow Chain, the physical structure of the products is analyzed. Functional subassemblies are identified, and a functional analysis is performed. Moreover, a hybrid functional and physical architecture graph (HyFPAG) is the output which depicts the similarity between product families by providing design support to both, production system planners and product designers. An illustrative example of a nail-clipper is used to explain the proposed methodology. An industrial case study on two product families of steering columns of thyssenkrupp Presta France is then carried out to give a first industrial evaluation of the proposed approach.

Introduction

Due to the fast development in the domain of communication and an ongoing trend of digitization and digitalization, manufacturing enterprises are facing important challenges in today's market environments: a continuing tendency towards reduction of product development times and shortened product lifecycles. In addition, there is an increasing demand of customization, being at the same time in a global competition with competitors all over the world. This trend, which is inducing the development from macro to micro markets, results in diminished lot sizes due to augmenting product varieties (high-volume to low-volume production) [START_REF] Ranjan | Design for manufacturability in additive manufacturing using graph based approach[END_REF]. To cope with this augmenting variety as well as to be able to identify possible optimization potentials in the existing production system, it is important to have a precise knowledge of the product range and characteristics manufactured and/or assembled in this system. In this context, the main challenge in modelling and analysis is now not only to cope with single products, a limited product range or existing product families, but also to be able to analyze and to compare products to define new product families. It can be observed that classical existing product families are regrouped in function of clients or features. However, assembly oriented product families are hardly to find.

On the product family level, products differ mainly in two main characteristics: (i) the number of components and (ii) the type of components (e.g. mechanical, electrical, electronical).

Classical methodologies considering mainly single products or solitary, already existing product families analyze the product structure on a physical level (components level) which causes difficulties regarding an efficient definition and comparison of different product families. Addressing this 1. Introduction

Motivation

Design of a product requires not only the satisfaction of requirements related to the functionality of the part, but also the realization of manufacturing process-related constraints. Since manufacturing is no more about building mere physical products, market drivers such as economics of production, mass customization, shorter lead times, improved business models, changes in consumer demands, nature of products, and broad areas of applications (aerospace, automotive, health care, etc.), have guided the stakeholders to take important decisions in early stages of design, and subsequently reduce the costs of assembly and logistics [START_REF] Ranjan | Design for manufacturability in additive manufacturing using graph based approach[END_REF].

Additive Manufacturing (AM), over the years, has displayed the potential to build anything. AM is defined by American Society for Testing Materials (ASTM) as the "process of joining materials to make objects from 3D model data usually layer upon layer, as opposed to subtractive manufacturing technologies like traditional machining" [START_REF]Standard Terminology for Additive Manufacturing Technologies, Standard F2792-12a[END_REF]. Moreover, Monzon et al. [START_REF] Monzon | Standardization in Additive Manufacturing: Activities Carried out by International Organizations and Projects[END_REF] divided AM in to 7 distinct areas; vat photopolymerization, material jetting, powder bed fusion, binder jetting, material extrusion, sheet lamination, and directed energy deposition. For each of these 7 areas, the associated AM processes are many (Stereolithography, Fused Deposition Modeling, Selective Laser Sintering, etc.). Therefore, the constraints involved in 3D modeling procedures for each process, digitization of ideas [START_REF] Huang | Additive Manufacturing: Current State, Future Potential, Gaps and Needs, and Recommendations[END_REF], discretization of the produced parts, choosing the right balance of materials, manufacturing processes and AM machines, limitations in material selection, longer design cycle compared to manufacturing cycle, post processing and surface finish issues, etc., are few of the many factors the designers working in the AM industry must concentrate on today [START_REF] Vaezi | Multiple material additive manufacturing-part 1: a review[END_REF][START_REF] Cozmei | Additive Manufacturing Flickering at the Beginning of Existence[END_REF].

With various constraints, functional requirements, and design tradeoffs related to product performance, the AM designers have been convinced to use the Design for Additive

Introduction

Motivation

Design of a product requires not only the satisfaction of requirements related to the functionality of the part, but also the realization of manufacturing process-related constraints. Since manufacturing is no more about building mere physical products, market drivers such as economics of production, mass customization, shorter lead times, improved business models, changes in consumer demands, nature of products, and broad areas of applications (aerospace, automotive, health care, etc.), have guided the stakeholders to take important decisions in early stages of design, and subsequently reduce the costs of assembly and logistics [START_REF] Ranjan | Design for manufacturability in additive manufacturing using graph based approach[END_REF].

Additive Manufacturing (AM), over the years, has displayed the potential to build anything. AM is defined by American Society for Testing Materials (ASTM) as the "process of joining materials to make objects from 3D model data usually layer upon layer, as opposed to subtractive manufacturing technologies like traditional machining" [START_REF]Standard Terminology for Additive Manufacturing Technologies, Standard F2792-12a[END_REF]. Moreover, Monzon et al. [START_REF] Monzon | Standardization in Additive Manufacturing: Activities Carried out by International Organizations and Projects[END_REF] divided AM in to 7 distinct areas; vat photopolymerization, material jetting, powder bed fusion, binder jetting, material extrusion, sheet lamination, and directed energy deposition. For each of these 7 areas, the associated AM processes are many (Stereolithography, Fused Deposition Modeling, Selective Laser Sintering, etc.). Therefore, the constraints involved in 3D modeling procedures for each process, digitization of ideas [START_REF] Huang | Additive Manufacturing: Current State, Future Potential, Gaps and Needs, and Recommendations[END_REF], discretization of the produced parts, choosing the right balance of materials, manufacturing processes and AM machines, limitations in material selection, longer design cycle compared to manufacturing cycle, post processing and surface finish issues, etc., are few of the many factors the designers working in the AM industry must concentrate on today [START_REF] Vaezi | Multiple material additive manufacturing-part 1: a review[END_REF][START_REF] Cozmei | Additive Manufacturing Flickering at the Beginning of Existence[END_REF].

With various constraints, functional requirements, and design tradeoffs related to product performance, the AM designers have been convinced to use the Design for Additive Manufacturing (DfAM) guidelines to develop an integrated approach in the design stage which can not only integrate design and manufacturing [referred to as Integrated Product Process Design (IPPD) from now on] but also cater for many manufacturing constraints and factors linked with traditional machining, such as avoiding separate fasteners, developing a modular design, using standard components, and minimizing assembly directions, to obtain parts of any geometric complexity without traditional machining aids such as tooling [START_REF] Hague | Design Opportunities with Rapid Manufacturing[END_REF][START_REF] Hopkinson | Emerging Rapid Manufacturing Processes[END_REF][START_REF] Kuo | Design for Manufacture and Design for 'X': Concepts, Applications and Perspectives[END_REF]. It is also imperative to understand here that as the design matures from preliminary level to full scale production, the freedom to change the design is decreased considerably. It further implies that DfAM becomes increasingly significant in the early phases of design as it can help in avoiding manufacturing pitfalls and maximize utilization of AM capability [START_REF] Rosen | Computer-aided design for additive manufacturing of cellular structures[END_REF][START_REF] Adam | Design for Additive Manufacturing-Element transitions and aggregated structures[END_REF].

DfAM has been used in researches related to tool path optimization, part orientation, light weight design, and dimensional accuracy [START_REF] Ponche | A novel methodology of design for Additive Manufacturing applied to Additive Laser Manufacturing process[END_REF]. Yazdi et al. [START_REF] Yazdi | Integrated design for additive manufacturing based on skin-skeleton approach[END_REF] used skin-skeleton model to integrate design and manufacturing via DfAM for topological optimization. Salonitis and Zarban [START_REF] Salonitis | Redesign optimization for manufacturing using additive layer techniques[END_REF] used multi criteria decision analysis to find one final design from a set of optimized designs via DfAM, too. Hence, considerable work can be seen in literature on the optimization of products and authors following either of the 'function-driven design strategy' or 'manufacturing-driven design strategy' [START_REF] Klahn | Design Strategies for the Process of Additive Manufacturing[END_REF] wherein more emphasis is put on the modification of Design for Manufacturing (DFM) for AM (DfAM) using a combination of design criteria (cost, environment, etc.). But such guidelines just provide a starting point and do not provide information related to AM machines and their production capabilities [START_REF] Thompson | Design for Additive Manufacturing: Trends, Opportunities, Considerations and Constraints[END_REF].

Resource Selection (RS) is an important phase of DFM. As AM itself has a healthy blend of processes, the suggestion of the correct combination of materials, manufacturing processes and associated machines, becomes an interdisciplinary effort considering AM's capability to be both inclined towards concurrent development (i.e., IPPD) and governing multiple areas of applications. Although AM has unlimited potential, but it doesn't guarantee having unlimited capability. The selected AM resource must satisfy requirements pertaining to product's lifecycle by considering factors such as design engineering, manufacturing, marketing, esthetics, reliability and quality [START_REF] Zaman | Integrated Product-Process Design to Suggest Appropriate Manufacturing Technology: A Review[END_REF].

As a usual practice, RS is conducted based on experience or knowledge of the designing/manufacturing personnel. Since, knowing everything by a person or a team is difficult, a comprehensive and robust selection system is necessary for the users to select a compromised AM material-process-machine combination(s) [START_REF] Wang | Selection of additive manufacturing processes[END_REF]. For the case of traditional or conventional manufacturing, various multi-criteria decision making (MCDM) methods have been proposed for RS, such as, Cambridge Engineering Selector (based on Ashby's material and process selection charts) [START_REF] Ashby | Material and Process Charts. The CES EduPack Resource Booklet 2[END_REF], Case-based Reasoning (CBR) [START_REF] Berman | Application of Case-Based Reasoning and Multi-Criteria Decision Making Methods for Material Selection in Petrochemistry[END_REF], material selection programs [START_REF] Kesteren | Materials in product selection: tools for including user-interaction in materials selection[END_REF], Knowledgebased Systems (KBS) [START_REF] Ipek | An expert system based material selection approach to manufacturing[END_REF], etc. However, for the case of AM, different processes not only display considerable overlap in terms of possible applications, but also significant differences exist in terms of suitable materials and quality of printed parts [START_REF] Wang | Selection of additive manufacturing processes[END_REF]. Therefore, being a relatively new technology, most users do not have enough knowledge and experience to make good judgments for RS in AM. Mancanares et. al [START_REF] Mancanares | Additive Manufacturing Process Selection Based on Parts' Selection Criteria[END_REF] used Analytical Hierarchy Process (AHP) to select AM processes based on the requirements generated from a part. Similarly, Armillotta [START_REF] Armillotta | Selection of Layered Manufacturing Techniques by an Adaptive AHP Decision Model[END_REF] selected a suitable AM process from a set of alternatives for prototypes by using an adaptive AHP decision model. The attributes considered included fast build, good accuracy, and reduced material cost. This also open a window of opportunity to apply AHP in RS for AM since it is the most widely and successfully used MCDM method [START_REF] Zaman | Integrated productprocess design: material and manufacturing process selection for additive manufacturing using multi-criteria decision making[END_REF]. It is apparent from the literature reviewed that AHP has been applied extensively on not only small-and large-scale problems but also on those areas that have multiple criteria. Moreover, it is suitable for areas such as manufacturing sector since it relies on the innate human inclination to conduct comparison by considering both subjective and objective attributes [START_REF] Emrouznejad | The state of the art development of ahp (1979-2017): a literature review with a social network analysis[END_REF]. It is applied to material selection in gears [START_REF] Yazdani | Analysis in material selection: influence of normalization tools on COPRAS-G[END_REF], selection of best material for design of lightweight aircraft metallic structures [START_REF] Adhikari | Study of knowledge-based system (KBS) and decision-making methodologies in materials selection for lightweight aircraft metallic structures[END_REF], selection of nontraditional machining processes in conceptual design stage for the body of modular hip joint endoprosthesis [START_REF] Lukic | Multi-criteria selection of manufacturing processes in the conceptual process planning[END_REF], and in material and process selection for a drilling grid in aerospace industry [START_REF] Zaman | Integrated productprocess design: material and manufacturing process selection for additive manufacturing using multi-criteria decision making[END_REF].

Having discussed the avenues of IPPD and RS for AM, it is now imperative to shed some light on the design cycle. The design activity can be divided into 3 main stages; conceptual, embodiment and detail designs. For each of the stages, the coupling between materials, component size and processes, cost interaction among processes, and sustainability indicators of materials, need to be considered [START_REF] Gupta | A system for generating process and material selection advice during embodiment design of mechanical components[END_REF]. Therefore, strategies such as rule-based system approach [START_REF] Zarandi | A material selection methodology and expert system for sustainable product design[END_REF] have been widely used to help in knowledge acquisition, choosing the selection criteria, building hierarchical definition of knowledge, selection of a user interface, and finally the implementation. But prior to all this, it is necessary to capture the voice of the customer in terms of needs, specifications, aesthetic preferences, and constraints, to formulate requirements and functionality [START_REF] Deng | The role of materials identification and selection in engineering design[END_REF]. Few studies [START_REF] Gupta | A system for generating process and material selection advice during embodiment design of mechanical components[END_REF][START_REF] Albinana | A framework for concurrent material and process selection during conceptual product design stages[END_REF] have worked on the RS in conceptual and embodiment design stages, respectively.

Considering the literature reviewed, it is evident that there is a need to rely on a systematic approach for RS in AM to capture the design requirements and structure the design knowledge for each stage of design process. Secondly, since RS is best dealt by decision making and as multiple criteria and attributes related to both product and process are involved, guidelines or axioms need to be defined from the literature reviewed to aid in decision making for each of the design criteria. Moreover, the approach in this paper compliments the existing methodologies and further attempts to develop a knowledge management system that can gather and provide feasible material-machineprocess combinations for AM designers. The research is a first step towards a holistic approach to support RS in AM.

Structure

The remainder of the paper is divided as follows: Section 2 explains the proposed methodology by focusing on structuring the design knowledge for decision making in each of the conceptual and embodiment design stages; Section 3 attempts to validate the 'decision-making' part of the framework by using an example; and finally, Section 4 discusses the conclusions drawn.

Integrated design-oriented framework for RS in AM

The integrated design-oriented framework proposed in this paper follows a step by step procedure for RS (material, machine and process selection) in AM. The framework is globally impacted by the DfAM guidelines and application type, and locally follows three major steps; translation, screening and ranking. It considers 3 design criteria; function, cost and environment. It also shows an interaction with 2 independent databases; one for the AM materials and the second for AM process-machine combinations. To decrease the cost related to manufacturability of a part, the proposed framework further works deeply in the conceptual and embodiment design phases with respect to decision-making in each phase. Furthermore, to structure the decision hierarchy, the framework explores the potentials of AM and suggests respective measures with the help of reviewed literature (see Fig. 1). The inner circle represents the potentials of AM such as complexity for free, individualization, etc., while the outer circle shows the measures that need to be taken to achieve each of the shown potentials. The text to follow will explore avenues of decision making in each of the design stages. Also, an example of how axioms are defined for one design criteria; environment, will be shown along with the mechanism to extract rules.

RS in Conceptual Design

In the context of the proposed framework, once the set of requirements are generated (design-, production-, and/or process-related), the information is routed to the "Conceptual design decision space" (see Fig. 2). Since decision-making in the development of products requires collaboration among different teams, the knowledge generated must be managed in a way that a compromised, yet 'win-win' solution is available for all stakeholders [START_REF] Albinana | A framework for concurrent material and process selection during conceptual product design stages[END_REF]. The decision space is shown in Fig. 3.

Each of the design criteria are associated with unique decisions that act as selection fronts for the RS. They are termed as technical decisions, economic decisions and sustainability decisions, respectively. Technical decisions are related to the performance of the product; viability and cost preferences are governed by the economic decisions; while sustainability decisions are related to environmental impact of AM materials in terms of landfill waste and recyclability.

As shown in Fig. 3, the AM materials and processes impact all 3 decisions. However, AM machines only impact technical and economic decisions since the current study is analyzing environmental aspects related to materials and processes only. Moreover, the generated product-process requirements directly impact the decisions and AM materials, while the constraints structure the selection procedures to determine the resources.

The decision hierarchy of Fig. 3 is based on Fig. 1 and shows how some of the many possibilities can interact amidst multiple criteria, attributes and deliverables. The result of the conceptual design stage includes the release of preliminary materials and machine-process combinations for AM to the embodiment design stage.

RS in Embodiment Design

The embodiment design stage is referred to as the 'detailed inspiration' wherein a design is developed in accordance with engineering and economic criteria. However, in the proposed framework, the subject stage governs the screening and ranking of the obtained results from Sec 2.1. The databases, design attributes (along with design guidelines for each design criteria) and MCDM tool(s) are further used to get the final combination(s) of AM materials, machines and processes along with the application of a cost model. The cost model was used to calculate the overall material cost and was adopted by Yim and Rosen [START_REF] Yim | Build Time and Cost Models for Additive Manufacturing Process Selection[END_REF] as given below:

M = Ks . Kr . N . v . Cm . ρ (1)
where, M = overall material cost (US$), Ks = support structure factor, Kr = recycling factor, N = number of parts, v = part volume (mm 3 ), Cm = material rate per unit weight (US$/kg) and  = material density (kg/mm 3 ). Ks is used to capture cost of additional material usage for building support structures and is usually in the range of 1.1 -1.5 while Kr is used to find the cost contribution of wasting loose powder which is not recycled after the build. Kr usually lies in the range of 1 -7.

Axioms for design criteria: Environment (Example)

The axioms defined in the framework related to environment as a design criterion are many, but few are listed below [START_REF] Albinana | A framework for concurrent material and process selection during conceptual product design stages[END_REF]:

• Avoid toxic or harmful materials • Avoid additives that emit harmful and toxic substances • Avoid materials which emit harmful and toxic substances during disposal • Use renewable materials • Use materials with low energy consumption The axioms stated above are very generic in nature and may not help the decision makers to reach a feasible solution. Therefore, rules were extracted based on IF-THEN representation. For example, for the first axiom "Avoid toxic or harmful materials", the decision tree looks like as shown in Fig. 4. The acceptance and rejection rules for the shaded strings in Fig. 4 are as shown below: • IF the AM material is not toxic and harmful AND the material is recyclable OR the material is renewable OR the material is biodegradable THEN use the material

• IF the material is not toxic and harmful AND the material is not recyclable AND the material is not renewable AND the material is not biodegradable THEN the material can be scraped/retrieved

• IF the material can be scraped/retrieved AND the material has high energy consumption THEN avoid the material

Example

To validate the decision-making system of the proposed framework, an example of a crank was chosen (see Fig. 5). This part is used in high speed motor applications for mechanical power (low power applications). It is assumed that the working environment may be rough. Since volume of most of the building chambers of AM machines are in X, Y and Z dimensions, the volume of the part was taken as a cuboid (60x16x17) mm 3 . The objective was to minimize weight, maximize strength and make the product recyclable. Two design criteria; function and environment, were hence, considered. The constraints included dimensional tolerance of 1/10 mm and part build up using a polymer. The part can be modeled as a 'column' for ease of application during conceptual design phase. The potential, 'Complexity for free', was used in this case (see Fig. 1) and the measure, 'reduce weight', was further explored in the decision space of conceptual design. Two decision fronts; technical and sustainability, were considered for the example part (see Fig. 3).

The material attributes included material strength properties, surface finish, environmental impact and landfill waste, while the machine attributes included geometry complexity, accuracy, minimum layer thickness, build volume, and build speed. For the sake of simplicity, only the results of each stage are shown and discussed.

RS in Conceptual Design

In view of the functional requirements of the part, the product-process requirements were generated using Ashby's material and process indices related to weight reduction and increasing stiffness of a column [START_REF] Ashby | Materials Selection in Mechanical Design[END_REF]. As shown in Sec. 2, various axioms and rules were defined with respect to the two design criteria; function and environment. Table 1 shows the preliminary materials and machines-processes (see Fig. 2) for the RS of an illustrative example.

RS in Embodiment Design

This stage used AHP as a MCDM tool to screen and rank the material-machine and process combinations generated from Sec. 3.1. Table 2 shows the final RS which can be sent to the detailed design stage. Any of the obtained combinations can be used by the customer.

Conclusions

A generic integrated design oriented framework was presented in this paper to suggest the best compromise of material(s), manufacturing process(es) and machine(s) for AM. The concept of IPPD was used to assist in providing output in the form of reduced costs, increased functional performance, and sustainability. The proposed framework worked intensively in the conceptual and embodiment design stages via defined axioms and extracted rules to produce a healthy blend of RS for AM. An example case was also used to validate the framework along with the proposal of relevant material, process, and machine combinations. Furthermore, the framework followed the standard translation, screening and ranking procedures. It was an intensive design task that can be employed to implement procedures in conjunction with the DfAM guidelines, application type, functional constraints, and part requirements. Finally, the generated AM materials, processes and machines provided sufficient opportunity for the consumer to try multiple combinations as per constraining factors such as budget.

Fig. 1 .

 1 Fig. 1. Potentials and respective measures of AM (developed by authors)Fig.2shows the global view of the proposed framework with the shaded boxes containing avenues for decision-making with respect to RS. Conceptual design, in context of IPPD, is considered the key stage of design process where the designer explores the fundamental scientific principles, DfAM guidelines, constraints, and associated relations, to structure an embodiment that can realize later in a design that satisfies the floated need. Next, the embodiment design stage allows for the application of MCDM tools (AHP and Ashby material/process selection charts) and any associated cost models for selection

Fig. 2 .

 2 Fig. 2. Integrated design-oriented framework for RS in AM of the compromised set of resources for AM, based on design attributes and functional constraints.The text to follow will explore avenues of decision making in each of the design stages. Also, an example of how axioms are defined for one design criteria; environment, will be shown along with the mechanism to extract rules.

Fig. 3 .

 3 Fig. 3. Conceptual design decision space for RS in AM: An Example (developed by authors)

Fig. 4 .

 4 Fig. 4. Decision tree for axiom "Avoid toxic or harmful materials" (modified from Zarandi et al., 2011)

Fig. 5 .

 5 Fig. 5. Crank

  Available online at www.sciencedirect.com

	ScienceDirect
	Procedia CIRP 00 (2017) 000-000
	www.elsevier.com/locate/procedia
	28th CIRP Design Conference, May 2018, Nantes, France
	Integrated design-oriented framework for Resource Selection in Additive
	Manufacturing
	Uzair Khaleeq uz Zaman a *, Mickael Rivette a , Ali Siadat a , Aamer Ahmed Baqai b
	a Laboratoire de Conception Fabrication Commande, Ecole Nationale Supérieure d'Arts et Métiers, 4 Rue Augustin Fresnel, Metz, 57078, France
	b College of Electrical and Mechanical Engineering, National University of Sciences and Technology, Islamabad, 44000, Pakistan
	2212-8271 © 2017 The Authors. Published by Elsevier B.V.
	Peer-review under responsibility of the scientific committee of the 28th CIRP Design Conference 2018.

Table 1 .

 1 Preliminary AM materials, machines and processes Sr. No.

		Material	Process Machine
	1	ABflex	DLP	P4 Standard XL
	2	VisiJet FTX Green SLA	ProJet 1200
	3	RGD 430	MJM	Objet 30 Pro/350/500
				Connex3
	4	Invicta 977	SLA	XFAB
	5	Vitra 429	SLA	XFAB
	6	ABS M30	FDM	Fortus 380/450/900 mc
	7	VisiJet M5 Black	MJM	ProJet 5000
	8	ASA	FDM	Fortus 380/450/900 mc
	9	ABSPlus	FDM	Print SE/SE Plus,
				Dimension Elite
	10	VisiJet M3 Black	MJM	ProJet 3600
	11	ABS-ESD7	FDM	Fortus 380/450/900 mc
	12	Duraform EX	SLS	sPro 140/230/60 HD-HS
	13	Accura 25	SLA	ProX 800/950
	14	VisiJet SL Tough	SLA	Projet 6000/7000 HD
	15	PC ABS	FDM	Fortus 380/450/900 mc
	16	Accura PP White	SLA	ProX 800/950
	17	Duraform_PA	SLS	sPro 140/230/60 HD-HS
	18	RGD 450	MJM	Objet 30 Pro/350/500
				Connex3
	19	R11	DLP	P4 Standard XL
	20	R5 Gray	DLP	P4 Standard XL
	21	ABStuff	DLP	P4 Standard XL

SLA: Stereolithography, DLP: Digital Light Processing, FDM: Fused Deposition Modeling, MJM: Multi-jet Modeling, SLS: Selective Laser Sintering

Table 2 .

 2 Final RS for AM Sr. No.

		Material	Process Machine
	1	R11	DLP	P4 Standard XL
	2	Duraform_PA	SLS	sPro 140/230/60 HD-HS
	3	RGD450	MJM	Objet 30 Pro/350/500 Connex3
	4	Accura PP White SLA	ProX 800/950
	5	PC ABS	FDM	Fortus 380/450/900 mc