Emilie Chouzenoux
email: emilie.chouzenoux@univ-paris-est.fr.

Jean-Christophe Pesquet

J.-C Pesquet

Convergence Rate Analysis of the Majorize-Minimize Subspace Algorithm -Extended Version

Keywords: convergence rate, optimization, subspace algorithms, memory gradient methods, descent methods, majorization-minimization, online optimization, learning

State-of-the-art methods for solving smooth optimization problems are nonlinear conjugate gradient, low memory BFGS, and Majorize-Minimize (MM) subspace algorithms. The MM subspace algorithm which has been introduced more recently has shown good practical performance when compared with other methods on various optimization problems arising in signal and image processing. However, to the best of our knowledge, no general result exists concerning the theoretical convergence rate of the MM subspace algorithm. This paper aims at deriving such convergence rates both for batch and online versions of the algorithm and, in particular, discusses the influence of the choice of the subspace.

Introduction

The Majorize-Minimize (MM) subspace algorithm [START_REF] Chouzenoux | A majorize-minimize subspace strategy for subspace optimization applied to image restoration[END_REF] is based on the idea of constructing, at the current iteration, a quadratic majorizing approximation of the cost function of interest [START_REF] Hunter | A tutorial on MM algorithms[END_REF], and generating the next iterate by minimizing this surrogate function within a subspace spanned by few directions [START_REF] Elad | Coordinate and subspace optimization methods for linear least squares with non-quadratic regularization[END_REF][START_REF] Conn | On iterated-subspace minimization methods for nonlinear optimization[END_REF][START_REF] Yuan | Subspace techniques for nonlinear optimization[END_REF]. Note that the MM subspace algorithm can be viewed as a special instance of nonlinear conjugate gradient (NLCG) [START_REF] Hager | A survey of nonlinear conjugate gradient methods[END_REF] with closed form formula for the stepsize and conjugacy parameter, or as a particular low memory BFGS (L-BFGS) algorithm [START_REF] Liu | On the limited memory BFGS method for large scale optimization[END_REF] with a specific combination of memory directions. The MM subspace algorithm enjoys nice convergence properties [START_REF] Chouzenoux | A majorize-minimize subspace approach for ℓ 2 -ℓ 0 image regularization[END_REF], and shows good performance in practice, when compared with NLCG, L-BFGS, and also with graph-cut based discrete optimization methods, and proximal algorithms [START_REF] Chouzenoux | A majorize-minimize subspace strategy for subspace optimization applied to image restoration[END_REF][START_REF] Chouzenoux | A memory gradient algorithm for ℓ 2 -ℓ 0 regularization with applications to image restoration[END_REF][START_REF] Florescu | A majorizeminimize memory gradient method for complex-valued inverse problem[END_REF]. It has recently been extended to the online case when only a stochastic approximation of the criterion is employed at each iteration [START_REF] Chouzenoux | A stochastic majorize-minimize subspace algorithm for online penalized least squares estimation[END_REF]. All these works illustrate the fact that the choice of the subspace has a major impact on the practical convergence speed of the algorithm (see, for instance [1, Section 5], [START_REF] Chouzenoux | A majorize-minimize subspace approach for ℓ 2 -ℓ 0 image regularization[END_REF]Section 5.1]). In particular, it seems that the best performance is obtained for the memory gradient subspace [START_REF] Miele | Study on a memory gradient method for the minimization of functions[END_REF], spanned by the current gradient and the previous direction, leading to the so-called MM Memory Gradient (3MG) algorithm. However, only an analysis concerning the convergence rates of half-quadratic algorithms (corresponding to the case when the subspace spans the whole Euclidean space) is available [START_REF] Allain | On global and local convergence of half-quadratic algorithms[END_REF][START_REF] Nikolova | Analysis of half-quadratic minimization methods for signal and image recovery[END_REF]. Section 2 describes the general form of the MM subspace algorithm and its main known properties. In Section 3, a convergence rate analysis is performed for both batch and online versions of the algorithm for minimizing a wide class of strongly convex cost functions.

MM subspace algorithm 2.1 Optimization problem

In this paper, we will be interested in the minimization of the penalized quadratic cost function:

F : R N → R : h → 1 2 h ⊤ Rh -r ⊤ h + Ψ(h), (1)
where r ∈ R N , R ∈ R N ×N is a symmetric positive definite matrix, and Ψ is a lower-bounded twice-continuously differentiable convex function. In this paper, it will be assumed that F is only accessible through a sequence (F n ) n 1 of approximations estimated in an online manner, such that, for every n ∈ N * ,

F n : R N → R : h → 1 2 h ⊤ R n h -r ⊤ n h + Ψ(h), (2)
where the vector r n and the symmetric nonnegative definite matrix R n are approximations of r and R. For simplicity, we will suppose that Assumption 1.

(i) ( r n -r n+1 ) n 1 and ( R n -R n+1 ) n 1 are summable sequences, (ii) (r n ) n 1
, and (R n ) n 1 converge to r and R, respectively.

It is worth emphasizing that Assumption 1 encompasses the batch case when F n ≡ F . Moreover, it should be pointed out that all the results presented subsequently can be easily extended to a stochastic framework where r n and R n are consistent statistical estimates of r and R, and convergence arises almost surely.

Majorant function

At each iteration n ∈ N * of the MM subspace algorithm, the available estimate F n of F is replaced by a surrogate function Θ n (•, h n ) based on the current point h n (computed at the previous iteration). This surrogate function [START_REF] Jacobson | An expanded theoretical treatment of iterationdependent Majorize-Minimize algorithms[END_REF][START_REF] Zhang | Surrogate maximization/minimization algorithms and extensions[END_REF][START_REF] Hong | A unified algorithmic framework for block-structured optimization involving big data: With applications in machine learning and signal processing[END_REF] must be such that

(∀h ∈ R N ) F n (h) -F n (h n ) Θ n (h, h n ) -Θ n (h n , h n ). (3)
We assume that Θ n (•, h n ) is a quadratic function of the form

(∀h ∈ R N ) Θ n (h, h n ) = F n (h n ) + ∇F n (h n ) ⊤ (h -h n ) + 1 2 (h -h n ) ⊤ A n (h n )(h -h n ), (4)
where

A n (h n ) = R n + B(h n ) and B(h n ) ∈ R N ×N
is some symmetric nonnegative definite matrix (see [START_REF] Figueiredo | Majorization-minimization algorithms for wavelet-based image restoration[END_REF][START_REF] Fessler | A paraboloidal surrogates algorithm for convergent penalized-likelihood emission image reconstruction[END_REF][START_REF] Repetti | Euclid in a taxicab: Sparse blind deconvolution with smoothed l1/l2 regularization[END_REF][START_REF] Ning | Chromatogram baseline estimation and denoising using sparsity (beads)[END_REF][START_REF] Song | Sparse generalized eigenvalue problem via smooth optimization[END_REF] for examples).

MM subspace algorithm

The MM subspace algorithm consists of defining the following sequence of vectors (h n ) n 1 :

(∀n ∈ N * ) h n+1 ∈ Argmin h∈ran D n Θ n (h, h n ), (5)
where h 1 is set to an initial value, and ran D n is the range of matrix D n ∈ R N ×M n with M n 1, constructed in such a way that the steepest descent direction -∇F n (h n ) belongs to ran D n . Several choices have been proposed in the literature for matrices (D n ) n∈N * . On the one hand, if, for every n ∈ N * , rank(D n ) = N , Algorithm (5) becomes equivalent to a halfquadratic method with unit stepsize [START_REF] Allain | On global and local convergence of half-quadratic algorithms[END_REF][START_REF] Idier | Convex half-quadratic criteria and interacting auxiliary variables for image restoration[END_REF][START_REF] Charbonnier | Deterministic edgepreserving regularization in computed imaging[END_REF]. Half-quadratic algorithms are known to be effective optimization methods, but the resolution of the minimization subproblem involved in [START_REF] Yuan | Subspace techniques for nonlinear optimization[END_REF] requires the inversion of matrix A n (h n ) which may have a high computational cost. On the other hand, if for every n ∈ N * , D n reduces to [-∇F n (h n ), h n ], then (5) reads: for every n ∈ N * h n+1 = u n,2 h n -u n,1 ∇F n (h n ), where (u n,1 , u n,2 ) ∈ R 2 . In the special case when u n,2 = 1, we recover the form of a gradient-like algorithm with step-size u n,1 [START_REF] Labat | Convergence of conjugate gradient methods with a closed-form stepsize formula[END_REF][START_REF] Lange | A gradient algorithm locally equivalent to the EM algorithm[END_REF]. An intermediate size subspace matrix is obtained by choosing, for every n > 1,

D n = [-∇F n (h n ), h n , h n -h n-1 ].
This particular choice for the subspace yields the 3MG algorithm [START_REF] Chouzenoux | A majorize-minimize subspace approach for ℓ 2 -ℓ 0 image regularization[END_REF][START_REF] Chouzenoux | A stochastic majorize-minimize subspace algorithm for online penalized least squares estimation[END_REF].

Convergence result

The convergence of the MM subspace Algorithm (5) has been studied in [START_REF] Chouzenoux | A majorize-minimize subspace strategy for subspace optimization applied to image restoration[END_REF][START_REF] Chouzenoux | A majorize-minimize subspace approach for ℓ 2 -ℓ 0 image regularization[END_REF][START_REF] Chouzenoux | A stochastic majorize-minimize subspace algorithm for online penalized least squares estimation[END_REF] under various assumptions. We now provide a convergence result which is a deterministic version of the one in [START_REF] Chouzenoux | A stochastic majorize-minimize subspace algorithm for online penalized least squares estimation[END_REF]Section IV]. This result requires the following additional assumption:

Assumption 2. (i) For every n ∈ N * , {∇F n (h n ), h n } ⊂ ran D n ,
(ii) There exists a positive definite matrix V such that, for every

n ∈ N * , ∇ 2 Ψ(h n ) B(h n )
V , where ∇ 2 Ψ denotes the Hessian of Ψ, 1

(iii) At least one of the following statements holds:

(a) r n ≡ r and R n ≡ R, (b) h → B(h)h -∇Ψ(h) is a bounded function.
Remark 1. Note that the convexity of Ψ and Assumption 2(ii) implies that [START_REF] Bauschke | Convex Analysis and Monotone Operator Theory in Hilbert Spaces[END_REF]. However, better choices for the curvature matrix are often possible [START_REF] Repetti | Euclid in a taxicab: Sparse blind deconvolution with smoothed l1/l2 regularization[END_REF][START_REF] Song | Sparse generalized eigenvalue problem via smooth optimization[END_REF]. In particular, Assumption 2(iii)(b), required in the online case, is satisfied for a wide class of functions and majorants [START_REF] Chouzenoux | A majorize-minimize subspace strategy for subspace optimization applied to image restoration[END_REF][START_REF] Chouzenoux | A stochastic majorize-minimize subspace algorithm for online penalized least squares estimation[END_REF].

Ψ is Lipschitz dif- ferentiable on R N , with Lipschitz constant |||V |||. Conversely, if Ψ is β-Lipschitz differentiable with β ∈]0, +∞[, Assumption 2(ii) is satisfied with V = B(h n ) = βI N
Proposition 1. Assume that Assumptions 1 and 2 are fulfilled. Then, the following hold:

(i) ( ∇F n (h n ) ) n 1 is square-summable. (ii) (h n ) n 1 converges to the unique (global) minimizer h of F .
Proof. See Appendix A. 3 Convergence rate analysis

Convergence rate results

We will first give a technical lemma the proof of which is in the spirit of classical approximation techniques for the study of first-order optimization methods (see [28, Section 1]):

Lemma 1. Suppose that Assumptions 1 and 2 hold. Let ǫ ∈]0, +∞[ be such that ǫI N ≺ R.

Then, there exists n ǫ ∈ N * such that, for every

n n ǫ , ∇ 2 F n (h n ) R -ǫI N and F n (h n ) -inf F n 1 2 (1 + ǫ) ∇F n (h n ) ⊤ ∇ 2 F n (h n ) -1 ∇F n (h n ). (6)
Proof. See Appendix B.

We now state our main result which basically allows us to quantify how fast the proposed iterative approach is able to decrease asymptotically the cost function:

Proposition 2. Suppose that Assumptions 1 and 2 hold. Let ǫ ∈]0, +∞[ be such that ǫI N ≺ R. Then, there exists n ǫ ∈ N * such that, for every

n n ǫ , ∇ 2 F n (h n ) R -ǫI N and F n (h n+1 ) -inf F n θ n F n (h n ) -inf F n ( 7
)
where

θ n = 1 -(1 + ǫ) -1 θ n , θ n = ∇F n (h n ) ⊤ C n (h n )∇F n (h n ) ∇F n (h n ) ⊤ ∇ 2 F n (h n ) -1 ∇F n (h n ) , (8)
C n (h n ) = D n (D ⊤ n A n (h n )D n ) † D ⊤ n ,

and (•)

† denotes the pseudo-inverse operation. Furthermore, some lower and upper bounds on θ n are given by

θ n = 1 -(1 + ǫ) -1 κ -1 n > 0, (9)
θ n = 1 -(1 + ǫ) -1 κ -1 n 1 - σ n -σ n σ n + σ n 2 < 1, ( 10
)
where κ n 1 (resp. κ n ) is the minimum (resp. maximum) eigenvalue of

A n (h n ) 1 2 ∇ 2 F n (h n ) -1 A n (h n ) 1 2 , and σ n (resp. σ n ) is the minimum (resp. maximum) eigenvalue of ∇ 2 F n (h n ).
Proof. See Appendix C.

Discussion on the choice of the subspace

Let us make some comments about the above results. First, as enlightened by our proof, at iteration n n ǫ , the upper value of θ n (i.e. the slowest convergence) is obtained in the case of a gradient-like algorithm. As expected, θ n has a larger value when the eigenvalues of the Hessian of F n are dispersed. Note that, according to (50),

σ n -σ n σ n + σ n η -η + 2ǫ η + η , (11)
where η > 0 is the minimum eigenvalue of R and η is the maximum eigenvalue of R + V . Since

A n (h n ) 1 2 ∇ 2 F n (h n ) -1 A n (h n ) 1 2
n n ǫ is bounded, there exists κ max ∈ [1, +∞[ such that (∀n n ǫ ) κ n κ max . All these show that the decay rate is uniformly strictly lower than 1.

In contrast, when the search subspace is the full space, the lower value of θ n (i.e. the fastest convergence) is obtained. The expression θ n in [START_REF] Chouzenoux | A memory gradient algorithm for ℓ 2 -ℓ 0 regularization with applications to image restoration[END_REF] shows that the decay is then faster when the quadratic majorant constitutes a tight approximation of function F n at h n . Ideally, if A n (h n ) can be chosen equal to ∇ 2 F n (h n ) and D n is full rank, then θ n = O(ǫ). Such a behavior similar to Newton's method behavior leads to the best performance one can reasonably expect from the available data at iteration n.

Finally, when a mid-size subspace is chosen (as in the 3MG algorithm), an intermediate decay rate is obtained. Provided that D n captures the main eigendirections in A n (h n ), a behavior close to the one previously mentioned can be expected in practice with the potential advantage of a reduced computational complexity per iteration.

Batch case

The case when F ≡ F n is of main interest since it is addressed in most of the existing works. Then, Proposition 2 and (11) lead to

(∀n n ǫ ) F (h n ) -inf F µϑ n , (12)
where µ = F (h n ǫ ) -inf F /ϑ n ǫ and the worst-case geometrical decay rate ϑ ∈]0, 1[ is given by

ϑ = 1 - 1 (1 + ǫ)κ max 1 - η -η + 2ǫ η + η 2 . ( 13
)
Since F is an η-strongly convex function, the following inequality is satisfied [27, Definition 10.5], for every α ∈]0, 1[,

F αh n + (1 -α) h + 1 2 α(1 -α)η h n -h 2 αF (h n ) + (1 -α)F ( h), (14)
or, equivalently,

1 2 α(1 -α)η h n -h 2 α F (h n ) -F ( h) + F ( h) -F αh n + (1 -α) h . (15)
Thus,

1 2 (1 -α)η h n -h 2 F (h n ) -F ( h). (16)
Letting α tend to 0 in the latter inequality implies that

1 2 η h n -h 2 F (h n ) -F ( h) µϑ n . ( 17
)
This shows that the MM subspace algorithm converges linearly with rate √ ϑ.

Conclusion

In this paper, we have established expressions of the convergence rate of an online version of the MM subspace algorithm. These results help in better understanding the good numerical behaviour of this algorithm in signal/image processing applications and the role played by the subspace choice. Even in the batch case, the provided linear convergence result appears to be new. In future work, it could be interesting to investigate extensions of these properties to more general cost functions than [START_REF] Chouzenoux | A majorize-minimize subspace strategy for subspace optimization applied to image restoration[END_REF].

A Proof of Proposition 1

A.1 Boundedness of (h n ) n 1 (online case)

Assume that Assumption 2(iii)(b) holds. For every n ∈ N * , minimizing Θ n (•, h n ) is equivalent to minimizing the function

(∀h ∈ R N ) Θ n (h, h n ) = 1 2 h ⊤ A n (h n )h -c n (h n ) ⊤ h, (18)
with

c n (h n ) = A n (h n )h n -∇F n (h n ) = r n + B(h n )h n -∇Ψ(h n ) ( 19
)
According to Assumption 2(iii)(b), these exists η ∈]0, +∞[ such that

(∀n 1) c n (h n ) η, (20)
In addition, because of Assumption 1(ii), there exists ǫ ∈]0, +∞[ and n 0 ∈ N * such that

(∀n n 0 ) A n (h n ) R -ǫI N ≻ O N , (21)
Using now the Cauchy-Schwarz inequality, we have

(∀n n )(∀h ∈ R N ) 1 2 h ⊤ (R -ǫI N )h -h η Θ n (h, h n ). (22)
Since R -ǫI N is a positive definite matrix, the lower bound corresponds to a coercive function with respect to h. There thus exists ζ ∈]0, +∞[ such that, for every

h ∈ R N , h > ζ ⇒ (∀n n 0 ) Θ n (h, h n ) > 0. ( 23
)
On the other hand, since 0 ∈ ran D n , we have

Θ n (h n+1 , h n ) Θ n (0, h n ) = 0. ( 24
)
The last two inequalities allow us to conclude that

(∀n n 0 ) h n+1 ζ. (25)
A.2 Convergence of (F n (h n )) n 1

According to Assumption 2(i), the proposed algorithm is actually equivalent to

(∀n ∈ N * ) h n+1 = h n + D n u n (26)
u n = arg min u∈R M n Θ n (h n + D n u, h n ). ( 27
)
By using (4) and cancelling the derivative of the function

u → Θ n (h n + D n u, h n ), D ⊤ n ∇F n (h n ) + D ⊤ n A n (h n )D n u n = 0. (28)
Hence,

Θ(h n+1 , h n ) = F n (h n ) - 1 2 u ⊤ n D ⊤ n A n (h n )D n u n = F n (h n ) - 1 2 (h n+1 -h n ) ⊤ A n (h n )(h n+1 -h n ). ( 29
)
In view of ( 3) and ( 4), this yields

(∀n ∈ N * ) F n (h n+1 ) + 1 2 (h n+1 -h n ) ⊤ A n (h n )(h n+1 -h n ) F n (h n ). (30)
In addition, the following recursive relation holds

(∀h ∈ R N ) F n+1 (h) = F n (h) -(r n+1 -r n ) ⊤ h + 1 2 h ⊤ (R n+1 -R n )h. ( 31
)
It can thus be deduced that

F n+1 (h n+1 ) + 1 2 (h n+1 -h n ) ⊤ A n (h n )(h n+1 -h n ) F n (h n ) + χ n ( 32
)
where

χ n = -(r n -r n+1 ) ⊤ h n+1 + 1 2 h ⊤ n+1 (R n -R n+1 )h n+1 . ( 33
)
We have

|χ n | r n -r n+1 h n+1 + 1 2 |||R n -R n+1 ||| h n+1 2 . ( 34
)
If Assumption 2(iii)(b) holds, then, according to ( 25), (h n ) n 1 is bounded, so that Assumption 1(i) guarantees that

+∞ n=1 |χ n | < +∞. ( 35
)
Otherwise, if Assumption 2(iii)(a) holds, then χ n ≡ 0 and ( 35) is obviously fulfilled. The lowerboundedness property of Ψ entails that, for every n ∈ N * , F n is lower bounded by inf Ψ > -∞. Furthermore, (32) leads to

F n+1 (h n+1 ) -inf Ψ + 1 2 (h n+1 -h n ) ⊤ A n (h n )(h n+1 -h n ) F n (h n ) -inf Ψ + |χ n |. ( 36
)
Since, for every

n ∈ N * , F n (h n ) -inf Ψ and (h n+1 -h n ) ⊤ A n (h n )(h n+1 -h n ) are nonnegative, (h n+1 -h n ) ⊤ A n (h n )(h n+1 -h n ) n 1 is a summable sequence, and (F n (h n )) n 1 is convergent. A.3 Convergence of (∇F n (h n )) n 1
According to (4), we have, for every φ ∈ R and n ∈ N * ,

Θ n h n -φ∇F n (h n ), h n = F n (h n ) -φ ∇F n (h n ) 2 + φ 2 2 ∇F n (h n ) ⊤ A n (h n )∇F n (h n ). ( 37
) Let Φ n ∈ Argmin φ∈R Θ n h n -φ∇F n (h n ), h n . ( 38
)
The following optimality condition holds:

∇F n (h n ) ⊤ A n (h n )∇F n (h n ) Φ n = ∇F n (h n ) 2 . ( 39
)
As a consequence of Assumption 2(i), (∀φ ∈ R) h n -φ∇F n (h n ) ∈ ran D n . It then follows from ( 5) and (39) that

Θ n h n+1 , h n Θ n h n -Φ n ∇F n (h n ), h n = F n (h n ) - Φ n 2 ∇F n (h n ) 2 , (40)
which, by using ( 29), leads to

Φ n ∇F n (h n ) 2 (h n+1 -h n ) ⊤ A n (h n )(h n+1 -h n ). ( 41
)
Let ǫ > 0. Assumption 2(ii) yields, for every n ∈ N * ,

A n (h n ) (|||R n ||| + |||V |||)I N . (42)
Therefore, according to Assumption 1(ii),

(∃n 0 ∈ N * )(∀n n 0 ) O N ≺ A n (h n ) α -1 ǫ I N ( 43
)
where

α ǫ = (|||R||| + |||V ||| + ǫ) -1 > 0. ( 44
)
By using now (39), it can be deduced from (43) that, if n n 0 and ∇F n (h n ) = 0, then Φ n α ǫ . Then, it follows from (41) that

α ǫ +∞ n=n 0 ∇F n (h n ) 2 +∞ n=n 0 h n+1 -h n ⊤ A n (h n ) h n+1 -h n . (45)
By invoking the summability property of (

h n+1 -h n ) ⊤ A n (h n )(h n+1 -h n ) n 1 , we can con- clude that ( ∇F n (h n ) 2 ) n 1 is itself summable. A.4 Convergence of (h n ) n 1 We have shown that (h n+1 -h n ) ⊤ A n (h n )(h n+1 -h n ) n 1 converges to 0.
In addition, we have seen that ( 21) holds for a given ǫ ∈]0, +∞[ and n 0 ∈ N * . This implies that, for every n n 0 ,

|||R -ǫI N ||| h n+1 -h n 2 h n+1 -h n ⊤ A n (h n ) h n+1 -h n (46)
where |||R -ǫI N ||| > 0. Consequently, (h n+1h n ) n 1 converges to 0. In addition, (h n ) n 1 belongs to a compact set. Thus, invoking Ostrowski's theorem [START_REF] Ostrowski | Solution of Equations in Euclidean and Banach Spaces[END_REF]Theorem 26.1] implies that the set of cluster points of (h n ) n 1 is a nonempty compact connected set. By using (1)-( 2), we have (∀n

∈ N * ) ∇F n (h n ) -∇F (h n ) = (R n -R)h n -r n + r. (47) Since
(h n ) n 1 is bounded, it follows from that ∇F n (h n ) -∇F (h n ) n 1 converges to 0. Since ∇F n (h n ) n 1 converges to 0, this implies that ∇F (h n ) n 1 also converges to 0. Let h be a cluster point of h n n 1 . There exists a subsequence h k n n 1 such that h k n → h. As F is continuously differentiable, we have ∇F ( h) = lim n→+∞ ∇F h k n = 0. (48)
This means that h is a critical point of F . Since F is a strongly convex function, it possesses a unique critical point h, which is the global minimizer of F [START_REF] Bauschke | Convex Analysis and Monotone Operator Theory in Hilbert Spaces[END_REF]Prop.11.7]. Since the unique cluster point of (h n ) n 1 is h, this shows that h n → h.

B Proof of Lemma 1

Because R is positive definite, according to Assumption 1(ii), there exists n 0 ∈ N * such that, for every n n 0 ,

O N ≺ R -ǫI N R n R + ǫI N . (49)
Let n n 0 . Then, F n is a strongly convex continuous function. From standard results, this function possesses a unique global minimizer h n . According to Assumption 2(ii), and (49), ∇ 2 F n is such that

(∀h ∈ R N ) O N ≺ R -ǫI N R n + ∇ (2) Ψ(h) = ∇ 2 F n (h) R + ǫI N + V . ( 50
)
By using now the second-order Taylor formula with integral remainder, we get

F n ( h n ) = F n (h n ) + ∇F n (h n ) ⊤ h n -h n + 1 2 h n -h n ⊤ H (2) n (h n ) h n -h n , (51)
where

∇F n (h n ) = ∇F n ( h n ) + H (1) n (h n )(h n -h n ) = H (1) n (h n )(h n -h n ) (52)
and, for every h ∈ R N ,

H (1) n (h) = 1 0 ∇ 2 F n h n + t(h -h n ) dt = R n + 1 0 ∇ 2 Ψ h n + t(h -h n ) dt (53)
H (2) n (h) = 2 1 0 (1 -t)∇ 2 F n h n + t(h -h n ) dt = R n + 2 1 0 (1 -t)∇ 2 Ψ h n + t(h -h n ) dt. (54)
Because of the lower bound in (50),

(∀h ∈ R N ) O N ≺ R -ǫI N H (1) n (h) (55)
and H

n (h) is thus invertible. Therefore, combining (51) and (52) yields

F n ( h n ) = F n (h n ) -∇F n (h n ) ⊤ H (1) n (h n ) -1 ∇F n (h n ) + 1 2 ∇F n (h n ) ⊤ H (1) n (h n ) -1 H (2) n (h n ) H (1) n (h n ) -1 ∇F n (h n ). ( 56
)
According to Assumption 2(ii), for every t ∈ [0, 1],

|||∇ 2 Ψ h n + t(h n -h n ) ||| |||V |||, (57)
where ||| • ||| denotes the matrix spectral norm. As Proposition 1(ii) guarantees that (h n ) n 1 converges to the unique minimizer h of F , it follows from Proposition 1(i), (52), and (55) that ( h n ) n 1 also converges to h. By using the continuity of

∇ 2 Ψ, ∇ 2 Ψ h n + t(h n -h n ) n 1
converges to ∇ 2 Ψ( h) and, by invoking the dominated convergence theorem, it can be deduced that

1 0 ∇ 2 Ψ h n + t(h n -h n ) dt → ∇ 2 Ψ( h). (58)
Since (R n ) n 1 converges to R, this allows us to conclude that H

n (h n ) n 1 converges to ∇ 2 F ( h). Proceeding similarly, it can be proved that H [START_REF] Hunter | A tutorial on MM algorithms[END_REF] n (h n ) n 1 also converges to ∇ 2 F ( h). This entails that H (1) n (h n )

-1 - 1 2 H (1) n (h n ) -1 H (2) n (h n ) H (1) n (h n ) -1 → 1 2 ∇ 2 F ( h) -1 . (59) Besides, since ∇ 2 F n (h n ) n 1 = R n +∇ 2 Ψ(h n ) n 1 converges to ∇ 2 F ( h)
, there exists n ǫ n 0 such that, for every n n ǫ ,

H (1) n (h n ) -1 - 1 2 H (1) n (h n ) -1 H (2) n (h n ) H (1) n (h n ) -1 - 1 2 ∇ 2 F n (h n ) -1 (60) 1 2 ǫ(R + ǫI N + V ) -1 1 2 ǫ ∇ 2 F n (h n ) -1 , (61)
where the last inequality follows from (50). This implies that

H (1) n (h n ) -1 - 1 2 H (1) n (h n ) -1 H (2) n (h n ) H (1) n (h n ) -1 1 2 (1 + ǫ) ∇ 2 F n (h n ) -1 . (62)
By coming back to (56), we deduce that, for every n n ǫ , (6) holds.

C Proof of Proposition 2

Let n ∈ N * . If ∇F n (h n ) is zero, then h n is a global minimizer of F n and, according to (3)-( 5),

F (h n+1 ) Θ n (h n+1 , h n )-Θ n (h n , h n )+F (h n ) F (h n )
so that h n+1 is also a global minimizer of F n , and ( 7) is obviously satisfied. So, without loss of generality, it will be assumed in the rest of the proof that ∇F n (h n ) is nonzero. Because of Assumption 2(ii) and (49), there exists n 0 ∈ N * such that, for every n n 0 ,

O N ≺ R -ǫI N R n A n (h n ). (63)
Using (30) and the definition of C n (h n ),

F n (h n+1 ) F n (h n ) - 1 2 (h n+1 -h n ) ⊤ A n (h n )(h n+1 -h n ) = F n (h n ) - 1 2 (∇F n (h n )) ⊤ C n (h n )∇F n (h n ). (64)
Combining (63), (64) and (40) yields

∇F n (h n ) 4 ∇F n (h n ) ⊤ A n (h n )∇F n (h n ) ∇F n (h n ) ⊤ C n (h n )∇F n (h n ). (65)
In turn, we have

Θ n h n , h n Θ n h n+1 , h n , (66)
where h n is a global minimizer of Θ n (•, h n . If n n 0 , then (63) shows that A n (h n ) is invertible, and

h n = h n -A n (h n ) -1 ∇F n (h n ) (67)
which, by using (64) and (66), yields

∇F n (h n ) ⊤ C n (h n )∇F n (h n ) ∇F n (h n ) ⊤ A n (h n ) -1 ∇F n (h n ). ( 68
)
It can be noticed that the lower bound in (65) is obtained when D n = ∇F n (h n ), while the upper bound in (68) is attained when M n = N and D n is full rank.

Let us now apply Lemma 1. According to this lemma, there exists n ǫ n 0 such that, for every n n ǫ , (6) holds with ∇ 2 F n (h n ) ≻ O N . Let us assume that n n ǫ . By combining ( 6) and (64), we obtain

F n (h n ) -F n (h n+1 ) θ n 1 + ǫ F n (h n ) -inf F n ⇔ F n (h n+1 ) -inf F n 1 - θ n 1 + ǫ F n (h n ) -inf F n , (69)
which itself is equivalent to [START_REF] Liu | On the limited memory BFGS method for large scale optimization[END_REF]. The following lower bound is then be deduced from (65):

θ n ∇F n (h n ) 4 β n ∇F n (h n ) ⊤ A n (h n )∇F n (h n ) , (70)
by setting

β n = ∇F n (h n ) ⊤ ∇ 2 F n (h n ) -1 ∇F n (h n )
. Hence, we have

θ n ∇F n (h n ) 4 β n β ′ n ∇F n (h n ) ⊤ ∇ 2 F n (h n )∇F n (h n ) ∇F n (h n ) ⊤ A n (h n )∇F n (h n ) , ∇F n (h n ) 4 β n β ′ n sup g∈R N g =0 g ⊤ A n (h n )g g ⊤ ∇ 2 F n (h n )g -1 , (71)
where

β ′ n = ∇F n (h n ) ⊤ ∇ 2 F n (h n )∇F n (h n ).
θ n ∇F n (h n ) ⊤ A n (h n ) -1 ∇F n (h n ) ∇F n (h n ) ⊤ ∇ 2 F n (h n ) -1 ∇F n (h n ) sup g∈R N g =0 g ⊤ A n (h n ) -1 g g ⊤ ∇ 2 F n (h n ) -1 g . ( 74
)
The sup term in (74) is equal to κ -1 n . Altogether (69), (73), and (74) yield ( 7)- [START_REF] Florescu | A majorizeminimize memory gradient method for complex-valued inverse problem[END_REF], by setting θ n = 1 -(1 + ǫ) -1 θ n . In view of Assumption 2(ii) and the equality in (50), the Hessian of F n is such that (∀h ∈ R N ) ∇ 2 F n (h) A n (h), (75) and therefore κ n 1.

1 and≺

 1 denote the weak and strict Loewner orders, respectively,

  The sup term in (71) corresponds to the generalized Rayleigh quotient of A n (h n ) and ∇ 2 F n (h n ), which is equal to κ n . By invoking now Kantorovich inequality [28, Section 1.3.2], we getθ n 4σ n σ n κ n (σ n + σ n ) 2 ,since σ n σ n > 0. An upper bound on θ n is derived from (68) and (8):

				(72)
	which leads to
	1 -	θ n 1 + ǫ	θ n < 1	(73)

This work was supported by the CNRS Imag'In project under grant 2015 OPTIMISME, and by the CNRS Mastodons project under grant 2016 TABASCO.


Margherita Peruzzini
email: m.peruzzini@univpm.it

Michele Germani
email: m.germani@univpm.it

Eugenia Marilungo
email: e.marilungo@univpm.it

Product-service lifecycle management in manufacturing: an industrial case study

Keywords: Product Lifecycle Management, PSS (Product-Service System), PLM improvement concept, Collaboration, Virtual Enterprise

Product-Service is a recent concept based on a novel product understanding consisting of integrated product and service shares. It represents a new trend for industries to innovate their artefacts and create fresh business opportunities. However, moving from product to services requires the identification of the needed assets to create the new solution and the integration of both product-related and service-related activities into a unique product-service lifecycle. In practice, such an evolution can be defined theoretically but it is hard to implement since supporting tools are strongly product-centred yet. As a consequence, product-service is still a fascinating idea especially in manufacturing sector. This paper tells about a success story of product-service management in manufacturing industry; it describes how a household appliances' manufacturer shifted from traditional product lifecycle towards product-service lifecycle to manage the new service. The study starts from analysis of the AS-IS processes and mapping of the ecosystem tangible and intangible assets, and describes how the company was supported into the definition of an integrated product-service lifecycle.

Introduction

Numerous manufacturing enterprises are challenged by the transition from a traditional product-oriented model to a new extended service-oriented one, which can be realized by Product-Service Systems (PSS) [START_REF] Goedkoop | Product-Service Systems -Ecological and Economic Basic[END_REF]. This trend mainly consists of adding a wide range of services to increase the value perceived by the customers and better satisfy their needs. In particular, Product-Service Systems (PSSs) consist of the combination of tangible and intangible assets and their integration in Product-Service modules, which are mutually determinant throughout the whole lifecycle [START_REF] Thoben | Extended Products: Evolving Traditional Product Concepts[END_REF]. Additional services and added value move the product towards an extended product really tackling the customer's needs and offering a solution. In this context, technical services represent the easiest way to create a PSS in manufacturing industry: from maintenance to training, retrofitting and product monitoring. Indeed, they can be easily realized by improving the product communication capabilities in order to make data flow from product to external systems to realize supporting or differentiating services. This integrated understanding leads to new and customer-oriented solutions, and enable innovative functions and result-oriented business models. Furthermore, services can bring great advantages for industry: from the economical viewpoint, services can create higher profit margins and contribute to higher productivity by means of reduced investment costs along the lifetime as well as reduced operating costs for the final users [START_REF] Baines | State of the art in Product-Service System[END_REF]; from an ecological viewpoint, product-services can be more efficient thanks to a more conscious product usage, an increased resource productivity and a close loop-chain manufacturing as reported by some examples [START_REF] Favi | A lifecycle design approach to analyse the ecosustainability of industrial products and product-service systems[END_REF]; finally, services create socially advanced scenarios, as they secure knowledge intensive jobs and contribute to a more geographically balanced wellbeing distribution [START_REF] Stahel | The Utilization-Focused Service Economy, Resource Efficiency and Product-Life Extension[END_REF].

However, producing product-services opens a big issue in enterprise modelling and data management since they overcome the model of traditional products and the boundaries of the single company. Indeed, if tangible assets can found in the manufacturer company and its supply chain, intangible assets require medium-term collaboration with service companies and the creation of the so-called Virtual Ecosystems [START_REF] Camarinha-Matos | Collaborative networked organizations: Status and trends in manufacturing[END_REF]. These ecosystems are replacing the former single enterprise and take over the lifetime responsibilities for the new PSS; they were also called Virtual Manufacturing Enterprise (VME). As a result, the cohesion between the product, associated services and the life cycle ideas is expressed by the product-service concept. For this reason, product-service requires the management of all assets and entities involved as well as the PSS designed and by modelling the complex the interrelations between products and non-physical services. In this context, the paper describes an industrial case study focusing on product-services in household appliances sector.

Product-service management for manufacturing companies

A product-service consists of proposing a mix of tangible products and intangible services designed and combined to optimize the product use and increase the value for customers [START_REF] Furrer | Le rôle stratégique des services autour des produits[END_REF]. Value creation can be provided through an extended business network involving different stakeholders, which concur to create the services. Product-service idea starts from the concept of extended product [START_REF] Thoben | Extended Products: Evolving Traditional Product Concepts[END_REF], where intangible services are incorporated into a core product to add value. The term PSS usually includes the tangible product, the related services, the enterprise network and the infrastructures needed [8]. In manufacturing applications PSS are almost based on technical services for two main reasons: they have a concurrent lifetime to the product to which they relate since they support the product use and interact with the customers during the operation stage [START_REF] Mont | Clarifying the concept of Product-Service System[END_REF]; and they can be added to product with low impact for manufacturers since they are usually available at low-cost and have a shorter time-to-market [START_REF] Aurich | Life Cycle oriented design of technical Product-Service Systems[END_REF]. After-sales services are usually provided as PSS. Here are located the services that ensure and restore the availability of the material good. Furthermore, they can extend the life cycle of the product and reduce consumption, emissions and operating costs [START_REF] Homburg | Towards an improved understanding of industrial services: quality dimensions and their impact on buyer-seller relationships[END_REF]. Specifically, after-sales services encompass: maintenance, hotline services, tele-service, spare parts, service and customer training.

There are various forms of servitization with different features and, as suggested by [START_REF] Oliva | Managing the transition from products to services[END_REF], there is a transition line from pure product manufacturers to service provid-ers. At one extreme of the line there are product manufacturers, which produce core products and services purely as an add-on to the product, and services are used as a differentiating factor in product marketing strategy. At the other extreme, there are service providers, whose products are merely an add-on to services since products represent only a small part of total value creation. The transition is based on an extended service business, starting with a few product-related services and ending up with a large number of service offerings. In the evolution of servitization, many manufacturing companies have moved into services and that caused the boundaries between products and services to become blurred [START_REF] Baines | The servitization of manufacturing: A review of literature and reflection on future challenges[END_REF].

In parallel with servitization, industrial models in manufacturing have recently evolved from rigid, deterministic supply chains, to a next generation of completely flexible, open and dynamic ecosystems with new economic opportunities. In such ecosystems, members share knowledge, innovate and collaborate together, interact or connect with each other, design new products, communicate globally and develop projects [START_REF] Tapscott | Wikinomics: How Mass Collaboration Changes Everything[END_REF]. Ad-hoc selection of the most suitable partners for each customer demand is needed in order to become more flexible and reactive on the global dynamic marketplace [START_REF] Windahl | Manufacturing firms and integrated solutions: characteristics and implications[END_REF]. In fact, the collaboration between the main manufacturing company and its key partners is fundamental to propose a new value by the right delivery channels and the best strategies by developing new business models.

It is evident that the increasing importance of services represents big challenges for manufacturing companies. However, the biggest challenge is carrying out a reliable management of PSS and a successful coordination of all the activities related. In particular, small and medium businesses often do not have sufficient knowledge and resources to professionalise its service business.

Manufacturing companies have usually a clearly defined product development process, but they lack a sufficiently defined service development process as found in traditional service companies. Many companies recognise that the existing corporate structures and processes do not allow for efficient development and market positioning of innovative services. In addition, they are faced with the problem of being poorly equipped with appropriate approaches, methodologies and tools for an efficient development of services. In addition, due to the often unclear or unstructured portfolio of services in the firms, a lot of time and resources is spent for coordination and consultation work between the main actors.

In a nutshell, services represent an excellent opportunity for manufacturing companies, but companies need to be supported properly to fully catch this new opportunity. Several methods have been recently proposed to manage PSS (from modularization-focused approach, to stochastic and behaviour-focused approach, until lifecycle-focused methods) [START_REF] Garetti | Life Cycle Simulation for the design of Product-Service Systems[END_REF]. However, some of them are very theoretical and hard to implement in practice, while others are too specific and have a limited applicability. Several recent studies focused on requirement elicitation in product-services [START_REF] Peruzzini | Shift from PLM to SLM: A method to support business requirements elicitation for service innovation[END_REF][START_REF] Wiesner | Requirements Engineering for Servitization in Manufacturing Service Ecosystems (MSEE)[END_REF]: they highlights the main challenges of requirement engineering in virtual manufacturing ecosystems and proposes an approach based on Business Use Cases (BUC) and Serious Games to elicit PSS requirements and investigate the PSS lifecycle. Moreover, the existing business models in manufacturing are not well suited to servitization and collaboration, which are required to enter new markets [START_REF] Casadesus-Masanell | From Strategy To Business Models And To Tactics[END_REF]. Indeed the company business model must change from product to services: PSS requirements should be satisfied through the analysis of both internal and external factors (i.e. environmental, economic, social, technological and political), and the ecosystem factors. In this context, STEEP analysis well support the identification of internal (related the company structure) and external (related the company ecosystem) requirements and the definition of a new business model for a certain idea. Then, Canvas Business Model can straightforwardly schematize the model by providing a clear definition of how an organization creates, delivers, and captures value throughout its ecosystem [START_REF] Osterwalder | Business Model Generation[END_REF].

However, companies still have a lot of difficulties in selecting the most appropriate tools for supporting PSS and selecting the partners to create the Virtual Manufacturing Enterprise (VME), as well as to define a valuable business model and a simple but reliable PSS lifecycle to refer their activities and those of their partners in the VME.

Methodology to shift from product to product-service lifecycle management

The research methodology provides a set of tools and a straightforward workflow to support manufacturing companies in realizing and managing a product-service idea to innovate their business. The crucial point is the analysis of the company ecosystem and the correlation between the tangible and intangible assets offered and the assets necessary for the new PSS, and the definition of a valuable PSS business model.

The proposed method starts from the analysis of the current situation and the mapping of the company's ecosystem assets to answer these research questions: a) Is the company ready to create a PSS? b) How to select the right partners to realize the product-service VME? c) How to manage the PSS lifecycle?

The method steps are described below. 1. Analysis of the AS-IS processes and assets: it consists of two main activities;

the first is related to the analysis of the current Product Lifecycle Management (PLM) process, from its ideation, through design, manufacturing and delivery, to disposal; the second consists of the mapping all the involved assets of the company and its ecosystem, both tangible (i.e. machines, materials, devices, sensors) and intangible (e.g. competences, skills, knowledge, relations among tangible components needed to realize another product functionality). The first analysis is supported by a modelling tool able to represent the company processes, identify all the activities involved in each process and, for each activity, the related input and output, the needed resources (e.g. human, material or IT), the involved competences and skills, and the control unit (e.g. company's departments, process's managers). The second mapping is supported by interviews and ad-hoc questionnaires, which investigate different aspects of the value creation process. In particular, each question is directly linked to a specific key business activity or resource identified in the company business model. 2. Analysis of the servitization readiness: it allows understanding if the company ecosystem is ready to create and manage a PSS solution or which areas should be improved. Such an analysis is based on the assessment of four areas (i.e.

lifecycle management capability, ecosystem creation capability, innovation level, network management capability). Different questionnaires are defined to analyse the four areas. 3. Mapping of tangible and intangible assets: it is based on the correlation between the new PSS items (i.e. product, services, ecosystem and infrastructure) and the related assets, mapping of the company ecosystem [START_REF] Cassina | Development of an Extended Product Lifecycle Management through Service Oriented Architecture[END_REF]. In fact, the PSS idea is defined by a set of tangible and intangible assets, coming from the specific competences and skills of the ecosystem partners. The mapping consists in tracing the network relations within the ecosystem, in terms of who has a certain knowledge, who provides a certain components, who realize a certain software application, who produces a specific product, who implements a specific service, etc. 4. Business model definition: it is based on the adoption of the Canvas model and allows clearly defining the company business model according to the previous results. It involves several areas within the company that should be defined in detail in order to identify which are the core areas where action is needed to implement the new business, such as key partners, key resources, key activities, value proposition, customer segments, channels, revenues streams, cost structure, and customer relationship. Business Model Canvas is used to analyse the building blocks of the new business model (service-centred), while STEEP analysis considers both internal and external factors related to environmental, political, economic, technological and social trends.

Definition of the TO-BE integrated product-service lifecycle and VME: it

finally identifies the new lifecycle to properly manage both product-related and service-related activities and the actors involved. Starting from the previous AS-IS processes modelling, it modifies the core factors individuated in points no. 2 and 3. Moreover, thanks to the definition of the new business model (point no. 4), the company can identify the partners and their own specific knowledge and skills that are necessary to realize the VME and develop the PSS desired.

After step no. 5, the company is ready to create, produce and manage the new product-service solution; in particular, it has defined a certain PSS and a precise VME to support its creation, delivery and operation. At this stage, the quality of the service implemented can be verified in different ways. This research focused on analysing the customer satisfaction and the sustainability index. Satisfaction can be analysed by satisfaction questionnaires and usability testing by directly involving final users, while sustainability is calculated by specific methods (i.e. LCA and LCC applied to the specific PSS solution) to compare the exist product with its related productservice to be developed.

The industrial case study in household appliances

The company ecosystem

The case study has been realized in collaboration with an Italian company producing household appliances and home care devices. The company ecosystem is guided by the manufacturer and is actually organized in a vertical supply-chain adopting a product-oriented development process. Collaboration between the manufacturer and its partners and suppliers is limited to design stages and components' supply. The leader company recently designs and produces "smart appliances", defined as a manufactured product embedded with a hardware (e.g. sensors, Zigbee module, router Wi-Fi) and software (e.g. web service, data repository, web/mobile application) infrastructure to connect the product to an external network.

The case study focuses on services for smart appliances; in particular, it aims at realizing a new maintenance service, called "Smart Maintenance Service": it consists of providing ad-hoc maintenance services to the consumers for a set of smart appliances connected at home. The PSS involved in the case study consists of the appliances, a web/mobile application able to provide personalized messages for coaching purposes and helping the final users in case of appliances' faults. Figure 1 expresses the underpinning idea of the Smart Maintenance Service and shows the main workflow. Data related to the appliances functioning are monitored by specific sensors and are collected in a database for data storage; here, e set of elaboration algorithms analyse these data according to two policies (i.e. coaching and fault management) in order to recognize the specific use scenario and support the user with personalized and tailored suggestions and advices. For the coaching function, the application gives best practices according to the product usage; for the fault management function, the system controls the appliances' parameters, detects dangerous situations and supports the user when some critical values occur in order to carry out the recommended actions or some specific checking actions.

AS-IS process analysis and assets

According to the methodology defined before, the company processes has been analysed. In particular, they refer to the following stages: "Product Conception", "Product Design", "Product Manufacturing", "Product Delivery and Commercialization", and "Product Use". For each process stage, experts from both company and academia detailed the main activities and identified the departments involved, selected the key resources and modelled the activity flows. Moreover, interviews and questionnaires allows tracing the current assets and defining the product items in terms of functionalities, features, number of sensors, brand, production line, materials, any hardware supports (tangible assets) and intangible assets like partners involved in the ecosystem, brand resources, and customer relationships after sales.

Analysis of the servitization readiness

The analysis shows if the company is ready to implement the specific productservice solution and highlights the strengths and the possible weaknesses of the company in designing and managing the "Smart Maintenance Service" process. In the case study the analysis revealed that the company already has an ecosystem made up of different partners, suppliers, consortia and research centres and theoretically able to support product-service lifecycles, but such ecosystem is still strongly productcentred; services are perceived almost as tools to differentiate the product and do not represent the core business. The analysis also highlighted that the company is interested in understanding how to model and fully exploit PSS innovation projects. The case study defined also those areas (and related processes) to be improved to efficiently move to product-service solutions; they are: "Service Ideation", "Requirements analysis", and "Ecosystem governance and decision making".

Mapping of the PSS assets and creation of the VME

According to the PSS idea to develop (i.e. Smart Maintenance Service), a set of partners in the ecosystem are selected for their competences, skills, supplies, devices

External) Information

Customer) relationship

PRODUCT) Conception

PRODUCT) Design

PRODUCT) Manufacturing

PRODUCT)

Delivery&Comm. and knowledge, and their relationships are mapped in order to identify the better configuration of virtual manufacturing enterprise. The result of this study has given the following VME configuration: a research centre to support the PSS modelling, a technological partners to develop the IT infrastructure, a smart home provider able to share its competences about the household appliances connection into a network, a consortium for disposal, a consortium that studies innovative solutions, a partners to support the company during the delivery phase (e.g. during the best practices proposal) of the PSS, and the customers themselves.

Internal) Information

STRATEGIC

Business model definition

A value proposition has been identified for the specific case study according to the Business Model Canvas. It consists of monitoring the household appliances and supports the maintenance process by web or mobile application and involving a group of selected partners (i.e. VME) performing the key activities to carry out preventive maintenance and coaching actions, PSS delivery and management. Figure 4 shows the identified business model: it addresses people that love smartness and convenience, and aims at creating making a long-term relationship between the customers and the company through on-line feedback, dynamic proposition and loyalty programs. Such a model can be delivered by local retailers as well as on-line shops, and can be implemented by different payment modes, from leasing contract to mobile payment.

4.6

The integrated product-service lifecycle and TO-BE processes

Finally, an integrated product-service lifecycle has been defined to organize and manage all the information gathered in the previous steps. It starts from a recent model proposed in literature [START_REF] Peruzzini | Product-Service Sustainability Assessment in Virtual Manufacturing Enterprises[END_REF], that considers how the lifecycle phases are affected by changes due to transition from product to service (e.g. ideation, design, delivery). Figure 4 represents the global view of the TO-BE process by detailing the process activities, data input (e.g. target customer, partners in the ecosystem, competitors

Conclusions

The paper proposed a methodology to support manufacturing companies to achieve new product-service solutions by managing all the Product-Service lifecycle phases, from ideation to delivery. At the same time, the method allows analysing the company ecosystem in order to identify the virtual enterprise able to develop the specific PSS.

Such a methodology has been applied to an industrial case study focused on realizing services in the household appliances sector, the so-called Smart Maintenance Service. It demonstrated how the shift from a product-oriented to a service-oriented lifecycle can be properly managed and enhancing effective collaboration within the virtual enterprise. The results demonstrated how the Smart Maintenance Service is investigated and modelled, and what are the criticalities to face and solve to move for the AS-IS process until the TO-BE process.

Fig. 1 .

 1 Fig. 1. The Smart Maintenance Service workflow

Fig. 2 .

 2 Fig. 2. The product maintenance process analysis (AS-IS scenario)

Fig. 3 .

 3 Fig. 3. Example of PSS assets mapping

  Delivery#through# analysis), data output (e.g. from the PSS ideated to the creation of the Smart Maintenance), exploited resources (e.g. brainstorming activity, procedures, software infrastructure), and finally the involved competences and skills needed (e.g. coming from company departments, research centre, or technological partners).

Fig. 4 .

 4 Fig. 4. Product-service lifecycle management process (TO-BE scenario)