G C Bento

S D B Bitar

J X Cruz Neto

Antoine Soubeyran
email: antoine.soubeyran@gmail.com

J C O Souza

Glaydston De
email: glaydstonc@gmail.com

Carvalho Bento

Sandro Dimy

Barbosa Bitar

João Xavier
email: jxavier@ufpi.edu.br

Cruz Neto

João Carlos De Oliveira

DM Souza Ccn

A proximal point method for difference of convex functions in multi-objective optimization with application to group dynamic problems

Keywords: Multi-objective programming, proximal point method, DC function, variational rationality, behavioral sciences

We consider the constrained multi-objective optimization problem of finding Pareto critical points of difference of convex functions. The new approach proposed by Bento et al. [SIAM J. Optim., 28 (2018), pp. 1104-970] to study the convergence of the proximal point method is applied. Our method minimizes at each iteration a convex approximation instead of the (non-convex) objective function constrained to a possibly non-convex set which assures the vector improving process. The motivation comes from the famous Group Dynamic Problem in Behavioral Sciences where, at each step, a group of (possible badly informed) agents tries to increase his joint payoff, in order to be able to increase the payoff of each of them. In this way, at each step, this ascent process guarantees the stability of the group. Some encouraging preliminary numerical results are reported.

Introduction

In this paper, we consider the multi-objective optimization problem of finding a Pareto critical point for a special class of non-convex vector functions which can be written as difference of two convex vector functions (called DC functions). Multi-objective optimization is the process of simultaneously optimizing two or more real-valued objective functions. Usually, no single point will minimize all given objective functions at once (i.e., there is no ideal minimizer), and so some concepts have to be adapted. There is a wide research program consisting of extension to the vector setting of several iterative methods for scalar functions, for instance the steepest descent method (see Graña Drummond and Svaiter [START_REF] Drummond | A steepest descent method for vector optimization[END_REF] and Bento et al. [START_REF] Bento | The self regulation problem as an inexact steepest descent method for multicriteria optimization[END_REF]), projected methods (see Fukuda and Graña Drummond [START_REF] Fukuda | On the convergence of the projected gradient method for vector optimization[END_REF], Graña Drummond and Iusem [START_REF] Drummond | A projected gradient method for vector optimization problems[END_REF] and Brito et al. [START_REF] Brito | A relaxed projection method for solving multiobjective optimization problems[END_REF]), subgradient method (see Bello Cruz [START_REF] Bello Cruz | A subgradient method for vector optimization problems[END_REF][START_REF] Neto | A subgradient method for multiobjective optimization[END_REF]), Newton's method (see Fliege et al. [START_REF] Fliege | Newton's method for multiobjective optimization[END_REF] and Qu et al. [START_REF] Qu | Nonsmooth multiobjective programming with quasi-Newton methods[END_REF]) and proximal methods (see Bonnel et al. [START_REF] Bonnel | Proximal methods in vector optimization[END_REF], Ceng and Yao [START_REF] Ceng | Approximate proximal methods in vector optimization[END_REF], Bento et al. [START_REF] Bento | A proximal point-type method for multicriteria optimization[END_REF], Apolinário et al. [START_REF] Apolinário | A scalarization proximal point method for quasiconvex multiobjective minimization[END_REF] and Qu et al. [START_REF] Qu | A new algorithm for linearly constrained c-convex vector optimization with a supply chain network risk application[END_REF]). Multi-objective optimization has application in economics, industry, agriculture, location of public facilities, among others. For more details see, for example, Jahn [START_REF] Jahn | Vector Optimization: Theory, Applications and Extensions[END_REF], Luc [START_REF] Luc | Theory of Vector Optimization[END_REF], Miettinen [START_REF] Miettinen | Nonlinear Multiobjective Optimization[END_REF] and Ross and Soland [START_REF] Ross | A multicriteria approach to the location of public facilities[END_REF].

There are a lot of works devoted to the theory of DC functions in different contexts; for instance, duality and optimality conditions (e.g. Hartman [START_REF] Hartman | On functions representable as a difference of convex functions[END_REF], Hiriart-Urruty [START_REF] Hiriart-Urruty | Generalized differentiabity, duality and optimization for problems dealing with difference of convex functions, Convexity and Duality in Optimization[END_REF]), proximal algorithms (e.g. Moudafi and Maingé [START_REF] Maingé | Convergence of new inertial proximal methods for DC programming[END_REF], Souza and Oliveira [START_REF] Souza | A proximal point algorithm for DC functions on Hadamard manifolds[END_REF], Sun et al. [START_REF] Sun | Proximal point algorithm for minimization of DC Functions[END_REF], Wen et al. [START_REF] Wen | A proximal difference-of-convex algorithm with extrapolation[END_REF]), subgradient methods (e.g. Tao and Souad [START_REF] Tao | Algorithms for solving a class of nonconvex optimization problems: methods of subgradient[END_REF]), cutting angle and branch-and-bound methods (e.g. Ferrer et al. [START_REF] Ferrer | Solving DC programs using the cutting angle method[END_REF], Tuy and Horst [START_REF] Tuy | Convergence and restart in branch-and-bound algorithms for global optimization. Application to concave minimization and dc optimization problems[END_REF]), vector optimization (e.g. Flores-Bazán and Oettli [START_REF] Flores-Bazán | Simplified optimality conditions for minimizing the difference of vector-valued functions[END_REF], Guo and Li [START_REF] Guo | Optimality conditions for vector optimization problems with difference of convex maps[END_REF]), variational inequality problems (e.g. Muu and Quoc [START_REF] Muu | One step from DC optimization to DC mixed variational inequalities[END_REF]), equilibrium problems (e.g. Dinh et al. [START_REF] Dinh | Duality and optimality conditions for generalized equilibrium problems involving DC functions[END_REF]), trust-region problem (e.g. Tao and An [START_REF] Tao | A DC Optimization algorithm for solving the trust region subproblem[END_REF]), transportation problem (e.g. Holmberg and Tuy [START_REF] Holmberg | A production-transportation problem with stochastic demand and concave production costs[END_REF]).

The aim of this paper is to study convergence properties of a proximal point method, for DC vector functions, adding a vectorial improving constraint which is not convex. In a finite dimensional multi-objective setting our non-convex multi-objective proximal descent method generalizes the (exact) convex multi-objective proximal point descent method introduced by Bonnel et al. [START_REF] Bonnel | Proximal methods in vector optimization[END_REF]. Our non-convex descent property also implies the one obtained by Ji et al. [START_REF] Ji | Proximal point algorithms for multi-criteria optimization with the difference of convex objective functions[END_REF]. It is worth to mention that even if DC function is a locally Lipschitz function our method is different to the one considered by Bento et al. [START_REF] Bento | The proximal point method for locally Lipschitz functions in multiobjective optimization with application to the compromise problem[END_REF] because our algorithm minimizes at each iteration a linear approximation of the objective function instead of minimizing the objetive function as [START_REF] Bento | The proximal point method for locally Lipschitz functions in multiobjective optimization with application to the compromise problem[END_REF] does. As we will see in Section 6 our approach have important applications in Behavioral Sciences.

A motivation to study the proximal point method involving DC vector functions and a step by step vectorial improving constraint comes from the very important "group dynamic" problem in Behavioral Sciences, where at each step, a group of agents tries to increase his joint payoff, in order to be able to increase the payoff of each of them. In this way, our ascent method guarantees, at each step, the stability of the group. There is a huge literature in Behavioral Sciences on "group dynamics" (see, for instance, the force field theory in Lewin [START_REF] Lewin | Frontiers in group dynamics: concept, method and reality in social science; social equilibria and social change[END_REF][START_REF] Lewin | Field Theory in Social Science[END_REF] in Psychology, organization change theories in Management Sciences, see Poole and Van de Ven [START_REF] Poole | Handbook of Organizational Change and Innovation[END_REF], and dynamic cooperative games in Economics). As emphasized by Lewin [START_REF] Lewin | Frontiers in group dynamics: concept, method and reality in social science; social equilibria and social change[END_REF][START_REF] Lewin | Field Theory in Social Science[END_REF], a major aspect of the group dynamic problem is the presence of resistance to change, given that becoming able to improve and then improving is costly. In order to modelize this resistance to change aspect, we consider an original vectorial proximal algorithm where the perturbation term includes the square of a quasi distance. This model of resistance to change comes from the recent variational rationality approach of human dynamics (see Soubeyran [START_REF] Soubeyran | Variational rationality, a theory of individual stability and change: worthwhile and ambidextry behaviors[END_REF][START_REF] Soubeyran | Variational rationality and the "unsatisfied man": routines and the course pursuit between aspirations, capabilities and beliefs[END_REF][START_REF] Soubeyran | Variational rationality. A theory of worthwhile stay and change approach-avoidance transitions ending in traps[END_REF][START_REF] Soubeyran | Variational rationality.1. An adaptive theory of the unsatisfied man[END_REF][START_REF] Soubeyran | Variational rationality. 2. A general theory of goals and intentions as satisficing worthwhile moves[END_REF]).

The paper is organized as follows. Section 2 presents the definition and basic properties of quasi distances. Section 3 states formally the problem and introduces some required definitions and results in multi-objective optimization. The algorithm is presented in Section 4. Section 5 contains the convergence analysis of the algorithm. An application to "group dynamic" problems in Behavioral Sciences is given in Section 6. Finally, some numerical experiments are presented in Section 7 to show the efficiency of the method and its performance compared to another proximal method for DC functions.

Quasi distance

The motivation to handle with quasi distances as the regularization term in proximal methods comes from applications. In Management Sciences costs of moving (being able to move and move) from a position x to a position y is not symmetric, i.e., C(y, x) = C(x, y). Thus, cost functions involving quasi distances are better adapted to applications in Behavioral Sciences.

A quasi metric space is a pair (X, q) such that X is a non-empty set and q : X × X → R + , called a quasi metric or quasi distance, is a mapping satisfying:

1. For all x, y ∈ X, q(x, y) = q(y, x) = 0 ⇔ x = y; 2. For all x, y, z ∈ X, q(x, z) ≤ q(x, y) + q(y, z).

Therefore, metric spaces are quasi metric spaces satisfying the symmetric property q(x, y) = q(y, x). Quasi distances are not necessarily convex, continuously differentiable or coercive functions. Examples of quasi distances can be found in [START_REF] Moreno | A proximal point algorithm with quasi distance. Application to habit's formation[END_REF] and references therein. In this paper, we consider quasi distances satisfying the following condition: Condition 1: There are positive real numbers α > 0 and β > 0 such that

α||x -y|| ≤ q(x, y) ≤ β||x -y||, ∀x, y ∈ R n . ( 1
)
Remark 1 This condition has been used to prove convergence of proximal point algorithms for non-convex and non-smooth functions, see for instance [START_REF] Bento | Generalized inexact proximal algorithms: routine's formation with resistance to change, following worthwhile changes[END_REF][START_REF] Bento | A generalized inexact proximal point method for nonsmooth functions that satisfies Kurdyka-Lojasiewicz inequality[END_REF][START_REF] Moreno | A proximal point algorithm with quasi distance. Application to habit's formation[END_REF]. Moreno et al. [START_REF] Moreno | A proximal point algorithm with quasi distance. Application to habit's formation[END_REF] present several examples of quasi distances highlighting some that satisfy Condition 1.

Proposition 1 Let q : R n × R n → R + be a quasi distance that verifies (1).

Then, for each z ∈ R n the functions q(•, z), q(z, •) are Lipschitz and q 2 (•, z), q 2 (z, •) are locally Lipschitz on R n .

Proof See [44, Propositions 3.6 and 3.7].

Multi-objective optimization

In this section we recall some basic definitions and properties of multi-objective optimization which can be found, for instance, in Jahn [START_REF] Jahn | Vector Optimization: Theory, Applications and Extensions[END_REF] and Luc [START_REF] Luc | Theory of Vector Optimization[END_REF]. We state and prove some results that will allow to define the algorithm and study its convergence properties. Let R m be the m-dimensional Euclidean space with the partial order " " induced by the Paretian cone R m + , given by y z (or z y) iff z -y ∈ R m + with its associate relation " ≺ ", given by y ≺ z (or z y) iff z -y ∈ R m ++ , where

R m + = {x ∈ R m : x j ≥ 0, j ∈ I} , R m ++ = {x ∈ R m :
x j > 0, j ∈ I} , and I = {1, . . . , m}. Given a vector function F : R n → R m , the problem of finding a weak Pareto point of F consists of finding a point x * ∈ R n such that there exists no other x ∈ R n with F (x) ≺ F (x * ). More generally, given a non-empty set Ω ⊂ R n , a point x * ∈ Ω is a weak Pareto point of the vector function F constrained to the set Ω iff there is no x ∈ Ω with F (x) ≺ F (x * ). We denote this problem by

min w {F (x) : x ∈ Ω}, (2)
and the set of all weak Pareto points of F in Ω by arg min w {F (x) : x ∈ Ω}. Let F be a vector function given by F (x) := (f 1 (x), . . . , f m (x)). The Jacobian m × n matrix of F at x ∈ R n is defined by JF (x) := (∇f 1 (x), . . . , ∇f m (x)) .

For a vector function F : R n → R m we say that F is R m + -convex if and only if, for every x, y ∈ R n and λ ∈ (0, 1)

F (λx + (1 -λ)y) λF (x) + (1 -λ)F (y).
Note that this concept is equivalent to component-wise convexity.

Let f : R n → R be a locally Lipschitz function at x ∈ R n and let d ∈ R n . The Clarke directional derivative of f at x in the direction of d, denoted by f • (x, d), is defined as follows

f • (x, d) := lim sup y→x t↓0 f (y + td) -f (y) t ,
and the Clarke subdifferential of f at x, denoted by ∂f (x), as follows

∂f (x) := {w ∈ R n : w, d ≤ f • (x, d), ∀ d ∈ R n } ,
see Clarke [START_REF] Clarke | Generalized gradients and applications[END_REF].

Proposition 2 Let f 1 , f 2 : R n → R be Lipschitz continuous at a point x ∈ R n . One has ∂(f 1 f 2 )(x) ⊂ f 2 (x)∂f 1 (x) + f 1 (x)∂f 2 (x).
Proof See [START_REF] Clarke | Optimization and Nonsmooth Analysis[END_REF]Proposition 2.3.13].

Given a locally Lipschitz vector function F : R n → R m , i.e., all component functions f i : R n → R are locally Lipschitz functions, the Clarke subdifferential of F at x ∈ R n , denoted by ∂F (x), is defined as

∂F (x) := {U ∈ R m×n : U d F • (x; d), ∀d ∈ R n }, (3)
where

F • (x; d) := (f • 1 (x; d), . . . , f • m (x; d)).
It is worth to point out that an equivalent definition has appeared, in a more general context, in Thibault [START_REF] Thibault | Subdifferentials of nonconvex vector-valued functions[END_REF]. If F is C-convex, for some ordering cone C, a similar definition can be found in Luc [START_REF] Luc | Convex Vector Functions and their Subdifferential[END_REF].

Let Ω be a closed and convex set. A point x ∈ Ω is said to be a Pareto-Clarke critical point (or Pareto critical point) of F in Ω iff, for any y ∈ Ω, there exists i ∈ I such that

f • i (x, y -x) ≥ 0. ( 4
)
Remark 2 Note that if m = 1 in the previous definition, we retrieve the (classical) definition of critical points for non-smooth functions: 0 ∈ ∂f (x). It is worth to notice that, combining (4) with Clarke [16, Proposition 1.4], we have the following alternative definition: a point x ∈ R n is a Pareto critical point of F in Ω if, for any y ∈ Ω, there exist i ∈ I and ξ ∈ ∂f i (x) such that ξ, y -x ≥ 0. Thus, if x is not a Pareto critical point of F in Ω, then there exists y ∈ Ω such that

U (y -x) ≺ 0, ∀U ∈ ∂F (x).
The next result gives a necessary condition for a point to be a Pareto critical point of a vector function.

Lemma 1 Let a, b ∈ R m + be such that at least one of the vectors is not equal to zero. Assume that Ω is a non-empty, closed and convex set. If -(U a+V b) ∈ N Ω (x), for some U, V ∈ ∂F (x), then x is a Pareto critical point of F . Proof We will prove just the case where a, b ∈ R m + \{0}. The proof of the case where one of the vectors a, b ∈ R m + is equal to zero follows the same argument. Take x ∈ Ω and a, b ∈ R m + \{0} such that -(U a + V b) ∈ N Ω (x). Let us suppose, by contradiction, that x is not a Pareto critical point of F . From Remark 2, there exists y ∈ Ω such that U (y -x) ≺ 0 and V (y -x) ≺ 0. Thus, we have a, U (y -x) < 0 and b, V (y -x) < 0 because a, b ∈ R m + \{0}. Therefore, we have that U a, y -x < 0 and V b, y -x < 0. Adding last two inequalities, we obtain U a + V b, y -x < 0, which contradicts the fact that -(U a+V b) ∈ N Ω (x), and hence the desired result is proved.

We denote the distance function d : R n → R of a point x ∈ R n to a non-empty set C ⊂ R n as

d C (x) := inf{||x -c|| : c ∈ C}. (5)
The next result presents a necessary condition for a point x * to be a solution of (2) in the particular case where D ⊂ R n is a non-empty, closed and convex set, and

Ω = {x ∈ D : s j (x) ≤ 0, j ∈ J} with s j : R n → R. (6)
This kind of results have been extended to more general contexts such as vector variational inequalities (see Mai and Luu [START_REF] Mai | Optimality conditions for weakly efficient solutions of vector variational inequalities via convexificators[END_REF]) and others minimal concept in vector optimization such as relative minimal points; see Mordukhovich [START_REF] Mordukhovich | Variational Analysis and Applications[END_REF]Definition 9.3].

Theorem 1 Let D ⊂ R n be a non-empty, closed and convex set. Assume that the set Ω in (2) is given as in [START_REF] Bento | The self regulation problem as an inexact steepest descent method for multicriteria optimization[END_REF], and the functions f j , s j : R n → R, j ∈ I, are locally Lipschitz. If x * ∈ Ω is a weak Pareto solution of (2), then there exist real numbers u j ≥ 0, v j ≥ 0, with j ∈ I, and τ > 0 such that

j∈I u j ∂f j (x * ) + j∈I v j ∂s j (x * ) + τ ∂d D (x * ) = 0, with j∈I (u j + v j ) = 1 and v j s j (x * ) = 0, j ∈ I.
Proof The proof follows from Minami [START_REF] Minami | Weak Pareto-optimal necessary conditions in a nondifferentiable multiobjective program on a Banach space[END_REF]Theorem 3.1]. Remark 4 It is worth to mention that a stronger version of the above theorem can be found in Mordukhovich [START_REF] Mordukhovich | Variational Analysis and Generalized Differentiation[END_REF]Theorem 5.61] and Mordukhovich [48, Theorem 9.22] by using the Mordukhovich's subdifferential; see also Mordukhovich [START_REF] Mordukhovich | Variational Analysis and Generalized Differentiation. I. Basic Theory[END_REF]. In [START_REF] Mordukhovich | Variational Analysis and Generalized Differentiation[END_REF][START_REF] Mordukhovich | Variational Analysis and Applications[END_REF], necessary optimality conditions for local minimizers are given for constrained multi-objective problems under different perspectives such as Pareto and weak Pareto points but also to other concepts of minimal points in multi-objective optimization, for instance, relative, intrinsic relative and quasi-relative minimal points. A visible disadvantage of weak Pareto points is the non-empty interior requirement on the order cone in both finite and infinite dimensional frameworks. In this case, the concept of relative interior points seems to be reasonable. The relative-type minimal points handle with a possible empty interior requirement of the order cone and they intermediate the position between Pareto and weak Pareto minimal points (equivalence between them holds for instance in finte dimensional spaces with a non-empty interior order cone).

The algorithm

The (scalar) proximal point method was introduced in the literature by Moreau [START_REF] Moreau | Proximité et dualité dans un espace Hilbertien[END_REF], Martinet [START_REF] Martinet | Regularisation d'inéquations variationelles par approximations succesives[END_REF] and later popularized by Rockafellar [START_REF] Rockafellar | Monotone operators and the proximal point algorithm[END_REF] who performs the method for the problem of finding zeros of operators. We recall that Bonnel et al. [START_REF] Bonnel | Proximal methods in vector optimization[END_REF] have proposed an extension of the proximal point method to vector optimization, i.e., when other underlying ordering cones are used instead of the non-negative orthant R m + . If we restrict the analysis to the finite dimensional multi-objective setting, the method proposed in [START_REF] Bonnel | Proximal methods in vector optimization[END_REF] generates a sequence satisfying

x k+1 ∈ argmin w F (x) + λ k 2 ||x -x k || 2 ε k : x ∈ Ω k , (7)
where

{λ k } ⊂ R ++ , {ε k } ⊂ R m + , Ω k = {x ∈ R n : F (x)
F (x k )} and "argmin w " denotes the set of weak Pareto solutions. The constrained set Ω k assures the vector improving process in the sense that F (x k+1 ) F (x k ).

Recently, Ji et al. [START_REF] Ji | Proximal point algorithms for multi-criteria optimization with the difference of convex objective functions[END_REF] have studied a proximal point method for vector functions F = (f 1 , . . . , f m ) : R n → R m such that, for each i ∈ I, f i (•) can be written as difference of two convex function, namely,

f i (x) = g i (x) -h i (x).
At the kth iterate, the algorithm proposed in [START_REF] Ji | Proximal point algorithms for multi-criteria optimization with the difference of convex objective functions[END_REF] computes a point x k+1 ∈ S solution of the following (scalar) subproblem

min x∈S max i∈I g i (x) -v k i , x -x k + θ k 2 ||x -x k || 2 , (8)
where v k i ∈ ∂h i (x k ), with ∂h i denoting a subdifferential of h i at x k , θ k > 0 for all k and, S is a compact and convex set. Note that (8) can be written as

min x∈S ξ F k (x) + θ k 2 ||x -x k || 2 e , (9)
where

F k (x) = G(x) -V k (x -x k ), V k ∈ ∂H(x k
), e = (1, . . . , 1) ∈ R m and ξ : R m → R is a scalarization function given by ξ(x) = max 1≤j≤m x, β j where {β j } is the canonical base of the space R m . The finite termination of an algorithm given by ( 9) was analyzed by Bento et al. [START_REF] Bento | A proximal point-type method for multicriteria optimization[END_REF]. Clearly, the sequence generated by [START_REF] Bolte | Clarke critical values of subanalytic Lipschitz continuous functions[END_REF] does not satisfies F (x k+1 ) F (x k ), for all k ∈ N as in [START_REF] Bento | Generalized inexact proximal algorithms: routine's formation with resistance to change, following worthwhile changes[END_REF]. This descent property plays an important role for a large class of application problems; see for instance Bento and Soubeyran [START_REF] Bento | Generalized inexact proximal algorithms: routine's formation with resistance to change, following worthwhile changes[END_REF]. Other authors have considered variants of the proximal method proposed by Bonnel et al. [START_REF] Bonnel | Proximal methods in vector optimization[END_REF] using the improving constraint F (x k+1 ) C F (x k ), for some ordering cone C; for instance Apolinário et al. [START_REF] Apolinário | A scalarization proximal point method for quasiconvex multiobjective minimization[END_REF], Bento et al. [START_REF] Bento | A proximal point-type method for multicriteria optimization[END_REF], Ceng and Yao [START_REF] Ceng | Approximate proximal methods in vector optimization[END_REF], Ceng et al. [START_REF] Ceng | Hybrid approximate proximal method with auxiliary variational inequality for vector optimization[END_REF], Choung et al. [START_REF] Choung | Hybrid approximate proximal algorithms for efficient solutions in vector optimization[END_REF], Villacorta and Oliveira [START_REF] Villacorta | An interior proximal method in vector optimization[END_REF]. However, the constrained set Ω k is convex in all these works. From now on, we assume the vector functions G, H : R n → R m are R m +convex with H continuously differentiable. Let F be a DC vector function given by F (x) := G(x) -H(x). Take a vector z ∈ R m + \{0} fixed, and auxiliar sequences {λ k } ⊂ R ++ and {ε k } ⊂ R m ++ such that ||ε k || = 1 for all k. In this case, the proximal point method generates a sequence {x k } as follows:

ALGORITHM 1. Initialization: Choose x 0 ∈ D. Stopping rule: Given x k , if x k is a Pareto critical point, then set x k+p = x k for all p ∈ N.
Iterative step: Take, as next iterate, x k+1 ∈ D such that

x k+1 ∈ arg min G(x) -JH(x k )(x -x k ) + λ k 2 q 2 (x k , x)ε k , z : x ∈ Ω k , (10)
where

Ω k = {x ∈ D : F (x) F (x k )}.
Next, we present some interesting comments and existing results about proximal point methods in vector optimization literature.

Remark 5 As mentioned before, the first extension of the proximal point method to vector optimization was proposed by Bonnel, Iusem and Svaiter [START_REF] Bonnel | Proximal methods in vector optimization[END_REF] (see [START_REF] Bento | Generalized inexact proximal algorithms: routine's formation with resistance to change, following worthwhile changes[END_REF]). They assume that each iterate of the sequence is a weak Pareto solution of a (constrained) vector optimization problem. Note that Algorithm 1 computes at each subproblem a solution of a scalar problem. This is the well-known scalarization approach for solving a multi-objective optimization problem. As mentioned in [START_REF] Bonnel | Proximal methods in vector optimization[END_REF]Remark 5] the difference between the presentation of the iterative steps in ( 10) and ( 7) is not substantial because every solution of the scalar subproblems ( 10) is a weak Pareto solution of the subproblem [START_REF] Bento | Generalized inexact proximal algorithms: routine's formation with resistance to change, following worthwhile changes[END_REF] with

F (x) = G(x) -JH(x k )(x -x k ); see [38, Proposition 2.2].
Remark 6 Note that in [START_REF] Bento | Generalized inexact proximal algorithms: routine's formation with resistance to change, following worthwhile changes[END_REF], for each k ∈ N, the mapping

F k (x) = F (x) + λ k 2 ||x -x k || 2 ε k is R m
+ -convex, and hence, its Pareto critical points are weak Pareto point; see [START_REF] Fukuda | A survey on multiobjective descent methods[END_REF]Lemma 3.5]. This is why the authors in [START_REF] Bonnel | Proximal methods in vector optimization[END_REF] take as next iterate a point x k+1 which is a weak Pareto point of F k (constrained to Ω k ) besides the fact that the set of weak Pareto contains the Pareto ones. In this context, it is proved that the algorithm [START_REF] Bento | Generalized inexact proximal algorithms: routine's formation with resistance to change, following worthwhile changes[END_REF] converges to a weak Pareto solution of F . In some applications it is often the case that only Pareto solutions (instead of weak Pareto ones) are of interest; see [START_REF] Jahn | Vector Optimization: Theory, Applications and Extensions[END_REF]. In order to obtain convergence of the proximal point method to a Pareto solution (instead of weak Pareto point), the authors in [START_REF] Bonnel | Proximal methods in vector optimization[END_REF] consider a more restrictive choice of x k+1 , namely, they take x k+1 a properly Pareto point of F k (see this definition in [10, page 957]). One has that a properly Pareto point is a Pareto point, and hence, it is a weak Pareto point.

Remark 7 Recall that Algorithm 1 satisfies x k+1 ∈ Ω k , for all k ≥ 0, i.e.,

F (x k+1 ) F (x k ), (11)
which is an important property for some application problems, as we will see in Section 6, while the algorithm in [START_REF] Ji | Proximal point algorithms for multi-criteria optimization with the difference of convex objective functions[END_REF] satisfies

i∈I α i (f i (x k+1 ) -f i (x k )) ≤ 0, (12)
for some α i ≥ 0 with i∈I α i = 1. Clearly, the descent property ( 11) implies [START_REF] Burke | On the Clarke subdifferential of the distance function of a closed set[END_REF].

Remark 8 It is known that every DC function is a locally Lipschitz function; see [START_REF] Hiriart-Urruty | Generalized differentiabity, duality and optimization for problems dealing with difference of convex functions, Convexity and Duality in Optimization[END_REF]. The proximal point method for locally Lipschitz functions in multiobjective optimization was analyzed by Bento et al. [START_REF] Bento | The proximal point method for locally Lipschitz functions in multiobjective optimization with application to the compromise problem[END_REF] who firstly considered a possibly non-convex vector improving constraints Ω k . However, we mention that Algorithm 1 (applied for DC functions) offers an additional flexibility for considering, at each step, a linear approximation

G(x) -JH(x k )(x -x k ) instead of minimize the non-convex function F (x) = G(x) -H(x) directly.
Minimizing a linear approximation is very important for instance for applications in Behavioral Sciences when agents are badly informed, i.e., agents have limited or partial knowledge. An interesting application by using a linear (local) approximation of each (scalar) cost function was considered by Cruz Neto et al. [START_REF] Neto | A generalized proximal linearized algorithm for DC functions with application to the optimal size of the firm problem[END_REF].

Remark 9 As mentioned in Huang and Yang [START_REF] Huang | Duality for multiobjective optimization via nonlinear Lagrangian functions[END_REF], the vector functions

F (•) and e F (•) := (e f1(•) , . . . , e fm(•) )
have the same set of weak Pareto points, where e α denotes the exponential map valued at α ∈ R. This result can be easily extended to the Pareto critical setting. Hence, concerning Pareto critical points, we can assume without loss of generality that F 0. On the other hand, any non-negative DC function admits a non-negative DC decomposition; see [START_REF] Hiriart-Urruty | Generalized differentiabity, duality and optimization for problems dealing with difference of convex functions, Convexity and Duality in Optimization[END_REF]. Therefore, we will assume throughout of this paper that G 0. Proposition 3 Algorithm 1 is well-defined.

Proof The proof will be made by induction on k. Let {x k } be a sequence defined in [START_REF] Bonnel | Proximal methods in vector optimization[END_REF] and φ : R n → R ∪ {+∞} given by

φ(x) = G(x), z -[JH(x k )(x -x k )] z + λ k 2 ε k , z q 2 (x k , x) + I Ω k (x). ( 13
)
Since G 0 and (1) holds, it follows that φ is coercive taking into account that (λ k /2) ε k , z > 0. Then, as Ω k is closed, there exists x ∈ Ω k such that x ∈ argmin x∈Ω k φ(x).

Therefore, we can take x k+1 := x and the induction is done, which ends the proof.

From now on, we consider {x k }, {λ k } and {ε k } the sequences appearing in Algorithm 1. Next, we explore deeply the structure of the vector problem by using the necessary condition for a weak Pareto point of a multi-objective problem given by Theorem 1. The following result will be used in our main convergence result. [START_REF] Ceng | Hybrid approximate proximal method with auxiliary variational inequality for vector optimization[END_REF] where

Proposition 4 For all k ∈ N, there exist A k , B k ∈ R m×n , u k , v k ∈ R m + with ||u k + v k || 1 = 1, w k ∈ R m and τ k ∈ R ++ such that [A k -JH(x k-1 )] u k + λ k-1 ε k-1 , u k q(x k-1 , x k )η k + B k v k + τ k w k = 0,
A k ∈ ∂G(x k ), B k ∈ ∂F (x k ), w k ∈ B[0, 1] ∩ N D (x k ), η k ∈ ∂q(x k-1 , •)(x k ). ( 15
)
Proof It follows from definition of the algorithm that, for each k ∈ N, x k is a solution of the scalar problem

min G(x) -JH(x k-1 )(x -x k-1 ) + λ k-1 2 q 2 (x k-1 , x)ε k-1 , z : x ∈ Ω k-1 .
Thus, from [38, Proposition 2.2], we have that x k is a weak Pareto solution of the vector problem

min w {G(x) -JH(x k-1 )(x -x k-1 ) + λ k-1 2 q 2 (x k-1 , x)ε k-1 : x ∈ Ω k-1 }.
Note that

(f k ) j (x) = g j (x) -∇h(x k ), x -x k + λ k 2 q 2 (x k , x)ε k j , with j ∈ I, (16)
and

(s k ) j (x) = f j (x) -f j (x k ), with j ∈ I, (17)
are locally Lipschitz functions, for each j ∈ I, because each function in ( 16) is a sum of the convex function g j (•) -∇h(x k ), • -x k with the locally Lipschitz function λ k 2 q 2 (x k , •)ε k j and the functions in [START_REF] Clarke | Optimization and Nonsmooth Analysis[END_REF] are DC functions, for each j ∈ I, and hence locally Lipschitz. Besides, taking into account that x k ∈ D, we have that ∂d D (x k ) = B[0, 1] ∩ N D (x k ); see Burke et al. [START_REF] Burke | On the Clarke subdifferential of the distance function of a closed set[END_REF]. For each k ∈ N fixed applying Theorem 1 for f j (x) = (f k ) j (x) and s j (x) = (s k ) j (x) given by ( 16) and [START_REF] Clarke | Optimization and Nonsmooth Analysis[END_REF], respectively, we obtain that there exist

A k , B k ∈ R m×n , u k , v k ∈ R m + with ||u k + v k || 1 = 1, w k ∈ R m and τ k ∈ R ++ such that 0 ∈ [A k -JH(x k-1 )] u k + B k v k + τ k w k + λ k-1 2 ε k-1 , u k ∂q 2 (x k-1 , •)(x k ).
(18) Applying Proposition 2 for f 1 (x) = f 2 (x) = q(x k , x), we have

∂q 2 (x k-1 , •)(x k ) ⊂ 2q(x k-1 , x k )∂q(x k-1 , •)(x k ).
Combining this fact with [START_REF] Neto | A generalized proximal linearized algorithm for DC functions with application to the optimal size of the firm problem[END_REF] we have that there exists η k ∈ ∂q(x k-1 , •)(x k ) such that [START_REF] Choung | Hybrid approximate proximal algorithms for efficient solutions in vector optimization[END_REF] holds and the proof is finished.

Remark 10 Note that from [START_REF] Choung | Hybrid approximate proximal algorithms for efficient solutions in vector optimization[END_REF], {u k }, {v k } and {w k } are bounded sequences. As mentioned by Bolte et al. [9, Remark 1], ∂f j is bounded on compact sets. So, we have that {A k }, {B k } and {η k } are bounded sequences as long as {x k } is bounded. Therefore, if {λ k } and {x k } are bounded sequences, it follows from (14) that {τ k } is also bounded.

As a consequence of the previous proposition, we obtain the following stopping rule for Algorithm 1.

Corollary 1 Let k 0 ∈ N be such that u k0 = 0. Then, x k0 is a Pareto critical point of F .

Proof If there exists k 0 ∈ N such that u k0 = 0, then [START_REF] Ceng | Hybrid approximate proximal method with auxiliary variational inequality for vector optimization[END_REF] implies

B k0 v k0 + τ k0 w k0 = 0.
As τ k0 > 0 and w k0 ∈ N D (x k0 ), last equality implies

-B k0 v k0 ∈ N D (x k0 ).
Hence, the desired result follows by using Lemma 1 and taking into account that v k0 ∈ R m + \{0}. As in [START_REF] Bonnel | Proximal methods in vector optimization[END_REF], the stopping rule in Algorithm 1 can be changed by the following, which is easier to check: after computing x k+1 the algorithm stops if x k+1 = x k , i.e., we set x k+p = x k for all p ∈ N. Similar to Corollary 1 last proposition combined with Lemma 1 allows to see that this condition is sufficient to get the stopping rule given in Algorithm 1.

Corollary 2 If x k+1 = x k , then x k is a Pareto critical point of F .

Convergence analysis

In this section, we analyze the convergence of Algorithm 1. Note that, if the algorithm terminates after a finite number of iterations, it terminates at a Pareto critical point. Throughout this paper, we suppose that {x k } is an infinite sequence. Thus, we assume that x k+1 = x k and u k = 0 for all k ∈ N, in view of Corollaries 2 and 1, respectively. Additionally, we assume that {λ k } is a bounded sequence such that lim inf

k λ k ε k , z > 0, (19)
which is easily verified if z ∈ R m ++ and inf k∈N λ k > 0.

Proposition 5 The following properties hold:

i) { F (x k ), z } is decreasing; ii) lim k→+∞ q(x k , x k+1 ) = 0.
Proof Since H is a differentiable and convex function, we have

JH(x k )(x k+1 -x k ), z ≤ H(x k+1 ) -H(x k ), z , ∀k ∈ N. (20)
On the other hand, from (10), we have, for each k ∈ N,

G(x k+1 ) -JH(x k )(x k+1 -x k ) + λ k 2 q 2 (x k , x k+1 )ε k , z ≤ G(x k ), z , (21)
which, combined with (20) and F (x) = G(x) -H(x), implies that

F (x k+1 ), z + λ k 2 ε k , z q 2 (x k , x k+1 ) ≤ F (x k ), z , ∀k ∈ N. (22)
Thus, using that (λ k /2) ε k , z > 0 and x k+1 = x k for all k ∈ N, item i) is proved. Now, taking into account that F 0 and z ∈ R m + \{0}, from item i), we have that { F (x k ), z } is a convergent sequence. From [START_REF] Fliege | Newton's method for multiobjective optimization[END_REF], we have

0 ≤ λ k 2 ε k , z q 2 (x k , x k+1 ) ≤ F (x k ), z -F (x k+1 ), z , ∀k ∈ N. (23)
Thus, taking k → +∞ in last inequality, we obtain that the right hand-side of ( 23) vanishes which combined with [START_REF] Neto | A subgradient method for multiobjective optimization[END_REF] leads to the desired result.

Theorem 2 Every cluster point of {x k }, if any, is a Pareto critical point of F .

Proof Let x be a cluster point of {x k } and consider a subsequence {x k l } of {x k } converging to x. From Proposition 4, there exist sequences [START_REF] Ceng | Hybrid approximate proximal method with auxiliary variational inequality for vector optimization[END_REF]. Since {x k l } is convergent and {λ k } is bounded, it follows from Remark 10 that {A k }, {B k }, {u k }, {v k }, {w k }, {τ k } and {η k } are bounded. Thus, we can assume without loss of generality that

{A k }, {B k } ⊂ R m×n , {u k }, {v k } ⊂ R m + , {w k } ⊂ R m , {τ k } ⊂ R ++ and {η k }, with η k ∈ ∂q(x k-1 , •)(x k ), satisfying
A k l → Â, B k l → B, u k l → û, v k l → v, w k l → ŵ
and τ k l → τ (we may extract other subsequences if necessary). From [START_REF] Ceng | Hybrid approximate proximal method with auxiliary variational inequality for vector optimization[END_REF] we have [START_REF] Fukuda | On the convergence of the projected gradient method for vector optimization[END_REF] where γ k l = λ k l -1 ε k l -1 , u k l . Note that {γ k } is bounded. This fact combined with Proposition 5 implies that {γ k l q(x k l -1 , x k l )η k l } converges to 0 as l → +∞. Note that ∂f i (x) = ∂g i (x) -∇h i (x), for each i ∈ I; see Bačák and Borwein [START_REF] Bačák | On difference convexity of locally Lipschitz functions[END_REF]. Thus, taking the limit as l → +∞ in [START_REF] Fukuda | On the convergence of the projected gradient method for vector optimization[END_REF] and having in mind the closedness property of ∂f (•) and N D (•) (in the sense that if w k → w, x k → x and w k ∈ X (x k ), then w ∈ X (x)), we have

[A k l -JH(x k l -1 )] u k l + γ k l q(x k l -1 , x k l )η k l + B k l v k l + τ k l w k l = 0,
Ĉ û + B v + τ ŵ = 0, (25)
where Ĉ := ( Â -JH(x)) ∈ ∂F (x), B ∈ ∂F (x) and τ ŵ ∈ N D (x). Hence, from (25) we obtain

-( Ĉ û + B v) ∈ N D (x). ( 26
)
Note that û, v ∈ R m + and from (15), we have that û = 0 or v = 0. Therefore, the desired result follows from (26) combined with Lemma 1.

The "group dynamic" problem

In this section, we have in mind a group of producers I = {1, 2, . . . , m}. The objective of each of them is a "to be increased" payoff (profit, utility), f i (x) = g i (x) -h i (x), where g i (x) ∈ R + and h i (x) ∈ R + refer to their scalar revenue and cost function. The decision variables of the group is the vector x ∈ R n which must satisfy some constraints. Each agent wants a payoff as high as possible. The goal of the group I is to approach and reach a Pareto point, or a Pareto critical point. The vectorial objective of the group is F (x) = G(x) -H(x) ∈ R m , where F (x) = ( f 1 (x), . . . , f m (x)), G(x) = ( g 1 (x), . . . , g m (x)) and H(x) = ( h 1 (x), . . . , h m (x)). In a dynamic cooperative setting, all agents of the group will accept to change from the last position x = x k to the next, y = x k+1 only if their payoffs do not decrease, i.e., if f i (x k+1 ) ≥ f i (x k ), for all i ∈ I. In the opposite case, some agents will quit the group or resist to change. This defines a cooperative improving dynamic x k+1 ∈ Ω(x k ), where

Ω(x k ) = y ∈ D : F (y) F (x k
) . The cooperative group dynamic problem is to find a cooperative improving dynamic x k+1 ∈ Ω(x k ) which converges to (approaches and reaches) a Pareto critical point.

In real life, in a dynamic setting, most of the time, increasing returns prevail, coming from fixed costs, learning by doing, .... This means that each marginal revenue and cost function is decreasing. Then, each revenue g i and cost h i function is concave. In this case the payoff of each agent is a difference between two concave functions. We consider "to be decreased" payoffs f i (y) = -f i (y), where g i (y) = -g i (y) and h i (y) = -h i (y), which are the difference of two convex functions f i (y) = g i (y) -h i (y). Then, we will provide a multi-objective DC algorithm which converges to a Pareto critical point, with a very important added requirement: the algorithm must follow a cooperative improving dynamic x k+1 ∈ Ω(x k ). In this behavioral setting, the papers of Bento et al. [START_REF] Bento | A proximal point-type method for multicriteria optimization[END_REF], Bonnel et al. [START_REF] Bonnel | Proximal methods in vector optimization[END_REF] and Choung et al. [START_REF] Choung | Hybrid approximate proximal algorithms for efficient solutions in vector optimization[END_REF] make "as if" they consider this case of a cooperative improving dynamic. However, the aforementioned works use convexity of the improving sets Ω(x k ), for all k ∈ N. A motivation, in a dynamic context, to consider the constrained set Ω(x k ) is given in Bento et al. [START_REF] Bento | A proximal point-type method for multicriteria optimization[END_REF]. In the present paper, we drop this important convexity hypothesis, allowing for non-convex improving sets with DC multi-objective payoff functions. The algorithm proposed in Ji et al. [START_REF] Ji | Proximal point algorithms for multi-criteria optimization with the difference of convex objective functions[END_REF] cannot be viewed as a cooperative improving dynamic and this is why algorithm [START_REF] Bonnel | Proximal methods in vector optimization[END_REF] is better adapted to applications, for instance in behavioral sciences, than algorithm [START_REF] Bento | A generalized inexact proximal point method for nonsmooth functions that satisfies Kurdyka-Lojasiewicz inequality[END_REF].

In this paper, we suppose that agents have limited (partial) knowledge. We also suppose that they do not know their cost function, while they are aware of their revenue function. This is why we consider approximate objective functions, using a linear local approximation of each cost function (see Cruz Neto et al. [START_REF] Neto | A generalized proximal linearized algorithm for DC functions with application to the optimal size of the firm problem[END_REF] for a more precise justification in the scalar case of one agent, with one objective).

Worthwhile to change processes

For applications, consider the behavioral context of the dynamics of human behavior, at the level of an agent with several objectives, or at the level of an organization where each agent has his own objective. We will use the recent variational rationality approach (see Soubeyran [START_REF] Soubeyran | Variational rationality, a theory of individual stability and change: worthwhile and ambidextry behaviors[END_REF][START_REF] Soubeyran | Variational rationality and the "unsatisfied man": routines and the course pursuit between aspirations, capabilities and beliefs[END_REF][START_REF] Soubeyran | Variational rationality. A theory of worthwhile stay and change approach-avoidance transitions ending in traps[END_REF][START_REF] Soubeyran | Variational rationality.1. An adaptive theory of the unsatisfied man[END_REF][START_REF] Soubeyran | Variational rationality. 2. A general theory of goals and intentions as satisficing worthwhile moves[END_REF]) where, at each period, agents accept to change if these changes are worthwhile. This is the case when their (vectorial) motivation to change M = U [A] ∈ R m are high enough with respect to their (vectorial) resistance to change R = D [I] ∈ R m + , i.e., M ≥ ξR. High enough means that the threshold level ξ > 0 is high enough (it represents a satisfying ratio).

Let us show very briefly how, in this variational rationality context, our algorithm [START_REF] Bonnel | Proximal methods in vector optimization[END_REF] represents a worthwhile stay and change process.

This paper considers the linear-quadratic case where the utilities U [A] = A of their advantages to change A ∈ R m are higher enough with respect to the disutilities D [I] = I 2 of their inconveniences to change I ∈ R m + , i.e., A ≥ ξI 2 , where I 2 is the vector of squares of each component of vector I. At each period, advantages to change moving from "repeating the last action x" to "doing a new action y" are defined as the difference A = A(x, y) = P (y) -P (x) between the current (to be increased) vectorial payoff P (y) ∈ R m to do the current action y and the current vectorial payoff P (x) ∈ R m to repeat the last action x. In this paper F (•) = -P (•) represents a vector of unsatisfied needs (to be decreased). Then, A(x, y) = F (x) -F (y). Inconveniences to change I(x, y) = C(x, y) -C(x, x) refer to the difference between costs C(x, y) ∈ R m + of being able to change from x to y and costs C(x, x) ∈ R m + of being able to repeat action x. In our paper C(x, y) = q 2 (x, y)ε, where ε ∈ R m ++ is a vector of shares. Then, a change from x to y is worthwhile if A(x, y) ≥ ξC(x, y) 2 , i.e., F (x) -F (y) ξq 2 (x, y)ε.

Let z ∈ R n ++ be a given vector of weights which helps to add different advantages to change and different inconveniences to change, and allow to compare their scalarized formulations. In this case a change is worthwhile if

F (x) -F (y), z ≥ ξq 2 (x, y) ε, z .
Let k and k + 1 be the previous and current periods, where x = x k and y = x k+1 . Then, in the current period, a change from repeating the previous action x k to perform the new action x k+1 is worthwhile if

F (x k ) -F (x k+1 ), z ≥ ξ k q 2 (x k , x k+1 ) ε k , z ,
where {ε k } and ξ k = λ k /2 are as in [START_REF] Bonnel | Proximal methods in vector optimization[END_REF].

This proves that our algorithm is a specific instance of a worthwhile stay and change process (see [START_REF] Fliege | Newton's method for multiobjective optimization[END_REF]). Our paper shows when, in the context of cooperative group dynamics where all agents require, at each period, that their payoffs do not decrease, a worthwhile stay and change transition approaches and reaches a Pareto critical point. Future research will consider the case of weak Pareto points, to better fits with applications to group dynamics ending in some trap. Of course the convex case ends in weak Pareto points which are traps; see [START_REF] Bonnel | Proximal methods in vector optimization[END_REF]Theorem 3.1].

Numerical experiments

In this section we present some numerical results to verify the practical efficiency of the proposed method. We consider some academic test problems with smooth and non-smooth multi-objective DC functions. These functions are introduced in the sequel and after that we report our numerical experiments analyzing our results under different point of view. We coded our simulations in MATLAB on a machine AMD Athlon(tm) X2 with a Dual Core Processor and 1.20 GHz CPU. In all tests we consider Algorithm 1 with D = R 2 , λ k = 1, for all k ∈ N, q(x, y) = ||x -y|| and the stopping rule

||x k+1 -x k || < 10 -4 .
The proximal point method has proved to be an efficient tool in several instances, be it through direct application or through methods which can be derived from it. The performance of the method strongly depends on the algorithm used to solve the subproblems. In this situation, it makes little sense to compare the proximal method with other methods in terms of computational efficiency, unless a specific procedure is chosen for solving the subproblems. In this paper, we refrain from discussing algorithms to solve the subproblems, and hence we skip a discussion of comparing it with other methods. However, we compare the performance of our method with the proximal-type method (8) for multi-objective DC function proposed in [START_REF] Ji | Proximal point algorithms for multi-criteria optimization with the difference of convex objective functions[END_REF] (called Algorithm 2). In our simulations the subproblems are solved using the MATLAB routine "fmincon".

Problem 1 Let F : R 2 → R 2 be a non-smooth multi-objective DC function given by F (x, y) = G(x, y) -H(x, y), where G(x, y) = (x -3) 2 + (y -4) 2 + 4, (x + 6) 2 + (y + 5) 2 + 4 and

H(x, y) = 0.3(x -3) 2 + 0.2(y -4) 2 , 0.01 (x + 6) 2 + (y + 5) 2 .
Problem 2 Let F : R 2 → R 2 be a multi-objective DC function given by F (x, y) = G(x, y) -H(x, y), where

G(x, y) = 1 2 (x 2 + y 2 ) 2 , x 2 + y 2 and H(x, y) = 1 2 (x 4 + y 4 ), 0.5x 2 + xy + 0.5y .
Problem 3 Let F : R 2 → R 2 be a multi-objective DC function given by F (x, y) = G(x, y) -H(x, y), where G(x, y) = x 2 + y 2 , x 2 + y 2 and H(x, y) = x 2 + y 2 -xy, 0.5x 2 + xy + 0.5y 2 .

Computing Pareto critical points

Next, we perform Algorithm 1 for 30 randomly generated starting points. In this subsection, we consider z = (1, 2) and

ε k = ( 1 √ 2 , 1 √
2 ), for all k ∈ N. The results are presented in Table 1. The column Iteration (k) refers to the number of iteration until the stopping rule is satisfied. The column Time(s) shows the time in second from the starting point is generated to the algorithm stoped. The other columns present the starting point used, the limit point obtained and its objective value, respectively. Algorithm 1 solves in each subproblem a multi-objective problem subject to the constraint set Ω k which imposes a descent process. Such a condition does not make the algorithm unpractical or even costly. Actually, our analysis shows that Algorithm 1 has better performance (in all the 30 experiments) in number of iterations and running time than Algorithm 2. The performance of Algorithm 2 is presented in Table 2. In Algorithm 2, we use the same MATLAB routine to solve the subproblems and the same initial points considered in Table 1. The subproblems are solved subject to the set S = {(x, y) ∈ R 2 : a ≤ x ≤ b, c ≤ x ≤ d} large enough to include the solution set.

x 0 x * F (x * ) = (F 1 (x * ), F 2 (x * )) 1 46
It is worth to mention that Algorithm 2 finds the same Pareto critical point of Problem 1 in all the 30 experiments. On the other hand, our method has been shown to be efficient to obtain an approximation of the solution set when it is performed many times for different initial points. This is illustrated in Figures 1, 2 and3 where the blue squares, red stars and blue stars denote the generated initial points, the points of the sequence and the limit points of the sequence generated by Algorithm 1, respectively. We also mention that Algorithm 2 does not necessarily decrease all the components of the objective function as our method does. This is shown in Table 3 and 4 where the algorithms are applied to solve Problem 1 with initial point x 0 = (20, -20). In last column of Table 4 we report the values F (x k+1 )-F (x k ) for Algorithm 2 proving that the method is not a descent process.

Analysis of the parameters ε k and z

In this subsection we analyze the performance of Algorithm 1 taking into account the variation of the parameters ε k and z for Problem 1. Firstly, we analyze the influence of the parameter ε k in the time of convergence of Algorithm 1 by considering it fixed and variable in all iterations. For this case, we run the algorithm 30 times with randomly generated initial points. Each time, it is considered

ε k = ( 1 √ 2 , 1 √
2 ) fixed, for all k ∈ N, and ε k randomly variable for each k ∈ N. The results are presented in Figure 4. One can note that considering the parameter ε k variable provided a better performance of the method.

Secondly, we study the influence of parameter z in the time of convergence of the algorithm. To this end, we run the algorithm 30 times for each one of the following four cases: case 1 the initial point is (-20, -20); case 2 the initial point is [START_REF] Dinh | Duality and optimality conditions for generalized equilibrium problems involving DC functions[END_REF][START_REF] Dinh | Duality and optimality conditions for generalized equilibrium problems involving DC functions[END_REF]; case 3 the initial point is (-20, 20); case 4 the initial point is (20, -20). In all these cases, we consider the parameter ε k randomly variable for each k ∈ N and for each one of the 30 times that the algorithm was performed we take z = (10 cos( nπ 60 ), 10 sin( nπ 60 )), for n = 1, 2 . . . , 30. One can note that the choice of the parameter z does not significantly change the performance of the method (worst performance) except if it is taken close to the y-axis.

Iteration (k) time (s) ||x k+1 -x k || x k F (x k ) = (F 1 (x k ), F 2 (x k )) 1

Final remarks

Convergence properties of a proximal point method for vector-valued DC functions adding a (non-convex) vectorial improving constraint have been proved. Bento et al. [START_REF] Bento | The proximal point method for locally Lipschitz functions in multiobjective optimization with application to the compromise problem[END_REF] handled with a proximal algorithm for multi-objective optimization involving locally Lipchitz vector functions and (non-convex) vectorial improving constraints. Although we use the same approach of convergence analysis our method is different to the one considered in [START_REF] Bento | The proximal point method for locally Lipschitz functions in multiobjective optimization with application to the compromise problem[END_REF] because it minimizes at each iteration a convex approximation instead of the (non-convex) objective function. Our method generalizes (in a finite dimensional multi-objective setting) the exact proximal point method introduced by Bonnel et al. [START_REF] Bonnel | Proximal methods in vector optimization[END_REF] and our descent property implies the one obtained by Ji et al. [START_REF] Ji | Proximal point algorithms for multi-criteria optimization with the difference of convex objective functions[END_REF] who do not consider vectorial improving constraints. Preliminary computational experiments have been conducted to check that the non-convex constraints does not make the method costly. The performance of the method has been compared to the proximal method considered by [START_REF] Ji | Proximal point algorithms for multi-criteria optimization with the difference of convex objective functions[END_REF]. It also has been shown how our proximal method can greatly help to solve a "group dynamic" problem by using the recent variational rationality approach. In view of Remarks 4 and 6, it seems to be interesting to extend the approach proposed in [START_REF] Bento | The proximal point method for locally Lipschitz functions in multiobjective optimization with application to the compromise problem[END_REF] to infinite dimensional space and consider the concepts of relative minimizers in multi-objective optimization given by Mordukhovich [START_REF] Mordukhovich | Variational Analysis and Generalized Differentiation[END_REF][START_REF] Mordukhovich | Variational Analysis and Applications[END_REF] which generalize the concepts of Pareto and weak Pareto points. We will consider this topic in a future research.

Remark 3

 3 As remarked by Minami [42, Remark 3.1], since D is a closed and convex set, the cone {w : w ∈ τ ∂d D (x * ), τ > 0} is the normal cone N D (x * ).

Fig. 1 Fig. 2 Fig. 3

 123 Fig. 1 Approximation of the solution set of Problem 1

Fig. 4 Fig. 5

 45 Fig. 4 Parameter ε k in Algorithm 1: fixed × variable

Table 1

 1 Running 30 times Algorithm 1 for Problem 1 with random starting points

	#	Iteration (k)	Time (s)	x 0	x *	F (x * ) = (F 1 (x * ), F 2 (x * ))
	1	31	8.564	(-23.1101,20.7594)	(1.801711408,2.933901786)	(5.914379229,13.88901642)
	2	8	4.025	(-19.657,-27.781)

Table 2

 2 Running 30 times Algorithm 2 for Problem 1 with the same starting points used

	in Algorithm 1
	#	Iteration (k)	Time (s)

Table 3

 3 Algorithm 1 applied to Problem 1 from the starting point x 0 =(20, -20)

Table 4

 4 Algorithm 2 applied to Problem 1 from the starting point x 0 =(20, -20)

	Iteration (k)	time (s)	||x k+1 -x k ||	x k	F (x k+1 ) -F (x k )
	1	2.008	540.2782	(12.562497,-8.000002916)	(-483.89099980,-5.73896609)
	2	1.406	396.3238	(8.378802864,-1.99999502)	(-130.15698383,-2.73676509)
	3	0.861	108.2335	(6.025582502,1.000009166)	(-35.44415573,-0.89777979)
	4	0.548	28.0215	(5.144387567,1.498727569)	(-5.38389099,-0.39665191)
	5	0.733	1.1728	(4.610249688,1.187252036)	(-0.07969592,-0.46260705)
	6	0.641	0.88719	(4.067198442,0.8705688819))	(0.48763633,-0.47816945)
	7	0.502	0.83687	(3.527535025,0.7120754666)	(0.21125337,-0.42110684
	8	0.504	0.68461	(3.043118796,0.7434488424)	(-0.35776162,-0.30941565)
	9	0.521	0.57117	(2.614787298,0.803043836)	(-0.20510666,-0.25709370)
	10	0.516	0.47313	(2.252940969,0.9022951263)	(-0.21300673,-0.19125228)
	11	0.578	0.38491	(1.952390871,1.016024702))	(-0.17576213,-0.13957153
	12	0.77	0.31081	(1.707198679,1.133397375)	(-0.14766312,-0.09763097)
	13	0.543	0.25051	(1.508999834,1.245808462)	(-0.11924930,-0.06655698
	14	0.439	0.2022	(1.349762745,1.348786032)	(-0.09516737,-0.04419338)
	15	0.484	0.16364	(1.2221836 ,1.440223332)	(-0.07503942,-0.02867450)
	16	0.464	0.13283	(1.120031828,1.519695262)	(-0.05888158,-0.01815865)
	17	0.42	0.10802	(1.038228736,1.587677475)	(-0.04610299,-0.01117334)
	18	0.394	0.087993	(0.9726313697,1.645176892)	(-0.03611252,-0.00662644)
	19	0.405	0.071745	(0.9199493651,1.693397949)	(-0.02835212,-0.00371806)
	20	0.393	0.058524	(0.877569491,1.733578141)	(-0.02232555,-0.00189737)
	21	0.394	0.047747	(0.8434215179,1.766894321)	(-0.01764201,-0.00078698
	22	0.417	0.038954	(0.8158644645,1.794414643)	(-0.01399127,-0.00013417)
	23	0.408	0.031775	(0.7935956034,1.817080566)	(-0.01113498,0.00022864)
	24	0.385	0.025915	(0.7755783156,1.835705368)	(-0.00889078,0.00041115)
	25	0.396	0.021131	(0.7609856084,1.850981736)	(-0.00711987,0.00048440)
	26	0.345	0.017228	(0.749155217,1.863494331)	(-0.00571666,0.00049383)
	27	0.36	0.014042	(0.7395576356,1.873730913)	(-0.00460079,0.00046820)
	28	0.359	0.011444	(0.7317663605,1.882097829)	(-0.00370959,0.00042505)
	29	0.343	0.0093247	(0.7254380348,1.888931632)	(-0.00299615,0.00037520)
	30	0.323	0.0075974	(0.7202956245,1.894510011)	(-0.00242332,0.00032481)
	31	0.345	0.0061895	(0.7161152716,1.899061535)	(-0.00196234,0.00027728)
	32	0.345	0.0050422	(0.7127158586,1.902773916)	(-0.00159064,0.00023429)
	33	0.374	0.0041075	(0.7099506166,1.905801125)	(-0.00129045,0.00019641)
	34	0.326	0.0033462	(0.7077005881,1.908269211)	(-0.00104770,0.00016369)
	35	0.361	0.0027263	(0.7058691266,1.910281391)	(-0.00085115,0.00013580)
	36	0.329	0.0022222	(0.7043774777,1.911922317)	(-0.00069195,0.00011227)
	37	0.375	0.0018085	(0.7031644616,1.913258462)	(-0.00056303,0.00009260)
	38	0.551	0.0014687	(0.7021800244,1.914343991)	(-0.00045720,0.00007604)
	39	0.55	0.0011961	(0.7013784133,1.915228136)	(-0.00037062,0.00006216)
	40	0.527	0.00097422	(0.7007257972,1.915948489)	(-0.00030196,0.00005102)
	41	0.532	0.00079357	(0.7001943229,1.916535355)	(-0.00024561,0.00004174)
	42	0.527	0.00064653	(0.699761428,1.917013555)	(-0.00019999,0.00003414)
	43	0.583	0.00052688	(0.6994087086,1.917403296)	(-0.00016283,0.00002791)
	44	0.567	0.00042954	(0.6991211918,1.91772106)	(-0.00013266,0.00002280)
	45	0.547	0.0003504	(0.6988866665,1.917980293)	(-0.00010812,0.00001864)
	46	0.296	0.00028612	(0.6986951772,1.918191971)	(-0.00008819,0.00001523)
	47	0.141	0.0013963	(0.6977027884,1.919174066)	(-0.00008819,0.00001523)
	48	0.205	0.00022594	(0.697905313,1.919118593)	(-0.00046806,0.00008155)
	49	0.153	5.0025e-005	(0.6978705211,1.919087548)	(0.000215490,-0.00003755)


Issam Mjejri

Stéphane Mornet

Manuel Gaudon
email: manuel.gaudon@icmcb.cnrs.fr

From nano-structured polycrystalline spheres with Zn1-xCoxO composition to core-shell Zn1-xCoxO@SiO2 as green pigments

Keywords: Polyol synthesis, Zinc-Cobalt oxide, Green Pigment, Core-shell I

ZnO doped with Co 2+ , a well-known green pigment which is an alternative to the chromium based inorganic pigments, has been prepared by a polyol process and investigated in terms of crystallographic structure and UV-visible properties. Thanks to the obtaining of nanometric crystallite size from our process, the incorporation of a very high concentration of Co 2+ in the ZnO matrix is achieved. Thus, different grades of more or less deep green pigments can be produced. Furthermore, the obtaining of spherical aggregates allows the easy preparation of ZnO:Co@SiO2 core-shells, minimizing hence the problems linked to the zinc oxide high solubility into slightly acidic conditions acidic conditions and the metal cation's toxicity.

Introduction

Zinc oxide doped with cobalt, ZnO:Co is a well-known green pigment which is largely developed, especially as an alternative to the chromium based inorganic pigments [START_REF] Šulcova | New green pigments; ZnO-CoO[END_REF][START_REF] Zhou | Synthesis and characterization of Zn1-xCoxO green pigments with low content cobalt oxide[END_REF][START_REF] Karasu | Effects of cobalt, copper, manganese and titanium oxide additions on the microstructures of zinc containing soft porcelain glazes[END_REF][START_REF] Cui | Nanoparticle Synthesis of Willemite Doped with Cobalt Ions (Co0.05Zn1.95SiO4) by an Epoxide-Assisted Sol-Gel Method[END_REF][START_REF] Gaudon | Green Coloration of Co-Doped ZnO Explained from Structural Refinement and Bond Considerations[END_REF][START_REF] Milao | Zn0.97M0.03O (M = Co, Fe, and V) pigments: thermal, structural, and optical characterization[END_REF]. Indeed, the chromium has adverse effects on environment and the chroma a* value is often porrer than the cobalt green materials [START_REF] Zhou | Synthesis and characterization of Zn1-xCoxO green pigments with low content cobalt oxide[END_REF]. Furthermore, chromium exist in oxides with various oxidation states [START_REF] Yeo | Enhanced Activity of Gold-Supported Cobalt Oxide for the Electrochemical Evolution of Oxygen[END_REF], and the change of the Cr-O valence bond with the effect of oxygen partial pressure and/or temperature makes it difficult to uniform the hue of such Cr-based pigments; it is not easy to control in the actual production. The green cobalt pigments can meet the requirements of the color stability, lower toxicity for, as illustration, building materials. Nevertheless, to get different grades of green hues, the solubility of cobalt inside the ZnO matrix must be varied with great amplitude [START_REF] Zhou | Synthesis and characterization of Zn1-xCoxO green pigments with low content cobalt oxide[END_REF]. New synthesis process allowing the introduction of high content of cobalt inside zinc oxides must be found [START_REF] Chanda | Study of structural, optical and magnetic properties of cobalt doped ZnO nanorods[END_REF][START_REF] Hammad | Structure, optical properties and synthesis of Codoped ZnO superstructures[END_REF][START_REF] Gandhi | Effect of Cobalt Doping on Structural, Optical, and Magnetic Properties of ZnO Nanoparticles Synthesized by Coprecipitation Method[END_REF][START_REF] Woo | Co-Doped Branched ZnO Nanowires for Ultraselective and Sensitive Detection of Xylene[END_REF][START_REF] Senapati | Designing Dual Emissions via Co-doping or Physical Mixing of Individually Doped ZnO and Their Implications in Optical Thermometry[END_REF]. Also, the toxicity and the chemical stability of Zn1-xCoxO pigments (the oxide is quite soluble in slightly acidic aqueous solution) has to be increased. That is why, during recent past years, there are many methods for preparation of these pigments such as solid state reaction [START_REF] Kim | Magnetoresistance in laser-deposited Zn1-xCoxO thin films[END_REF], hydrothermal technique [START_REF] Bouloudenine | Bulk Zn1-xCoxO magnetic semiconductors prepared by hydrothermal technique[END_REF] co-precipitation method [START_REF] Xu | Effects of temperature and atmosphere on the magnetic properties of Co-doped ZnO rods[END_REF], combustion synthesis [START_REF] Birajdar | Synthesis, structural, morphological, optical and magnetic properties of Zn1-xCoxO (0 ≤ x ≤ 0.36) nanoparticles synthesized by sol-gel auto combustion method[END_REF], High Energy Ball milling processes... But at the exception of solid state reaction, those methods are easy to introduce impurities, and experimental process can be expansive and complex. The solid state route, due to the obtaining of large particle size, gets the disadvantage to lead to the appearing of cobalt secondary phases for moderate cobalt concentration (depending on the authors); so, the preparation of deep green hues is compromised. To prevent metal oxide dissolution and their subsequent toxicity effects, numerous coating strategies involving a thin silica layer have been developed to create colloidally stable surface-protected nanoparticles [START_REF] Liu | Silica-coated metal nanoparticles[END_REF]. For instance, ZnO nanoparticles have shown a significant reducing of dissolution in acidic digestive media such as gastric media after coating with a 28 nm thick silica shell [START_REF] Chia | Reducing ZnO nanoparticles toxicity through silica coating[END_REF]. This increased chemical stability was directly related to the decrease in cytotoxicity when ZnO @ SiO2 nanoparticles were incubated with colorectal cell lines.

In this study, we propose a simple and low cost process: polyol co-precipitation route in order to obtain pure nanometric Zn1-xCoxO with x that can be varied in a wide range. The easy obtaining of ZnO:Co@SiO2 core-shell powers is also reported, these latter materials exhibiting a decreased solubility into acidic solution and avoids the possibility of environment pollution with Co 2+ ions.

II. Experimental

II.1. Powder from polyol synthesis

All of the chemical reagents were purchased from Aldrich and used without further purification steps. Zinc acetate dihydrate: Zn(CH3OO)2.2H2O and cobalt nitrate :Co(NO3)2,6H2O were used as zinc and cobalt sources and diethylene glycol (H10C4O3) as template. In a typical synthesis, zinc and cobalt precursors are introduced in stoichiometric proportion in order to get 1 g. of Zn1-xCoxO powder (2.924 g) and added to 250 mL H10C4O3.

The resulting mixture was heated under continuous stirring to obtain a green sol refluxed at 180 °C for 1 h. At the end of the reaction, the precipitate was separated by centrifugation, washed several times with ethanol to remove the organic product and dried in an oven at 80 °C.

In a second step Zn1-xCoxO@SiO2 core-shells were obtained by controlled hydrolysiscondensation of tetraethoxysilane (TEOS, Si(OC2H5)4) in alcohol medium (EtOH) from a derived Stöber process. Typically, 100 mg of powder were dispersed in 150 mL of EtOH, and 6.4 mL of NH4OH as hydrolysis-condensation catalyst. TEOS with various quantities calculated to obtain different SiO2 shell thicknesses was then added drop by drop to the asprepared suspension. The suspension was then maintained under agitation at room temperature, 2hrs. Additionally, few drops of HCL aqueous solution (0.2M) can be added at the end of the maturation step in order to dissolve ZnO cores in case the silica shell is porous.

II.2. Characterization techniques

The powder structure was characterized by X-ray diffraction analysis (Philips PW 1820, PANalytical X'Pert instrument, 2θ range from 8 to 60° and λ-Cu (Kα1-Kα2). The unit cell parameters were refined by structural pattern matching using the Fullprof® program package.

The powder patterns were firstly analyzed using the Caglioti function (function N°5), i.e. the isotropic peak profile function for which the u, v, w and shape parameters are refined; in a more appropriate model, the powder patterns were also refined using anisotropic peak profile (function N°7) that allows the crystallite size determination for each (hkl) crystallographic directions.

ICP measurements allow the determination of the Co/(Zn+Co) molar ratio of the Co-doped ZnO powder samples. Measurements were carried out on aqueous solutions obtained after complete dissolution of the powder into HCl with high purity using the Varian ICP-OES 720ES apparatus.

Scanning electron microscope used for morphological investigations of the Zn1-xCoxO asprepared particles is a Jeol-6700F SEM apparatus (using for chemical characterization, 15 kV as probe potential -15 mm distance). For the Zn1-xCoxO@SiO2 morphological characterization, transmission electron microscopy (TEM) images were recorded with JEOL JSM-6700F (operating at 5 kV) microscope.

The diffuse reflectance of our pigment particles were measured using a Varian Cary 5000 UV-Vis-NIR spectrophotometer in 250 nm -2500 nm range. From the diffuse reflectance spectra, the determination of colorimetric parameters in the CIE L*a*b* colorimetric space for each compound was made using x();y();z() color matching functions which are the numerical description of the chromatic response of the observer (herein, CIE 1931 Standard Observer function).

III. Results and Discussion

III.1. Physico-chemical characterizations of as-prepared Co 2+ -doped ZnO powder

All of the chemical reagents were purchased from Aldrich and used without further purification steps. Zn1-xCoxO compounds with targets for x concentrations equal to 0.5 mol%, 10 mol%, 20mol% and 30 mol% are prepared by our polyol-mediated precipitation process.

These five compounds named here after ZC0, ZC05, ZC1, ZC2, and ZC3 were analyzed on a structural point of view by X-ray diffraction and on a chemical point of view by ICP measurements. In Figure 1 are reported the X-ray diffraction patterns of the 5 as-prepared compounds. The compounds exhibit a pure würtzite structure, i.e. a single ZnO-type solid solution is obtained whatever the target composition. The efficient doping concentration, which was previously shown to differ from the target concentration in such a process used for the obtaining of Zn1-xAlxO or Zn1-xGaxO phases, [START_REF] Trenque | Visible-transparent and UV/IR-opaque colloidal dispersions of Ga-doped zinc oxide nanoparticles[END_REF][START_REF] Dengpan | Synthesis and structure analysis of aluminum doped zinc oxide powders[END_REF] was titrated using ICP measurements. Full pattern matching was performed on the X-ray pattern of the various Zn1-xCoxO compounds. Two models were used for the full-pattern matching process: (i) a model using for the peak profile an isotropic peak profile function (Caglioti function, N°5 in FullProf package);

(ii) an anisotropic peak profile function (N°7 in FullProf package). Indeed, previous works have already shown that ZnO particles obtained from the polyol process are constituted of anisotropic crystallites shape: olive shapes, with a (001) crystallographic orientation (large diameter direction). Consequently, the peak width has to be correlated to the peak's Miller indexes, using an anisotropic model for X-ray peak profile description. For illustration, the results obtained using these two different refinement processes on the ZC2 compound are reported in Figure 3.

One can see that the correlation factors: Rp, Rwp, Rexp are both significantly decreased while an be observed. This unit-cell volume decrease is due to the decrease of the a cell parameter (Figure 4a) and c cell parameter (Figure 4b) into the same proportion (i.e., an homothetic decrease of both unti-cell parameters is observed). Indeed, the c/a ratio remains in a very narrow range: in between 1.6023 and 1.6026 values (Figure 4c). From anisotropic peak profile refinements, it is possible to extract directly the crystallite shape of the as-prepared Zn1-xCoxO crystallites with great accuracy. The crystal shapes are drawn in Figure 5. The pure ZnO compounds, already studied using same polyol route and using same experimental condition exhibits the same crystal shape and size than the first ZC05 compound (i.e. the impact of the doping for such low cobalt concentration is negligible).

However, a significant evolution of the crystal size (volume) and shape (needle-like anisotropic distortion) is shown with increasing cobalt concentration. From an olive shape with a large diameter along the (001) axis, a length of 11 nm/ a width of 8.6 nm, the particles with increasing the cobalt concentration tend to become more spherical and with a smaller crystallite size. Even, a reversal of the anisotropy is observed for the ZC3 compound, with a width along (kk0) direction becoming larger than the length along (001) direction. Besides the monotonic evolution of the cell parameters versus the cobalt concentration, the monotonic modification in the crystallite shapes supports the efficient and homogeneous introduction of cobalt ions inside the würtzite crystallographic network. SEM images were also made on various ZCX powders. The images reveal Co 2+ -doped ZnO particles are made of spherical aggregates with sub-micron diameter in good agreement with our previous reports on the synthesis of Ga 3+ -doped ZnO particles from polyol synthesis (Figure 6) [START_REF] Trenque | Visible-transparent and UV/IR-opaque colloidal dispersions of Ga-doped zinc oxide nanoparticles[END_REF][START_REF] Dengpan | Synthesis and structure analysis of aluminum doped zinc oxide powders[END_REF]. The obtaining in a reproducible way of spherical aggregates is shown, i.e. whatever the cobalt concentration very similar powder architecture is observed. Nevertheless, an accurate study of the aggregate (particle) size distribution from image treatment has allowed a significant evolution versus the cobalt concentration to be shown. Indeed, in the case of low cobalt concentration a bimodal distribution for the particle size, with approximatively half-half concentration, with centers located at 225 nm and 425 nm, is evidenced whereas, for high cobalt concentration, a single mode centered on 225 nm is observed (i.e. the biggest aggregates have disappeared) (Figure 7). Maybe, the change of the aggregate size can be correlated positively with the change of crystallite size versus cobalt distribution, since for both scales a slight tendency to the size decrease is noted. Considering that the aggregates have an average diameter of about 200-400 nm and are constituted of a random pack (with about 40% of porosity) [START_REF] Trenque | Visible-transparent and UV/IR-opaque colloidal dispersions of Ga-doped zinc oxide nanoparticles[END_REF] of olive-like crystallites (about 10 nm size), the aggregates are constituted of about 10 4 crystallites.

III.2. Co 2+ -doped ZnO@SiO2 core-shell powder

Three different amounts of TEOS were used in order to vary the shell thickness deposited via sol-gel process on the Co-doped ZnO powder, the experiments being performed on the ZC20 sample. The three as-prepared core@shell samples, from the thinner to the thicker silica shell, are noted ZC20@SiO2-1, ZC20@SiO2-2 and ZC20@SiO2-3. TEM images for the ZC20@SiO2-1 sample, before and after acid attack are reported on Figure 8. The silica shells are evidenced by the acid attack which leads to a total dissolution of the ZC20 cores. It can be seen, that quite homogeneous shells are obtained with just few silica nuclei due to the competition between shell growth and secondary germination. TEM pictures on the three ZC20@SiO2-1, ZC20@SiO2-2 and ZC20@SiO2-3 samples are compared in Figure 9, all the pictures being taken after acid attack. Three main results can be extracted from these TEM observations: (i) the silica shell thickness well increases from ZC20@SiO2-1 to ZC20@SiO2-2 and ZC20@SiO2-3, (ii) the agglomeration of the spherical aggregates increases with the amount of TEOS used, certainly in correlation with the appearing of more and more silica colloids, (iii) the silica shell becomes resistant to the acid attack and protect the core particle while the thickness of the shell increases. Figure 9. TEM photographs on the ZC2@SiO2 compounds with various average shell thickness (after acid attack): ZC2@SiO2-1, ZC2@SiO2-2, ZC2@SiO2-3 from top to bottom.

III.3. Optical properties

In a first step, the optical properties were studied on the Co-doped ZnO powders. The diffuse reflectivity K/S Kubelka-Munk transforms are plotted in Figure 10. For tetrahedral symmetry, with regard to Tanabe-Sugano diagrams established for a 3d 7 electronic configuration, three transitions corresponding to ν1 ( 4 A2(F)→ 4 A1(F)), ν2( 4 A2(F)→ 4 T1(F)), and ν3( 4 A2(F)→ 4 T1(P)) are allowed [START_REF] Robertson | Cobalt/Zinc Molybdates as New Blue Pigments Involving Co 2+ in Distorted Trigonal Bipyramids and Octahedra[END_REF]. For cobalt(II) ions the first two transitions are known to be located in the infrared with a wavelength of about 1400 and 1600 nm, respectively. The third one is known to appear as a triplet due to the L-S Russel-Sanders coupling for which the maximal absorptions, for instance, in CoAl2O4 where the punctual symmetry of the tetrahedral site is Td, are around 540, 590, and 640 nm. 12 The visible triple transition, centered in the orange region, leads to blue coloration when the material color is only due to the cobalt (II) ion located in a tetrahedral site as a unique chromophore. The reported diffuse reflectance spectra for the different cobalt rates clearly show that the visible range triple transition as well as a large infrared peak between 1200 and 1700 nm associated to the convolution of ν1( 4 A2(F)→ 4 A1(F))

(located here at 1640 nm: Figure 10b), ν2 ( 4 A2(F) → 4 T1(F) (located here at 1310 nm) low energy transitions, both characteristic of cobalt in a tetrahedral site, are well taking place whatever the cobalt rate studied. Furthermore, the positions of the triple ν3 transition, calculated from raw the K/S transform, remain about the same whatever the cobalt content at 565, 610, and 660 nm (Figure 10b). Hence, one can assume that from the remarkable stability of the 3 (ν1, ν2, ν3) absorption bands positions versus cobalt concentration, that cobalt adopts in all cases, at room temperature, its thermodynamically stable environment; i.e., as already assumed from Xray diffraction analyses, cobalt is located in a nearly isotropic tetrahedral site. The evolution of the K/S intensity of the cobalt d-d absorption bands (focusing on ν1,and ν3 transitions) are plotted in Figure 11 (evolution of the 660 nm peak intensity reported in Figure 11a , evolution of the 1640 nm peak intensity reported in Figure 11b). Co-doped ZnO@SiO2 core-shell compounds prepared in this publication (corresponding to the ZC2@SiO2-1, ZC2@SiO2-2, ZC2@SiO2-3 samples).

The diffuse reflectance focusing on the UV-visible range of the various core@shell materials are joined in Figure 13. It can be seen an increase in the visible reflectivity while the silica shell thickness increase. The comparison of the obtained colours of the three core@shell compounds and the core compound are also reported in Figure 12. Consequently to the addition of silica shells, the Luminosity parameter tends to increase versus the shell thickness. However, the green hue is only slightly impacted by the shell, and very beautiful green hue: a* = -18.9, b* = 1.7 remains for the ZC2@SiO2-3 sample. For this sample, the combination of optimal color properties and of a chemical stability (for illustration besides acid attack) is so reached.

Figure 13. Diffuse reflection spectra (Rdiff%) of the ZC2@SiO2 compounds with various average shell thickness : a)ZC2@SiO2-1, b) ZC2@SiO2-2, c)ZC2@SiO2-3.

IV. Conclusion

In this paper, the polyol synthesis route has allowed the obtaining of Zn1-xCoxO compounds, with nanometric crystallites (olive shape) / micrometric spherical aggregates, a large range of cobalt concentration associated to a large deep greenpale green scale of colorations. The color of the as-prepared powders is attributed to the efficient substitution of the zinc ions by cobalt (II) ions inside the tetrahedral sites of the würtzite phase. The spherical aggregates can be easily coated via sol-gel process by silica shells with tuned thickness in order to limit pigment's toxicity; this, without a significant decrease of the colorimetric properties.

Figure 1 . 2 .

 12 Figure 1. X-ray diffraction patterns for various Co 2+ -doped ZnO oxides obtained by hydrolysis-

Figure 2 .

 2 Figure 2. Co 2+ concentration efficiently introduced in Co-doped ZnO particles versus the Co 2+ target concentration (corresponding to the ratio between Zn 2+ and Co 2+ ions introduced in the

  y  x/2 anisotropic peak profile is taken into account, in comparison pattern matching with the use of an isotropic peak profile. Hence, the cell parameters and correlation factors (Rp, Rwp and Rexp) reported in the Table1are extracted from the refinements using the anisotropic peak profile function. Also, the evolutions of the unit-cell parameters versus the cobalt concentration (efficient concentration in the mixed oxide) are plotted in the Figure 4. The proximity of Co 2+ and Zn 2+ Shannon ionic radii: Co 2+ in (IV) coordination site equals 0.58 Å while ionic radius of Zn 2+ in (IV) coordination site equals 0.60 Å, makes the variation of the cell parameters with composition very light. Nevertheless, a clear tendency of a decrease of the unit-cell volume can

Figure 3 .

 3 Figure 3. (a) Full-pattern matching using isotropic peak profile function on the Zn0.88Co0.12O

Figure 4 .

 4 Figure 4. Evolution of the unit-cell-parameters of Co-doped ZnO powders versus the cobalt

  . amount (mol%) cobalt: exp. amount (mol%)

Figure 5 .

 5 Figure 5. Crystallite's average morphology obtained from X-ray diffraction pattern refinements

Figure 6 .Figure 7 .

 67 Figure 6. SEM photograhs of the various Co 2+ -doped ZnO spherical particles obtained by

Figure 8 .

 8 Figure 8. TEM photographs on the 12 mol% Co 2+ -doped ZnO core-shell compound

Figure 10 .Figure 11 .

 1011 Figure 10. Diffuse reflection spectra (Rdiff%) (a) and K/S absorption spectra (b) of the various Co 2+ -doped ZnO oxides.

Figure 12 .

 12 Figure 12. L*a*b* parameters and coloration of the different Co-doped ZnO compounds and

7 x

 7 Co =1.5 mol% x Co = 5.5 mol% x Co =12 mol% x Co =16 mol% x

Table 1 .

 1 Unit-cell parameters and reliability factors obtained on the various Co 2+ -doped ZnO oxides.

	Target compound Effective compound	a	c	Rp	Rwp	Rexp
	ZnO-Co 0%	ZnO-Co 0%	3.25286(16) 5.21277(30)	3.0	3.5	2.0
	ZnO-Co 5%	ZnO-Co 1.5%	3.25256(35) 5.21258(55)	3.8	3.8	2.1
	ZnO-Co 10%	ZnO-Co 5.5%	3.25269(17) 5.21204(28)	2.8	3.2	1.9
	ZnO-Co 20%	ZnO-Co 12%	3.25238(16) 5.21162(27)	2.6	3.9	1.9
	ZnO-Co 30%	ZnO-Co 16%	3.25238(22) 5.21182(22)	2.9	3.6	1.9

The authors are thankful to L'Agence Nationale de la Recherche for financial support, under the ChoCoComp project (No. ANR-13-RMNP-0011)