G C Bento

J X Cruz Neto

G L Ópez
email: glopez@us.es

A Soubeyran ¶

J C O Souza

THE PROXIMAL POINT METHOD FOR LOCALLY LIPSCHITZ FUNCTIONS IN MULTIOBJECTIVE OPTIMIZATION WITH APPLICATION TO THE COMPROMISE PROBLEM *

Keywords: proximal point method, multiobjective optimization, locally Lipschitz function, Pareto critical point, compromise problem, variational rationality AMS subject classifications. 49J52, 65K05, 90C26, 90C29, 91E99

This paper studies the constrained multiobjective optimization problem of finding Pareto critical points of vector-valued functions. The proximal point method considered by Bonnel, Iusem, and Svaiter [SIAM J. Optim., 15 (2005), pp. 953-970] is extended to locally Lipschitz functions in the finite dimensional multiobjective setting. To this end, a new (scalarization-free) approach for convergence analysis of the method is proposed where the first-order optimality condition of the scalarized problem is replaced by a necessary condition for weak Pareto points of a multiobjective problem. As a consequence, this has allowed us to consider the method without any assumption of convexity over the constraint sets that determine the vectorial improvement steps. This is very important for applications; for example, to extend to a dynamic setting the famous compromise problem in management sciences and game theory.

this case, distances are locally Lipschitz vector functions; see, for instance, [START_REF] Opricovic | Compromise solution by MCDM methods: A comparative analysis of VIKOR and TOPSIS[END_REF][START_REF] Opricovic | Extended VIKOR method in comparison with outranking methods[END_REF]. In a broad range of applications (see location theory, utility theory, consumer theory, . . .) such distance functions are used as objectives. Second, taking advantages of a nice specificity of the algorithm proposed in Bonnel, Iusem, and Svaiter [START_REF] Bonnel | Proximal methods in vector optimization[END_REF], it allows us to examine a dynamic version of the compromise problem in the context of human dynamics in behavioral sciences. This comes from the fact that the algorithm in [START_REF] Bonnel | Proximal methods in vector optimization[END_REF] is a descent process. Thus, it can model in a crude but precise way a transition that a group of agents accepts to follow before reaching a compromise solution as an ascent process, each agent required not to decrease his payoff from one period to the other one.

The beginning of the story starts with the (scalar) proximal point method introduced in the literature by Moreau [42], Martinet [START_REF] Martinet | Régularisation d'inéquations variationelles par approximations successives[END_REF], and later popularized by Rockafellar [START_REF] Rockafellar | Monotone operators and the proximal point algorithm[END_REF] who performs the proximal point method for the problem of finding zeros of operators. A brief description of this method can be found in Bonnel, Iusem, and Svaiter [START_REF] Bonnel | Proximal methods in vector optimization[END_REF]. We also refer to Lemaire [START_REF] Lemaire | The Proximal Algorithm[END_REF] who surveys the literature on proximal point algorithms for real-valued functions up to 1989.

We consider multiobjective (or multicriteria) optimization in the following context. Let R m be the m-dimensional Euclidean space with the partial order " " in R m induced by the Paretian cone R m + , given by y z (or z y) if and only if zy ∈ R m + with its associate relation " ≺," given by y ≺ z (or z y) if and only if zy ∈ 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}. We recall that the cone R m + is Daniell, i.e., any decreasing sequence having a lower bound converges to its infimum, and R m + is correct in the sense that (cl R m + )+R m + \l(R m + ) ⊂ R m + , where cl denotes the closure and l(R m + ) stands for the set R m + ∩-R m + ; see Luc [START_REF] Luc | Theory of Vector Optimization[END_REF]. Given a vector-valued function F : R n → R m , we analyze the proximal point method for finding a Pareto critical point of F := (f 1 , . . . , f m ). A point x ∈ R n is a Pareto critical point of F if there exists a component function f i of F for which the Clarke directional derivative of f i at x in the direction of yx is nonnegative for all y ∈ R n with f i : R n → R and i ∈ I; see the details of this concept in section 2.

Two different strategies have been used for solving multiobjective optimization problems: scalarization techniques (see, for instance, [START_REF] Burachik | A new scalarization technique to approximate Pareto fronts of problems with disconnected feasible sets[END_REF][START_REF] Cruz Neto | A subgradient method for multiobjective optimization[END_REF][START_REF] Das | Normal-boundary intersection: A new method for generating the Pareto surface in nonlinear multicriteria optimization problems[END_REF][START_REF] Durea | Some remarks on proximal point algorithm in scalar and vectorial cases[END_REF][START_REF] Eichfelder | An adaptive scalarization method in multiobjective optimization[END_REF][START_REF] Fliege | Newton's method for multiobjective optimization[END_REF][START_REF] Graña Drummond | On the choice of parameters for the weighting method in vector optimization[END_REF][START_REF] Jahn | Scalarization in vector optimization[END_REF]) and nonscalarization approaches (see [START_REF]Practical approaches to multiobjective optimization[END_REF] for an overview on this subject). In the first case, for finding a (weak or not) Pareto optimal point of F (see these concepts in section 2), the scalarization approach finds a minimizer of the scalar function ζ(F (•)), for some scalarization functional ζ which has the property that a solution of the scalar problem is a Pareto solution of the vectorial problem. In the second case, multiobjective optimization algorithms that do not scalarize have been developed and some of these techniques are extensions of scalar optimization algorithms, e.g., steepest descent method [START_REF] Fliege | Steepest descent methods for multicriteria optimization[END_REF][START_REF] Drummond | A steepest descent method for vector optimization[END_REF], projected gradient method [START_REF] Fukuda | On the convergence of the projected gradient method for vector optimization[END_REF][START_REF] Drummond | A projected gradient method for vector optimization problems[END_REF], subgradient method [START_REF] Bello | A subgradient method for vector optimization problems[END_REF], and Newton's method [START_REF] Fliege | Newton's method for multiobjective optimization[END_REF], while others borrow heavily from ideas developed in heuristic optimization; see, e.g., [START_REF] Laumanns | Combining convergence and diversity in evolutionary multiobjective optimization[END_REF][START_REF] Mostaghim | Multi-Objective Particle Swarm Optimization on Computer Grids[END_REF] and references therein. For the latter, no convergence proofs are known.

Our paper extends the work of Bonnel, Iusem, and Svaiter [START_REF] Bonnel | Proximal methods in vector optimization[END_REF] to a nonconvex setting without using a scalarization method. Thus, it will be useful for the reader to describe the exact method analyzed in Bonnel, Iusem, and Svaiter [START_REF] Bonnel | Proximal methods in vector optimization[END_REF] who proposed an extension of the proximal point method to vector optimization, i.e., when other underlying ordering cones are used instead of the nonnegative orthant R m + . They actually perform a similar approach for the case of the proximal point method for scalar functions. They use this method in order to find a weak Pareto optimal (or weak Pareto) of a map F : X → Y from a real Hilbert space X to a real Banach space Y containing a closed, convex, and pointed cone C with nonempty interior, where "pointed" means that C ∩ (-C) = {0} with respect to the partial order " C " induced by the cone C. In this context, weak Pareto point means a point x ∈ X such that there exists no y ∈ X satisfying F (y) ≺ C F (x). For orders induced by non-Paretian cones, the problem of finding Pareto points (weak or not) is certainly not as frequent as the one concerning the pointwise partial order, but, nevertheless, it is not just an extension of the Paretian case and has its own importance. The (exact) method analyzed in [START_REF] Bonnel | Proximal methods in vector optimization[END_REF] takes as the (k + 1)th iteration a weak Pareto solution of F k : X → Y defined as

F k (x) = F (x) + λ k ||x -x k || 2 ε k subject to the constrained set Ω k = {x ∈ X : F (x) C F (x k )},
where {λ k } is a bounded sequence of positive scalars and ε k is an exogenously selected vector belonging to the interior of C such that ||ε k || = 1 for each k ≥ 0. The idea underlying the convergence results is based on the first-order optimality condition of the scalar problem

(1) min

x∈Ω k η k (x),
where

η k (x) = F (x), z k + λ k 2 ε k , z k ||x-x k || 2
and {z k } is an exogenous sequence belonging to the positive polar cone C + ⊂ Y * given by C + = {z ∈ Y * : y, z ≥ 0, for all y ∈ C} such that ||z k || = 1 for all k ≥ 0, and Y * is the topological dual space of Y , where •, • : Y × Y * → R is the duality pairing. Thus, x k+1 is a solution of (1) and, hence, it satisfies the following inclusion:

(2) 0 ∈ ∂ψ k (x k+1 ) + λ k ε k , z k (x k+1 -x k ),
where

ψ k (x) = F (x), z k + δ Ω k (x)
, ∂ψ k denotes the subdifferential of ψ k in the sense of convex analysis, and

δ Ω k (•) is the indicator function, that is, δ Ω k (x) = 0 if x ∈ Ω k ,
and δ Ω k (x) = +∞, otherwise. Bonnel, Iusem, and Svaiter [START_REF] Bonnel | Proximal methods in vector optimization[END_REF] establish that any sequence generated by this algorithm converges (in the weak topology of X) to a weak Pareto point of F under the following two assumptions: (A1) (convexity and lower semicontinuity). F is C-convex with respect to the order " C ," i.e., F ((1-t)x+ty) C (1-t)F (x)+tF (y) for all x, y ∈ X and t ∈ [0, 1], and F is positively lower semicontinuous which means that for every z ∈ C + , the scalar function

x → F (x), z is lower semicontinuous. (A2) (completeness). The set (F (x 0 ) -C) ∩ F (X) is C-complete, i.e., for every sequence {a k } ⊂ X with a 0 = x 0 , such that F (a k+1 ) C F (a k ) for all k ∈ N, there exists a ∈ X such that F (a) C F (a k
) for all k ∈ N. Assumption (A1) guarantees that the constrained set Ω k is closed and convex for all k ∈ N. Thus, (2) can be viewed as

α k (x k -x k+1 ) ∈ ∂( F (•), z k )(x k+1 ) + N Ω k (x k+1 ), where α k = λ k ε k , z k and N Ω k (x k+1
) stands for the normal cone to Ω k at x k+1 ∈ Ω k in the classical sense of convex analysis. In this approach, convexity of each set Ω k plays an important role. The set Ω k forces the algorithm to be a descent process. A motivation, in a dynamic context, to consider the constrained set Ω k is given in Bento, Cruz Neto, and Soubeyran [START_REF] Bento | A proximal point-type method for multicriteria optimization[END_REF]. They mention that the set Ω k characterizes a vector improvement process where a vectorial minimizing solution x k of the current proximal problem moves to a next one such that it improves the current solution, which is essential to justify the process at a behavioral level where a risk averse agent accepts change only if the change is improving on all aspects (all components of the vector). If we consider a group of agents, as we do here, this constraint set is even more important. It imposes that the payoff of each agent of the group does not decrease; see section 3.1.

Other authors have proposed variants of the algorithm considered by Bonnel, Iusem, and Svaiter [START_REF] Bonnel | Proximal methods in vector optimization[END_REF] for convex vector or multiobjective problems; see, for instance, Ceng and Yao [START_REF] Ceng | Approximate proximal methods in vector optimization[END_REF], Ceng, Mordukhovich, and Yao [START_REF] Ceng | Hybrid approximate proximal method with auxiliary variational inequality for vector optimization[END_REF], Choung, Mordukhovich, and Yao [START_REF] Choung | Hybrid approximate proximal algorithms for efficient solutions in vector optimization[END_REF], Gregório and Oliveira [START_REF] Gregório | A logarithmic-quadratic proximal point scalarization method for multiobjective programming[END_REF], and Villacorta and Oliveira [START_REF] Villacorta | An interior proximal method in vector optimization[END_REF]. Recently, the R m + -quasi-convex case was discussed in Bento, Cruz Neto, and Soubeyran [START_REF] Bento | A proximal point-type method for multicriteria optimization[END_REF] and Apolinário, Papa Quiroz, and Oliveira [START_REF] Apolinário | A scalarization proximal point method for quasiconvex multiobjective minimization[END_REF]; see the definition of R m + -quasi-convexity on section 2. In these works, their corresponding algorithms, at the (k + 1)th iteration, compute a point

x k+1 satisfying 0 ∈ ∂ζ(F (x k+1 )) + α k (x k+1 -x k ) + N Ω k (x k+1 ),
where ζ : R m → R is a scalarization function, ∂ζ denotes some subdifferential of ζ, and {α k } is a sequence of positive real numbers; see section 4 for more details about these algorithms. In both [START_REF] Apolinário | A scalarization proximal point method for quasiconvex multiobjective minimization[END_REF] and [START_REF] Bento | A proximal point-type method for multicriteria optimization[END_REF], convexity of Ω k comes from the R m + -quasi-convexity of F .

The aim of this paper is twofold. First, we present an approach (which does not use a scalarization), for convergence of the proximal point algorithm in (finite dimensional) multiobjective optimization. In [START_REF] Bonnel | Proximal methods in vector optimization[END_REF], the authors use an optimality condition for the scalarized problem (1), while here we establish convergence results without using any scalarization method combining the fact that each iteration of the algorithm is a weak Pareto solution for a constrained multiobjective problem with a necessary condition for a point to be a weak Pareto solution of a constrained multiobjective problem. We mention that our approach does not use convexity assumption of the constraint sets as the previously mentioned works do. As a second contribution, we expand the application of proximal methods in (finite dimensional) multiobjective optimization for locally Lipschitz vector-valued functions with nonconvex constraints. We mention that the C-convex case analyzed by Bonnel, Iusem, and Svaiter [START_REF] Bonnel | Proximal methods in vector optimization[END_REF], restricted to the finite dimensional multiobjective framework, is indeed a particular instance of our locally Lipschitz case. In light of our approach, the R m + -quasi-convex case is also analyzed and convergence results as proved in [START_REF] Apolinário | A scalarization proximal point method for quasiconvex multiobjective minimization[END_REF] and [START_REF] Bento | A proximal point-type method for multicriteria optimization[END_REF] are presented.

As an application, we give a dynamic formulation of the well-known static group compromise problem to model, in a crude way, how, starting from an initial situation, a group of agents with interrelated payoffs are able to approach and reach, following an acceptable transition, a desired end, defined as a compromise solution. This is a very important problem related to cooperative dynamical games.

This paper is organized as follows. Section 2 introduces some notations as well as some basic concepts and results in multiobjective optimization. In section 3, we define the proximal method and state and prove some of its properties. Section 4 is devoted to the convergence analysis of the algorithm. Finally, some remarks and future works are discussed in section 5.

Multiobjective optimization.

In this section, we discuss some basic definitions and properties of multiobjective optimization which can be found, for instance, in Luc [START_REF] Luc | Theory of Vector Optimization[END_REF].

Given a nonempty set Ω ⊂ R n and F = (f 1 , . . . , f m ) : R n → R m a vector-valued function, a point x * ∈ Ω is called a Pareto optimal point (or Pareto point) of F in Ω if there exists no other x ∈ Ω with F (x) F (x * ) and F (x) = F (x * ). A point x * ∈ Ω is called weak Pareto optimal (or weak Pareto) of F in Ω if there exists no x ∈ Ω with F (x) ≺ F (x * ). This means that for all x ∈ Ω there exists an index j(x) = j ∈ {1, . . . , m} such that F j (x) -F j (x * ) ≥ 0. We call (VP) the problem of finding a weak Pareto optimal point which we denote by [START_REF] Bento | Generalized inexact proximal algorithms: Routine's formation with resistance to change, following worthwhile changes[END_REF] min w {F (x) : x ∈ Ω}.

The set of all weak Pareto points of F in Ω is denoted by arg min w {F (x) : x ∈ Ω}.

Remark 1. 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.

For a vector function F : R n → R m , we say that i) F is R m + -convex if, for every x, y ∈ R n , the following holds:

F ((1 -t)x + ty) (1 -t)F (x) + tF (y) ∀t ∈ [0, 1]; ii) F is R m + -quasi-convex if, for every x, y ∈ R n , the following holds: F ((1 -t)x + ty) max{F (x), F (y)} ∀t ∈ [0, 1],
where the maximum is taken componentwise. We recall now some concepts involving locally Lipschitz functions and nonconvex constrained sets. The definitions and notation are taken from [START_REF] Custódio | Direct multisearch for multiobjective optimization[END_REF]. A scalar-valued function f : R n → R is locally Lipschitz at a point x ∈ R n if there exists a neighborhood U of this point and some real number L > 0 such that

|f (y) -f (y )| ≤ L||y -y || ∀y, y ∈ U.
A function f is locally Lipschitz when it is locally Lipschitz at all points of its domain.

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 .
We denote the distance function

d : R n → R of a point x ∈ R n to a set C ⊂ R n as (4) d C (x) := inf{||x -c|| : c ∈ C}.
Let Ω ⊂ R n be a nonempty and closed set. As in [START_REF] Custódio | Direct multisearch for multiobjective optimization[END_REF], we say that a point

x ∈ Ω is a Pareto-Clarke critical point (or Pareto critical point) of F in Ω if, for any v ∈ T Ω (x), there exists i ∈ I such that (5) f • i (x, v) ≥ 0,
where

T Ω (x) := {v ∈ R n : d • Ω (x, v) = 0}
denotes the set of all tangent vectors to Ω at x. As mentioned in [15, page 11], a vector v belongs to T Ω (x) if and only if it satisfies the following property: for every sequence {x k } in Ω converging to x and every sequence t n in (0, ∞) converging to 0, there is a sequence v n converging to v such that x n + t n v n belongs to Ω for all n. Having defined a tangent cone, the likely candidate for the normal cone is the one obtained from T Ω (x) by polarity. Accordingly, we define N Ω (x), the normal cone to Ω at x, as follows:

N Ω (x) := {ξ ∈ R n : ξ, v ≤ 0 ∀v ∈ T Ω (x)}.
If Ω is convex, N Ω (x) coincides with the cone of normals in the sense of convex analysis; see [START_REF] Clarke | Optimization and Nonsmooth Analysis[END_REF]Proposition 2.4.4].

The Clarke subdifferential of f at x, denoted by ∂f (x), is defined 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]. Given a locally Lipschitz vector-valued 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 },
where

F • (x; d) := (f • 1 (x; d), . . . , f • m (x; d)).
It is worth pointing 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, Tan, and Tinh [START_REF] Luc | Convex vector functions and their subdifferential[END_REF].

Remark 2. Note that if m = 1 in the previous definition of a Pareto critical point, we retrieve the (classical) definition of critical points for nonsmooth functions: 0 ∈ ∂f (x). It is worth noticing that, combining [START_REF] Chrétien | Generalized Proximal Point Algorithms and Bundle Implementations[END_REF] with Clarke [14, 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 v ∈ T Ω (x), there exist i ∈ I and ξ ∈ ∂f i (x) such that ξ, v ≥ 0. Thus, if x is not a Pareto critical point of F in Ω, then there exists v ∈ T Ω (x) such that U v ≺ 0 ∀U ∈ ∂F (x).

The next result gives a necessary condition for a point to be a Pareto critical point of a vector-valued function.

Lemma 1. Let w ∈ R m + \{0} and assume that Ω is a nonempty and closed set. If -U w ∈ N Ω (x) for some U ∈ ∂F (x), then x is a Pareto critical point of F . Proof. Take x ∈ Ω such that -U w ∈ N Ω (x) and let us suppose, by contradiction, that x is not a Pareto critical point of F . From Remark 2, there exists v ∈ T Ω (x) such that U v ≺ 0.
Since w ∈ R m + \{0}, we have w, U v < 0, but this contradicts the fact that -U w ∈ N Ω (x) and U w, v = w, U v . Hence, the desired result is proved.

Consider the problem (3) of finding a weak Pareto point of a vector-valued function F subject to the following constrained set [START_REF] Bolte | Clarke critical values of subanalytic Lipschitz continuous functions[END_REF] Ω := {x ∈ D :

g s (x) ≤ 0, s ∈ J },
where D ⊂ R n is a nonempty and closed set, and g s : R n → R is a locally Lipschitz function for each s ∈ J := {1, . . . , p}. The next result presents a necessary condition for a point x * ∈ Ω to be a weak Pareto solution of (3).

Theorem 1. Let D ⊂ R n be a nonempty and closed set. Assume that the set Ω in (3) is given as in [START_REF] Bolte | Clarke critical values of subanalytic Lipschitz continuous functions[END_REF], and the functions f j , g s : R n → R, j ∈ I, and s ∈ J , are locally Lipschitz. If x * ∈ Ω is a weak Pareto solution of (3), then there exist real numbers u j ≥ 0, v s ≥ 0, with j ∈ I and s ∈ J , and τ > 0 such that As remarked by Minami [41, Remark 3.1], if D is additionally convex (which we do not assume), then the cone

j∈I u j ∂f j (x * ) + s∈J v s ∂g s (x * ) + τ∂d D (x * ) = 0 with j∈I u j + s∈J v s = 1 and v s g s (x * ) = 0, s ∈ J .
{w : w ∈ τ∂d D (x * ), τ > 0}
is the normal cone in the classical sense of convex analysis. For the nonconvex case, a formula for the Clarke subdifferential of the distance function (4) can be found in Burke, Ferris, and Qian [START_REF] Burke | On the Clarke subdifferential of the distance function of a closed set[END_REF]. We present a proof just for the sake of clarity.

Theorem 2. Let C be a nonempty and closed subset of R m . If x ∈ C, then ∂d C (x) ⊂ B[0, 1] ∩ N C (x),

The proximal point method.

In this section, we prove some facts related to our approach for convergence of the proximal method for vector-valued functions. As an application, we show how this method can be a nice tool to solve the famous compromise solution problem.

Compromise problem.

Let us consider a group of producers i ∈ I = {1, . . . , m}. The decision variables of the group form the vector x ∈ R n which must satisfy some constraints x = (x 1 , . . . , x n ) ∈ D ⊂ R n . The objective of each of them is a "to be increased" payoff (profit, utility), h i (x) ∈ R + . The vectorial objective of the group is H(x) ∈ R m , where H(x) = (h 1 (x), . . . , h m (x)). Thus, the subset of feasible vectorial payoffs of the group, i.e., the payoff subspace of the group is

H(D) = {H(x) : x ∈ D} ⊂ R m .
Each agent wants a payoff as high as possible.

Suppose that the maximum payoff of each agent of the group is bounded above, i.e., h i = sup {h i (x) : x ∈ D} < +∞, i ∈ I. Then, the vectorial payoff H = (h 1 , . . . , h m ) is the ideal (or utopian) vectorial payoff of this group. Usually, the ideal vectorial payoff is not feasible, which means, H / ∈ H(D). Let us consider "to be decreased" payoffs

f i (x) = h i -h i (x) ≥ 0, i ∈ I,
which refer, in psychology, to unsatisfaction gap functions f i . They measure how much each individual payoff h i (x) with x ∈ D, fails to reach its maximum (ideal or utopian) value h i . These vectorial unsatisfaction gaps

F (x) = (f 1 (x), . . . , f m (x)) = H -H(x) 0
generate individual regrets or unsatisfactions with respect to ideal payoffs. The compromise solution (with respect to a norm) is some feasible alternative x * ∈ D which minimizes the whole unsatisfaction of the group, in other words, it minimizes the distance between the ideal vectorial payoff H and the payoff subspace H(D). For the "compromise" problem in multicriteria decision making, see the well known references Gearhart [START_REF] Gearhart | Compromise solutions and estimation of the non inferior set[END_REF] and Goetzmann, Busing, and Matuschke [START_REF] Goetzmann | The Power of Compromise. Reference Point Methods and Approximation in Multicriteria Optimization[END_REF].

Using this (static) compromise model, let us consider a simple group dynamic model. It includes a starting point, an acceptable transition, and some desired ends. This simple group dynamics model considers that transitions are acceptable if, each period, all members of the group improve their payoffs. In the opposite case, some agents will quit the group or resist change. The desired end of the group I is to approach and reach an end point, which itself approaches as much as possible the ideal point. In a dynamic cooperative setting, all agents of the group will accept change from the last position x = x k to the next, y = x k+1 only if their payoff does not decrease, i.e., if [START_REF] Bonnel | Proximal methods in vector optimization[END_REF] h

i (x k ) ≤ h i (x k+1 ) ∀i ∈ I ⇐⇒ H(x k ) H(x k+1 ).
This defines a cooperative improving dynamic

x k+1 ∈ Ω(x k ), where Ω(x k ) = {x ∈ D : H(x k ) H(x)}.
The cooperative group dynamic problem is to find a cooperative improving dynamic x k+1 ∈ Ω(x k ) which approaches and reaches (converges to) a desired end position close enough to the ideal point; see 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] for the details of "group dynamics" and "organizational change" management problems in psychology and management sciences.

The algorithm.

Throughout this paper, we consider D ⊂ R n a nonempty and closed set, and F = (f 1 , . . . , f m ) : R n → R m such that each component function f i : R n → R, i ∈ I, is a locally Lipschitz function. From Remark 1, we can assume without loss of generality that F 0.

Next, we consider the proximal point algorithm for finding a Pareto critical point of F in D. Let {λ k } be a sequence of positive real numbers and let {ε k } ⊂ R m ++ be a sequence such that ||ε k || = 1 for all k ≥ 0. The method generates a sequence {x k } ⊂ D 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 (8)

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

Ω k = {x ∈ D : F (x) F (x k )}.
We would like to mention that this method finds separate solutions at time and not the whole solution set. It has been noticed by Fukuda and Graña Drummond [START_REF] Fukuda | A survey on multiobjective descent methods[END_REF], and Fliege, Graña Drummond, and Svaiter [START_REF] Fliege | Newton's method for multiobjective optimization[END_REF] that we can expect to somehow approximate the solution set by just performing this method for different initial points. In the well-known weighting method, this kind of idea also appears; see Burachik, Kaya, and Rizvi [START_REF] Burachik | A new scalarization technique to approximate Pareto fronts of problems with disconnected feasible sets[END_REF]. More precisely, the method can be performed for different weights in order to find the solution set, or a reasonable approximation of this set. However, in some cases, arbitrary choices of the weighting vectors may lead the weighting method to unbounded problems. The Pareto front, i.e., the objective values of these solutions, is in general an infinite set. Thus, in practice, only an approximation of the Pareto front is obtained.

Next, we prove that Algorithm 1 is well defined. To this end, we consider the concept of completeness as in (A2).

Proposition 1. Algorithm 1 is well defined. Proof. The starting point x 0 ∈ D is chosen in the initialization step. Assuming that the algorithm reached iteration k, we show next that the (k + 1)th iteration exists. Denote by

F k (x) := F (x) + λ k 2 ||x -x k || 2 ε k . Note that x k ∈ Ω k which implies that F k (Ω k ) is nonempty. It is straightforward to check that F k (x) 0 and F k (Ω k ) is closed. Since the cone R m + is Daniell, it follows from [38, Lemma 3.5] that F k (Ω k ) is R m + -complete. Thus, from [38, Theorem 3.3] the set arg min w {F k (x) : x ∈ Ω k } is nonempty.
From now on, {x k }, {λ k }, and {ε k } denote the sequences considered in Algorithm 1. Next, we explore deeply the structure of the vector problem by using the necessary condition for a weak Pareto optimal point of a multiobjective problem given by Theorem 1. The following result will be used in our main convergence results. Proposition 2. For all k ∈ N, there exist

A k ∈ R m×n , u k , v k ∈ R m + , w k ∈ R m , and τ k ∈ R ++ such that (9) A k (u k + v k ) + λ k-1 ε k-1 , u k (x k -x k-1 ) + τ k w k = 0, where (10) w k ∈ B[0, 1] ∩ N D (x k ) and ||u k + v k || 1 = 1 ∀k ∈ N.
Proof. It follows from the definition of the algorithm that x k is a weak Pareto solution of the problem

min w {F k-1 (x) : x ∈ Ω k-1 }, where F k-1 (x) = F (x)+ λ k-1 2 ||x -x k-1 || 2 ε k-1 . Denoting G k-1 (x) = F (x) -F (x k-1
), it is easy to verify, from the locally Lipschitz continuity of F , that all component functions [START_REF] Ceng | Approximate proximal methods in vector optimization[END_REF] (

g k-1 ) j (•) = f j (•) -f j (x k-1 ) with j ∈ I, and (12)
(f k-1 ) j (•) = f j (•) + λ k-1 2 || • -x k-1 || 2 ε k-1 j with j ∈ I,
are locally Lipschitz functions. Hence, the desired result follows by applying Theorem 1, for each k ∈ N fixed, with g j and f j given by ( 11) and ( 12), respectively, and taking into account that, from Theorem 2, we have

∂d D (x k ) ⊂ B[0, 1] ∩ N D (x k ) ∀k ∈ N. In this case, A k = [a k 1 . . . a k m ]
, where a k j ∈ ∂f j (x k ) with j ∈ I, u k = (u k 1 , . . . , u k m ) , and v k = (v k 1 , . . . , v k m ) . Remark 3. Note that from [START_REF] Burke | On the Clarke subdifferential of the distance function of a closed set[END_REF], {u k }, {v k }, and {w k } are bounded sequences. From Bolte et al. [START_REF] Bolte | Clarke critical values of subanalytic Lipschitz continuous functions[END_REF]Remark 1] ∂f j is bounded on compact sets. So, we have that {A k } is bounded as long as {x k } is bounded because a k j ∈ ∂f j (x k ), j ∈ I. Therefore, if {λ k } and {x k } are bounded sequences, it follows from (9) 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 from ( 9), we have

A k0 v k0 + τ k0 w k0 = 0.
As τ k0 > 0 and w k0 ∈ N D (x k0 ), the last equality is equivalent to

-A k0 v k0 ∈ N D (x k0 ).
On the other hand, from the second assertion in [START_REF] Burke | On the Clarke subdifferential of the distance function of a closed set[END_REF], we can say that v k0 ∈ R m + \{0}. Since A k0 ∈ ∂F (x k0 ), the desired result follows by using Lemma 1 with U = A k0 , w = v k0 , and x = x k0 .

As in [START_REF] Bonnel | Proximal methods in vector optimization[END_REF], the stopping rule in Algorithm 1 can be changed by the following rule, 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 ≥ 1. Proposition 2 combined with Lemma 1 allows us to see that this condition is sufficient getting the stopping rule given in Algorithm 1. However, even in the convex case, it is possible to note that this rule might fail to recognize weak Pareto solutions; see [START_REF] Bonnel | Proximal methods in vector optimization[END_REF]Proposition 3.2].

Corollary 2. If x k+1 = x k , then x k is a Pareto critical point of F . Remark 4. Note that Algorithm 1 generates an infinite sequence {x k } which remains constant from a Pareto critical point on, i.e., if x k is a Pareto critical point, then x k+p = x k for all p ∈ N. In view of Corollaries 1 and 2, we can suppose without loss of generality that u k = 0 and x k+1 = x k for all k ∈ N, respectively.

Convergence analysis.

As mentioned before, the idea underlying the results of the scalarization approach is to solve a scalar problem in order to obtain a solution of the related vectorial problem. In our approach, the main result does not solve a scalar problem to obtain convergence of the sequence but it uses a necessary condition for weak Pareto points of a multiobjective problem instead. Hence, it allows us to analyze convergence properties of proximal methods for both locally Lipschitz vector and scalar functions with nonconvex constraints.

Locally Lipschitz case.

Before presenting our convergence results let us go back to our group compromise motivation. The nature of the group dynamic problem depends heavily on the nature of the objective functions which determine the properties of the improving sets and other relevant constraints. Objective functions can be convex or concave, quasi-convex or quasi-concave, difference of convex or concave functions, and, more generally, Lipschitz or locally Lipschitz functions. In the recent variational rationality (VR) approach of human behaviors (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]), Lipschitz and locally Lipschitz payoffs are very interesting for two reasons: they mean that when inconveniences to change are low, you cannot expect large advantages to change; there is no free lunch, which is a reasonable hypothesis. Furthermore, these functions are easy to estimate locally. This helps badly informed agents, who know their payoff functions only at some given points, to be able to find, at each step, some improving changes. Lipschitz functions f have concave underestimating functions

y ∈ D -→ u 0 (y) = f (x 0 ) -L||y -x 0 || for each x 0 ∈ D.
Next, we prove our main convergence result for the locally Lipschitz case.

Theorem 3. Suppose that there exist scalars a, b, c ∈ R ++ such that 0 < a ≤ λ k ≤ b and 0 < c ≤ ε k j for all k ∈ N and j = 1, . . . , m. Then, every cluster point of {x k }, if any, is a Pareto critical point of F .

Proof. It follows from the definition of the algorithm that x k for each k ∈ N, is an optimal solution of the problem

min w F (x) + λ k-1 2 ||x -x k-1 || 2 ε k-1 : x ∈ Ω k-1 .
This implies

max 1≤j≤m F j (x k-1 ) -F j (x k ) - λ k-1 2 ||x k -x k-1 || 2 ε k-1 j ≥ 0.
Take j 0 (k) = j 0 ∈ {1, . . . , m} as the index where the maximum in the last inequality is attained. Then, from the lower boundedness assumption of {λ k } and {ε k }, we have

(13) ac 2 ||x k -x k-1 || 2 ≤ F j0 (x k-1 ) -F j0 (x k ).
Since {F (x k )} is nonincreasing and F 0, we obtain that the right-hand side of (13) converges to 0 as k → +∞. Hence,

(x k -x k-1 ) → 0 as k → +∞. (14)
Now, let x be a cluster point of {x k }, and let {x k l } be a subsequence of {x k } converging to x. Applying Proposition 2 for the sequence {x k l }, we have that there exist sequences

{A k l } ⊂ R m×n , {u k l }, {v k l } ⊂ R m + , {w k l } ⊂ R m , and {τ k l } ⊂ R ++ satisfying (15) A k l (u k l + v k l ) + λ k l -1 ε k l -1 , u k l (x k l -x k l -1 ) + τ k l w k l = 0.
Note that {λ k l } is bounded and {x k l } converges to x and, hence, {x k l } is bounded. Thus, from Remark 3, we can assume that the sequences

{A k l }, {u k l }, {v k l }, {w k l },
and {τ k l } are bounded. Without loss of generality, we may assume that

A k l → Â, u k l → û, v k l → v,
and τ k l → τ as l → +∞ (we will use the same notation for the index even if we need to extract other subsequences). Since

{λ k l -1 ε k l -1 , u k l } is bounded, it follows from (14) that λ k l -1 ε k l -1 , u k l (x k l -x k l -1
) vanishes as l → +∞. Therefore, we get, taking the limit in [START_REF] Clarke | Optimization and Nonsmooth Analysis[END_REF] as l → +∞, that

 ŷ + τ ŵ = 0, where R m + \ {0} ŷ := û + v,  ∈ ∂F (x), and ŵ ∈ N D (x), because ∂F (•) and N D (•) are closed. Thus, from [START_REF] Cruz Neto | A subgradient method for multiobjective optimization[END_REF], we obtain

-Â ŷ ∈ N D (x),
and this together with Lemma 1, enables us to say that x is a Pareto critical point of F . This completes the proof.

Remark 5. Note that, if Ω 0 = {x ∈ R n : F (x)
F (x 0 )} is bounded, then {x k } is bounded. It is worthwhile to mention that, if we take m = 1 and D = R n throughout this work, our method coincides with the one studied in [START_REF] Chrétien | Generalized Proximal Point Algorithms and Bundle Implementations[END_REF] and the exact version analyzed in [START_REF] Durea | Some remarks on proximal point algorithm in scalar and vectorial cases[END_REF] for the finite dimensional Euclidean setting with the square Euclidean norm as the regularization term.

4.2.

Quasi-convex case. In this section, we consider Algorithm 1 with the additional assumptions that F : R n → R m is a R m + -quasi-convex function, D is convex, and the well-known R m + -completeness assumption on the set (F (x 0 ) -R m + ) ∩ F (D): (H): For every sequence {a k } ⊂ D, with a 0 = x 0 , such that F (a k+1 ) F (a k ), for all k ∈ N, there exists a ∈ D such that

F (a) F (a k ) ∀k ∈ N.
The R m + -quasi-convex case was analyzed by Apolinário, Papa Quiroz, and Oliveira [START_REF] Apolinário | A scalarization proximal point method for quasiconvex multiobjective minimization[END_REF] in the unconstrained framework, i.e., D = R n . They compute the (k + 1)th iteration as follows: [START_REF] Custódio | Direct multisearch for multiobjective optimization[END_REF] 0

∈ ∂ F (•), z k + λ k 2 ε k , z k || • -x k || 2 (x k+1 ) + N Ω k (x k+1 ).
This case was also studied by Bento, Cruz Neto, and Soubeyran [START_REF] Bento | A proximal point-type method for multicriteria optimization[END_REF]. They consider the following iterative procedure [START_REF] Das | Normal-boundary intersection: A new method for generating the Pareto surface in nonlinear multicriteria optimization problems[END_REF] x k+1 ∈ arg min

x∈R n ϑ F (x) + δ Ω k + λ k 2 ||x -x k || 2 e ,
where e = (1, . . . , 1) ∈ R m , the scalarization function ϑ : R m → R is given by ϑ(y) = max 1≤i≤m y, e i , and {e i } is the canonical base of the space R m . The convergence analyses of both algorithms [START_REF] Custódio | Direct multisearch for multiobjective optimization[END_REF] and ( 18) are based on Fejér monotonicity (see definition below), using the same approach proposed by Bonnel, Iusem, and Svaiter [START_REF] Bonnel | Proximal methods in vector optimization[END_REF]. In these works, the scalarization plays an important role in their proofs because the vectorial subproblems are replaced by scalar optimality conditions using a scalarization function; see [START_REF] Bonnel | Proximal methods in vector optimization[END_REF]Theorem 3.1], [START_REF] Apolinário | A scalarization proximal point method for quasiconvex multiobjective minimization[END_REF]Proposition 3.4.1], and [START_REF] Bento | A proximal point-type method for multicriteria optimization[END_REF]Theorem 4.1].

We emphasize that, as in the previously mentioned works, our convergence analysis for the R m + -quasi-convex case is also based on the Fejér monotonicity of the sequence generated by the algorithm. However, it does not depend on any scalarization functional.

Before we give the main result of this section, let us recall that a sequence {y k } is said to be Fejér convergent (or Fejér monotone) to a nonempty set

U ⊂ R n if, for all k ∈ N ||y k+1 -y|| ≤ ||y k -y|| ∀y ∈ U.
The following result is well known and its proof is elementary.

Proposition 3. Let U ⊂ R n be a nonempty set and {y k } be a Fejér convergent sequence to U . Then, {y k } is bounded. Moreover, if a cluster point y of {y k } belongs to U , then {y k } converges to y.

The next theorem shows that for the R m + -quasi-convex case we have convergence of the (whole) sequence to a Pareto critical point. Proof. We divide the proof into five steps.

Step 1 (Fejér convergence). Define E ⊂ D as

E = {x ∈ D : F (x) F (x k ) ∀k ∈ N}.
From assumption (H), the set E is nonempty. Now, take an arbitrary point x * ∈ E, which means that

x * ∈ Ω k for all k ∈ N. Denote γ k+1 = λ k ε k , u k+1 . Note that γ k+1 > 0, for each k ∈ N, because λ k > 0, ε k ∈ R m ++ , and u k ∈ R m + \{0} for all k ∈ N. Since ||x k -x * || 2 = ||x k -x k+1 || 2 + ||x k+1 -x * || 2 + 2 x k -x k+1 , x k+1 -x * ,
we conclude from ( 9) that

||x k -x * || 2 = ||x k -x k+1 || 2 + ||x k+1 -x * || 2 + 2 γ k+1 A k+1 (u k+1 + v k+1 ) + τ k+1 w k+1 , x k+1 -x * = ||x k -x k+1 || 2 + ||x k+1 -x * || 2 + 2 γ k+1 m i=1 (u k+1 i + v k+1 i ) a k+1 i , x k+1 -x * + τ k+1 w k+1 , x k+1 -x * , ( 19
)
where a k+1 i ∈ ∂f i (x k+1 ) for all k and i = 1, . . . , m. On the other hand, since F is R m + -quasi-convex and x * ∈ Ω k and γ k > 0 for all k, we obtain

(20) 2 γ k+1 m i=1 (u k+1 i + v k+1 i ) a k+1 i , x k+1 -x * ≥ 0.
Moreover, w k+1 ∈ N D (x k+1 ), together with τ k > 0, leads to [START_REF] Fliege | Steepest descent methods for multicriteria optimization[END_REF] τ k+1 w k+1 , x k+1x * ≥ 0.

Thus, using ( 20) and ( 21) in ( 19), we have

||x k+1 -x k || 2 ≤ ||x k -x * || 2 -||x k+1 -x * || 2 ∀k ∈ N which means that ||x k+1 -x * || ≤ ||x k -x * || for any x * ∈ E. In other words, {x k } is Fejér convergent to E.
Step 2 (the cluster points of {x k } belong to E). Since {x k } is Fejér convergent to E, it follows from Proposition 3 that {x k } is bounded. Let x * be a cluster point of {x k }. It follows from the definition of the algorithm that F (x k+1 ) F (x k ) for all k. Thus, from the continuity of F , we can easily conclude that F (x * ) F (x k ) for all k, which means that x * ∈ E.

Step 3 (convergence of the sequence). This step directly follows from Proposition 3 combined with Steps 1 and 2.

Step 4 (proximity of consecutive iterates). Assume that {x k } converges to x. From the triangular inequality, we have

(22) ||x k+1 -x k || ≤ ||x k+1 -x|| + ||x k -x|| ∀k ∈ N.
Noting that the right-hand side of ( 22) vanishes as k → +∞ because x k → x as k → +∞, we conclude lim k→+∞ ||x k+1x k || = 0.

Step 5 (Pareto criticality of the limit point). The proof of this step uses the same argument as in the proof of Theorem 3 from ( 15) on. This establishes the result. Remark 6. It is well known that, under the R m + -convexity assumption, the concepts of weak Pareto and Pareto critical are equivalent. In this case, under the assumption of R m + -convexity, the last theorem ensures that any sequence generated from Algorithm 1 converges to a weak Pareto point of F as in [START_REF] Bonnel | Proximal methods in vector optimization[END_REF]. Now, regarding assumption (H), it is standard for ensuring existence of Pareto points for vector optimization problems and an interesting discussion on existence conditions of such points can be found in [START_REF] Luc | Theory of Vector Optimization[END_REF]Chapters 2 and 3]. Taking into account that F 0, if we suppose that F (D) is a closed set, then assumption (H) holds. Note that this occurs naturally when F (D) is a compact set; for more details, see [START_REF] Luc | Theory of Vector Optimization[END_REF]Lemma 3.5,page 47]. It is worth noting that assumption (H) is used just to ensure that the set E, defined in step 1 of the proof of Theorem 4, is a nonempty set. As observed in [START_REF] Bonnel | Proximal methods in vector optimization[END_REF], in the absence of (H), this could be obtained by considering, for example, that the sequence {x k } has a cluster point.

Final remarks.

In this paper, we have proposed a new approach for the convergence of the proximal point algorithm in finite dimensional multiobjective optimization. We proved that our approach can be successfully applied to obtain convergence properties of the proximal method for locally Lipschitz vector-valued maps. Although this approach can be applied to R m + -quasi-convex vector functions (and, in particular, R m + -convex vector functions), this new technique seems to be particularly useful for vector functions which make Ω k in (8) a nonconvex set. To the best of our knowledge, it was the first time that a possible nonconvex Ω k was considered in the proximal method [START_REF]Practical approaches to multiobjective optimization[END_REF].

The next steps as future works would be to propose inexact versions of Algorithm 1, as for instance in Ceng and Yao [START_REF] Ceng | Approximate proximal methods in vector optimization[END_REF] and Durea and Strugariu [START_REF] Durea | Some remarks on proximal point algorithm in scalar and vectorial cases[END_REF], following our approach of convergence as well as a dynamic formulation of the well-known static group compromise problem using the recent VR approach of human behaviors (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]). In the VR context, an extension of this paper would examine the case where inconveniences to change can be identified at a distance, i.e., costs able to change C(x, y) = C(y, x) and costs able to stay C(x, x) = 0. In the general case C(x, y) = C(y, x), and C(x, x) 0 is possible. Hence, future research will examine the case where the Euclidean norm in (8) is replaced by a "like-distance" as done for instance in Bento and Soubeyran [START_REF] Bento | Generalized inexact proximal algorithms: Routine's formation with resistance to change, following worthwhile changes[END_REF] and Moreno, Oliveira, and Soubeyran [START_REF] Moreno | A proximal algorithm with quasidistance, Application to habit's formation[END_REF] for scalar-valued functions. This is better adapted to applications in behavioral sciences.

Finally let us emphasize two advantages of our approach compared to the one in [START_REF] Bonnel | Proximal methods in vector optimization[END_REF] and [START_REF] Durea | Some remarks on proximal point algorithm in scalar and vectorial cases[END_REF].

Remark 7. Given a nonempty set Ω and a vector-valued function F . Let ζ : R m → R be a scalar function, here called a scalarization function. One can define a scalar optimization problem, corresponding to the vector problem VP as follows: [START_REF] Fukuda | On the convergence of the projected gradient method for vector optimization[END_REF] min{ζ(F (x)) : x ∈ Ω}.

Consider the family of scalar-valued functions z : R m → R for each z ∈ R m + \ {0}, given by [START_REF] Fukuda | A survey on multiobjective descent methods[END_REF] z (y) := y, z . arg min{ z (F (x)) : x ∈ Ω}.

If
The scalarization function [START_REF] Fukuda | A survey on multiobjective descent methods[END_REF] was used in the convergence analysis of the proximal methods proposed in [START_REF] Bonnel | Proximal methods in vector optimization[END_REF] for a convex context. However, the authors show that the solution set of the scalarized problem may be empty. They actually provide an example where the solution set of the scalarized problem is nonempty only if z belongs to a set of measure zero in R 2 ; see [START_REF] Bonnel | Proximal methods in vector optimization[END_REF]Remark 1]. This means that even the convexity assumption of the vector problem being sufficient to ensure the family given by ( 24) is a complete weak scalarization for the vector problem, it does not guarantee that the scalarized problem is nonempty. where z ∈ R m ++ . Such a scalarization is a well-known functional in the vectorial optimization literature; see [27, section 3]. A particular instance of this functional was used in [START_REF] Bento | A proximal point-type method for multicriteria optimization[END_REF] to study the convergence of a proximal-point-type method for multiobjective optimization problems in the quasi-convex setting. A more general version of this scalarization functional was used in [START_REF] Durea | Some remarks on proximal point algorithm in scalar and vectorial cases[END_REF] for finding a weak Pareto point of an F K-convex, for an ordering cone K by means of the following (exact) proximal method:

x k+1 ∈ arg min x∈X { e (F (x)) + λ k 2 ||x -x k || 2 },
where e ∈ int K and X is a Hilbert space. However, it is worth mentioning that in this kind of method, we have the following descent property e (F (x k+1 )) ≤ e (F (x k )) ∀k ∈ N which clearly does not imply the descent property F (x k+1 ) F (x k ) as proposed by [START_REF] Bonnel | Proximal methods in vector optimization[END_REF] whereas the converse implication holds. As mentioned in the introduction the vectorial improvement F (x k+1 ) F (x k ) plays an important role in applications.

  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].

  where B[0, 1] denotes the closed unit ball in R m . Proof. It is known that the distance function, d C (x), is globally Lipschitz with constant L = 1; see, for instance, [15, Proposition 2.4.1]. From [15, Proposition 2.1.2], we have that a Lipschitz function f of rank L near to x satisfies ||ξ|| ≤ L for every ξ ∈ ∂f (x). Thus, ∂d C (x) ⊂ B[0, 1]. On the other hand, it follows from [15, Proposition 2.4.2] that N C (x) = cl {∪ λ≥0 λ∂d C (x)}, where cl denotes closure. This implies that ∂d C (x) ⊂ N C (x) and the proof is completed.

Theorem 4 .

 4 The sequence {x k } converges to a Pareto critical point of F .

Remark 8 .

 8 Let z : R m → R be a scalarization given byz (y) := inf{t ∈ R : y ∈ tz -R m + },

  the corresponding vector problem (VP) is convex, it follows from [38, Proposition 3.2] that this family is a complete weak scalarization for the corresponding vector problem, i.e., arg min w {F (x) : x ∈ Ω} =

	z∈R m + \{0}

Acknowledgments. G. Bento, J.X. Cruz Neto, and J.C.O. Souza wish to express their gratitude to Professor Genaro López and Professor Antoine Soubeyran for their hospitality during the authors' visit to Universidad de Sevilla and Aix-Marseille University (Aix-Marseille School of Economics). The authors wish to express their gratitude to the anonymous referees for their helpful comments.

The research of first author was supported in part by FAPEG 201210267000909 -05/2012, CNPq grants 458479/2014-4, 471815/2012-8, 312077/2014-9, and CAPES (88881.117595/2016-01). The research of the second author was supported in part by CNPq grant 305462/2014-8. The research of the third author has been supported by grant MTM2015-65242-C2-1-P. The research of the fourth author was supported by ANR GREEN-Econ research project (ANR-16-CE03-0005).

The research of the fifth author was supported in part by CNPq-Ciências sem Fronteiras grant 203360/2014-1.


Tripodi P, Barchi L, Kale S, Esposito S, Rabanus-Wallace MT, Acquadro A, Schafleitner R, Prohens J, Diez MJ, Börner A, Bovy A, Salinier J, Boyaci F, Pasev G, Brandt R, Himmelbach A, Portis E, Finkers R, Lanteri S, Paran I, Lefebvre V, Giuliano G, Stein N. 2020. Genebank genomics highlights novel centres of diversity and routes of migration in pepper (Capsicum spp.). SOL International Meeting 2020, November 9-11, 2020, on

  Session opens at 13:55 and closes at 15:30  14:00-14:25 CET BIOop: Alain Goossens, VIB-Ghent University, Belgium.

Matthew Gibson, Indiana

  University, USA. Reconstructing the history and biological consequences of a plant invasion on the Galapagos islands  16:40-16:55 CET EVOo02: Pasquale Tripodi, CREA Research Center for Vegetable and Ornamental Crops, Italy. Genebank genomics highlights novel centers of diversity and routes of migration in pepper (Capsicum spp.).  16:55-17:10 CET EVOo03: Andrea Coccuci, Instituto Multidisciplinario de Biologia Vegetal (IMBIV-UNC), Universidad Nacional de Cordoba, Argentina. Flower color evolution in the florally diverse genus Salpichroa (Solanaceae) seen through the eyes of its pollinators.  17:10-17:25 CET EVOo04: Remco Stam, Technical University of Munich, Germany. Using wild tomato species to study diversity and evolution of biotic stress responses.

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	 16:25-16:40 CET EVOo01: 17:30 -17:45 CET: Break
	17:45 -18:30 CET: Poster session 1 (BIO1, EVO1, DEV1)
	November 10th, 2020
	12:55 -13:40 CET: Poster session 2 (BIO2, EVO2, DEV2).
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	View session recording
	Session opens at 13:55 and closes at 15:30
	14:40-14:55 CET BIOo02: Johan Hunziker, University of Tsukuba, Japan. Improvement of
	carotenoid accumulation in tomato fruits by CRISPR-Cas9 Cytidine Deaminase system.
	 14:55-15:10 CET BIOo03: Eva Knoch, RIKEN, Japan. The third DWF1 paralog in
	Solanaceae, sterol D24-isomerase, branches withanolide biosynthesis from the general
	phytosterol pathway.
	 15:10-15:25 CET BIOo04: Lorena Ramirez, Wageningen University, The Netherlands. A
	new insight of StCDF1 under drought stress in potato.
	15:30-16:00 CET -Break

Unraveling the regulation of tomato metabolism through interactomics analyses.

 14:25-14:40 CET BIOo01: Peter Waterhouse, Queensland Univ of Technology, Australia. Genetic, epigenetic and metabolomic insights from reference level genome assembly of a wild and a laboratory strain of Nicotiana benthamiana.   14:00 -14:25 CET DEVop: Sebastian Soyk, University of Lausanne, Switzerland. Dissecting cryptic genetic variation with impact on tomato domestication and breeding. 14:25 -14:40 CET DEVo01: Glenn Philippe, Cornell University, USA. Elucidating the mechanisms of plant cuticle deposition and restructuring using tomato fruit development as a model.  14:40 -14:55 CET DEVo02:

Bruno Silvestre Lira, Universidade

  de So Paulo, Brazil. Light and ripening-regulated B-BOX protein-encoding genes in Solanum lycopersicum.  14:55 -15:10 CET DEVo03: Matthias Benoit, CSHL, USA. Contribution of Transposons to Structural Variation and Phenotypic Diversity in Tomato.  15:10 -15:25 CET DEVo04: Ashley Snouffer, University of Georgia, USA.Regulation of cell division and cell shape by tomato OFPs and TRMs to control early ovary shape. Deciphering the epigenetic variability in S. lycopersicum accessions and their influence on hybrid vigor.  16:40 -16:55 CET NETo02: Lara Pereira, University of Georgia, USA. Natural genetic diversity in tomato flavor genes. Oregon State University, USA. StPIP1, a predicted PAMP-induced peptide in potato, elicits plant defenses and is associated with disease symptom severity in a compatible interaction with potato virus Y  14:55 -15:10 CET STRo03: Saloni Mathur, National Institute of Plant Genome Research New Delhi, India. Decoding the regulatory networks of diverse coding and noncoding RNAs towards heat stress response in tolerant and sensitive tomato cultivars from Indian subcontinent  15:10 -15:25 CET STRo04: Golam Jalal Ahammed, Henan University of Science and Technology, China. Nitric oxide is involved in the SlWRKY81 transcription factor-regulated stomatal movement and drought tolerance in Solanum lycopersicum L. 16:00 -16:25 CET BREop: Dani Zamir, Hebrew University of Jerusalem, Israel. Exotic Genetic map resistance to Tomato Brown Rugose Fruit Virus (ToBRFV) to the Tobacco Mosaic Virus locus, TM1.  16:25 -16:40 CET BREo01: Sean Fenstemaker, Ohio State University, USA. 17:10 -17:20 CET BREo04: Ernst-Jan Eggers, Wageningen University & Research, The Netherlands. The S-Locus Inhibitor gene encodes an F-box protein with a Lectin domain and crucially enables hybrid potato breeding  17:20 -17:30 CET BREo05: Nathalie Kaiser, Michigan State University, USA. Assessing the contribution of Sli to self-compatibility in North American diploid potato germplasm using KASP markers  17:30 -17:35 CET Joint Question period for diploid potato inbreeding talks 17:45 -18:15 CET SOLIOM Meeting closing

	Using
	Solanum galapagense to achieve tolerance to water deficit through introgression breeding
	 16:55 -17:10 CET NETo03: Gustavo A. Silva-Arias, Technical University of Munich, and grafting
	Germany. Ecological genomics in wild tomatoes to reveal mechanisms of adaptation to living  16:40 -16:55 CET BREo02: Yury Tikunov, Wageningen University & Research, The
	in the driest region of the world. Netherlands. The genetic and functional analysis of flavor in commercial tomato: a novel
	 17:10 -17:25 CET NETo04: Salvador Torres Montilla, CRAG, Spain. A novel system for the FLORAL4 gene underlies a QTL for floral aroma volatiles in tomato fruit
	artificial differentiation of chromoplasts from pre-existing chloroplasts in leaves  16:55 -17:10 CET BREo03: Jaime Prohens, Universitat Politecnica de Valencia,
	Spain. Towards the development of the first eggplant MAGIC population and QTL analysis of
	17:30 -17:45 CET: Break selected morphological traits
	17:45 -18:30 CET: Poster session 3 (NET1, STR1, BRE1).
	Diploid potato inbreeding mini-session:
	November 11th, 2020
	12:55 -13:40 CET: Poster session 4 (NET2, STR2, BRE2).
	13:40 -13:55 CET: Break
	Session 5: STRESS BIOLOGY (BIOTIC AND ABIOTIC)
	View session recording
	Session opens at 13:55 and closes at 15:30
	15:30 -16:00 CET Break
	15:30 -16:00 CET Break
	Session 6: BREEDING (TECHNOLOGIES AND TRAITS)
	Session 4: NETWORKS OF METABOLITES, GENES AND ORGANISMS
	View session recording
	View session recording
	Session opens at 15:55 and closes at 17:30
	Session opens at 15:55 and closes at 17:30

 16:00 -16:25 CET NETop: Asaph Aharoni, Weizmann Institute of Science, Israel. The Bitter -Sweet Sides of Solanaceae-Specialized Metabolism.  16:25 -16:40 CET NETo01: Luisa Fernanda Bermudez Salazar, IABIMO, INTA-CONICET, Argentina,  14:00 -14:25 CET STRiop: Parvinderdeep Kahlon, TUM School of Life Sciences, Technical University of Munich, Germany. Population studies of the wild tomato species Solanum chilense reveal geographically structured major gene-mediated pathogen resistance  14:25 -14:40 CET STRo01: Andrea Schrader, RWTH Aachen University, Germany. Resilience to salinity in tomato -root transcriptome data analyses.  14:40 -14:55 CET STRo02: Aymeric Goyer,  


Prohens J, Salinier Diez Mj

J Lefebvre

Boyaci Hf

Boerner A, Brandt R, Stein N, Finkers R, Bovy A, Schafleitner R, Scaglione D Rotino Gl

Di Centa

Giuliano G 2019 Single

Eggplant Germplasmlorenzo Barchi

Alberto Acquadro

Ezio Portis

Sergio Lanteri

Davis Alonso

Pietro Gramazio

Santiago Vilanova

Maria Josè Díez

Jaime Prohens

Jeremy Salinier

Veronique Lefebvre

Gancho Pasev

Stanislava Grozeva

Hatice Filiz Boyaci

Abdullah Unlu

Laura Toppino

Laura Bassolino

Giuseppe Leonardo Rotino

Andreas Boerner

Ronny Brandt

Nils Stein

Richard Finkers

Arnaud G Bovy

Roland Schafleitner

Davide Scaglione

Eleonora Di Centa

Sara Pinosio

Giuseppe Aprea

Paola Ferrante

Giovanni Giuliano

W983 Single Primer Enrichment Genotyping Highlights the Worldwide Population Structure of

G2P-SOL (http://www.g2p-sol.eu ) is an EU-funded project, bringing together the main European and international genebanks hosting germplasm of the four major Solanaceous crops: potato, tomato, pepper and eggplant. 23,900 tomato and 5,900 eggplant accessions, including wild relatives of both crops, have been inventoried within the project.

To gain information about population structure of the collections, the novel Single Primer Enrichment Technology (SPET, US Patent 9,650,628) developed by NuGEN was used for genotyping. An SNP/indel panel was developed by assaying 14k probes for tomato and 11k for eggplant, evenly distributed in the gene-rich regions, and selecting the 5k best performing probes for each species. DNA samples were prepared by the genebanks using pre-tested protocols, and genotyping partners (ENEA for tomato and University of Torino for eggplant) performed QC and sample normalization.

Genotyping and sequencing was performed by IGA Technology Services. Reads were aligned to the eggplant and tomato reference genomes using BWA-MEM and SNP calling was performed using GATK-4.0.

We report on the assessment of the genetic relationships in a wide set of tomato and eggplant accessions maintained in genebanks as well as the extent of duplications and possible mis-classifications. The results suggest that SPET genotyping is a reliable, high-throughput, low cost technology for genetic fingerprinting of crops, with a high degree of cross-transferability to their wild relatives. SPET-based higher density genotyping is being developed to characterize the core collections developed in G2P-SOL for GWAS analyses. In https://pag.confex.com/pag/xxvii/meetingapp.cgi/Home/0

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

  Workshop: Solanaceae  Program: Workshops  Day: Sunday, January 13, 2019