J X Cruz Neto

J O Lopes

Antoine Soubeyran
email: antoine.soubeyran@gmail.com

João Carlos

O Souza

J X Cruz Neto

• A Soubeyran

J C O Souza

Abstract regularized equilibria: application to Becker's household behavior theory

Keywords: Equilibrium problem, Variational rationality, Desires, Traps, Household behavior, Resource allocation problems B

Introduction

Let K ⊂ R m be a nonempty, closed and convex set and consider a bifunction F : K ×K → R such that (P1) F(x, x) = 0 for all x ∈ K ; (P2) F(•, y) : K -→ R is upper semicontinuous for all y ∈ K ; (P3) F(x, •) : K -→ R is convex and lower semicontinuous for all x ∈ K .

The equilibrium problem (EP) consists of finding x * ∈ K such that F(x * , y) ≥ 0, ∀ y ∈ K .

(1) This problem will be denoted by E P(F, K ) and its solution set by S(F, K ). Equilibrium problems have been widely studied both on the issue dealing with conditions for existence of solutions (e.g., [START_REF] Iusem | On certain conditions for the existence of solutions of equilibrium problems[END_REF][START_REF] Iusem | New existence results for equilibrium problems[END_REF][START_REF] Konnov | Nonmonotone equilibrium problems: Coercivity conditions and weak regularization[END_REF][START_REF] Chadli | Existence results for equilibrium problems with applications to evolution equations[END_REF] and algorithmic purposes (see, for instance [START_REF] Bento | Generalized proximal distances for bilevel equilibrium problems[END_REF][START_REF] Burachik | On a generalized proximal point method for solving equilibrium problems in banach spaces[END_REF][START_REF] Cruzneto | On a Bregman regularized proximal point method for solving equilibrium problems[END_REF][START_REF] Flåm | Equilibrium programming using proximal-like algorithms[END_REF][START_REF] Iusem | Inexact proximal point methods for equilibrium problems in Banach spaces[END_REF][START_REF] Iusem | On the proximal point method for equilibrium problems in Hilbert spaces[END_REF][START_REF] Konnov | Application of the proximal point method to nonmonotone equilibrium problems[END_REF][START_REF] Langenberg | Interior point methods for equilibrium problems[END_REF][START_REF] Mordukhovich | Hybrid proximal methods for equilibrium problems[END_REF][START_REF] Moudafi | On finite and strong convergence of a proximal method for equilibrium problems[END_REF][START_REF] Moudafi | Proximal point algorithm extended to equilibrium problems[END_REF][START_REF] Nguyen | The interior proximal extragradient method for solving equilibrium problems[END_REF][START_REF] Oliveira | A Tikhonov-type regularization for equilibrium problems in Hilbert spaces[END_REF][START_REF] Hieu | New extragradient method for a class of equilibrium problems in Hilbert spaces[END_REF][START_REF] Mashreghi | A proximal augmented Lagrangian method for equilibrium problems[END_REF].

It represents a rather general and suitable format for investigation and solution of various problems arising in mathematical physics, economics, operations research, game theory, and other fields; see for instance [START_REF] Blum | From optimization and variational inequalities to equilibrium problems[END_REF][START_REF] Muu | Convergence of an adaptive penalty scheme for finding constrained equilibria[END_REF][START_REF] Panicucci | On solving generalized Nash equilibrium problems via optimization[END_REF]. The term equilibrium problem was coined in [START_REF] Blum | From optimization and variational inequalities to equilibrium problems[END_REF], but the problem itself was studied in much early work of [START_REF] Fan | A minimax inequality and applications[END_REF]. As pointed out in [START_REF] Blum | From optimization and variational inequalities to equilibrium problems[END_REF], EP includes as particular case scalar and vector optimization problems, saddle-point (minimax) problems, variational inequalities, Nash equilibria problems, complementarity problems and fixed point problems.

One of the most popular strategy for solving (EP) is the so-called regularization method. The Tikhonov regularization method for ill-posed problems is well-known for minimization, monotone inclusion and fixed-point problems. This approach was introduced for solving an equilibrium problems by [START_REF] Moudafi | Proximal point algorithm extended to equilibrium problems[END_REF], see also [START_REF] Moudafi | Proximal and dynamical approaches to equilibrium problems[END_REF] and [START_REF] Iusem | On the proximal point method for equilibrium problems in Hilbert spaces[END_REF]. This iterative method solves at each iteration the regularized E P(F k , K ), i.e., given x k it computes x k+1 as follows:

x k+1 ∈ S(F k , K ), ( 2
)
where

F k (x, y) = F(x, y) + γ k x -x k , y -x (3)
and {γ k } is a positive bounded auxiliar sequence. This kind of method was studied for instance in [START_REF] Bento | Generalized proximal distances for bilevel equilibrium problems[END_REF], [START_REF] Burachik | On a generalized proximal point method for solving equilibrium problems in banach spaces[END_REF], [START_REF] Blum | From optimization and variational inequalities to equilibrium problems[END_REF] and [START_REF] Konnov | Application of the proximal point method to nonmonotone equilibrium problems[END_REF] in different contexts and it consists of solving at each iteration a regularized (EP) instead of the original problem. [START_REF] Burachik | On a generalized proximal point method for solving equilibrium problems in banach spaces[END_REF] generalized the method (2) by using Bregman functions on the regularization term as follows:

F k (x, y) = F(x, y) + γ k ∇φ(x) -∇φ(x k ), y -x , ( 4
)
where φ is Bregman function; see [START_REF] Burachik | On a generalized proximal point method for solving equilibrium problems in banach spaces[END_REF] for more details. If φ(x) = 1 2 ||x|| 2 then (4) reduces to (3). In order to generalize (4), [START_REF] Bento | Generalized proximal distances for bilevel equilibrium problems[END_REF] proposed the following regularization in (2):

F k (x, y) = F(x, y) + λ k ∇ 1 d(x, x k ), y -x , ( 5
)
where ∇ 1 d stands to the gradient of the mapping of d with respect to the first variable. They established convergence results for a more general setting of bilevel equilibrium problems which include equilibrium problems. [START_REF] Oliveira | A Tikhonov-type regularization for equilibrium problems in Hilbert spaces[END_REF] considered existence of solutions of (EP) using a Tikhonov-type regularization given by F ˜ (x, y) = F(x, y) +λg(x, y),where g : K × K → R is a strongly monotone bifunction and λ> 0.

A more general class of problem than (EP) is the so-called Mixed Equilibrium Problem (MEP). This problem was studied by [START_REF] Ceng | A hybrid iterative scheme for mixed equilibrium problems and fixed point problems[END_REF], see also [START_REF] Yao | Mixed equilibrium problems and optimization problems[END_REF]and related literatures. The mixed equilibrium problem consists of finding x * ∈ K such that F(x * , y) + ψ(y) -ψ(x * ) ≥ 0, ∀ y ∈ K , (6) where ψ : K -→ R is a given function. This problem is denoted by M E P(F, K ). We refer to [START_REF] Muu | Dc-gap function and proximal methods for solving nash-cournot oligopolistic equilibrium models involving concave cost[END_REF], [START_REF] Oyewole | An extragradient algorithm for split generalized equilibrium problem and the set of fixed points of quasi-φ-nonexpansive mappings in Banach spaces[END_REF] and [START_REF] Shamshad | A hybrid iterative algorithm for a split mixed equilibrium problem and a hierarchical fixed point problem[END_REF] for a variants of this problem. Note that, if ψ ≡ 0, then the mixed equilibrium problem (6) reduces to the equilibrium problem (1). A more general version of ( 6) can be considered as finding x * ∈ K such that F(x * , y) + ϕ(x * , y) -ϕ(x * , x * ) ≥ 0, ∀ y ∈ K , (7) where ϕ : K × K -→ R is a given function. We call this problem of generalized mixed equilibrium problem (GMEP). An inexact version of problem ( 7), in the sense that each subproblem is solved inexactly, can be considered as follows: given δ ≥ 0, a point x * ∈ K is a δ-solution of the inexact mixed equilibrium problem if it satisfies

F(x * , y) + ϕ(x * , y) -ϕ(x * , x * ) ≥ -δ, ∀ y ∈ K . ( 8
)
In this case, if ϕ ≡ 0, then (8) becomes the inexact equilibrium problem studied by [START_REF] Hung | On inexact Tikhonov and proximal point regularisation methods for pseudomonotone equilibrium problems[END_REF]. We denote the inexact equilibrium problem (8) by I E P(F, K , δ). The aim of this work is twofold. Firstly, we analyze the existence and uniqueness of solutions for a mixed equilibrium problem given by ( 7) in order to propose an abstract regularized algorithm for solving equilibrium problems such that it generalizes some existing regularized methods in the literature. The motivation to consider this kind of regularization comes from application in behavioral sciences using recent Variational Rationality (VR) approach of human dynamics by [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]Soubeyran ( , 2019a, b), b). In this context, the regularization term can be seen as a crude formulation of the complex concept of resistance to change; see for instance [START_REF] Bento | Generalized inexact proximal algorithms: Routines formation with resistance to change, following worthwhile changes[END_REF][START_REF] Cruzneto | A generalized proximal linearized algorithm for DC functions with application to the optimal size of the firm problem[END_REF][START_REF] Bento | A proximal point method for difference of convex functions in multi-objective optimization with application to group dynamic problems[END_REF][START_REF] Hieu | New extragradient method for a class of equilibrium problems in Hilbert spaces[END_REF] and [START_REF] Soubeyran | General descent method using w-distance. Application to emergence of habits following worthwhile moves[END_REF]. As a second contribution, we give an application in behavioral sciences of our abstract regularization approach of equilibrium problems. We propose a dynamic version of the famous static Becker's "allocation of time problem" using the VR approach; see [START_REF] Becker | A theory of the allocation of time[END_REF].

In the seminal work [START_REF] Becker | A theory of the allocation of time[END_REF], "A theory of the allocation of time", the Nobel prize Becker [START_REF] Becker | A theory of the allocation of time[END_REF]) revolutionized the modeling of household behavior. Following the splendid comments of [START_REF] Chappori | A theory of the allocation of time[END_REF], " [START_REF] Becker | A theory of the allocation of time[END_REF] is not the first study to consider time use in the home, e.g., he cites Jacob Mincer (1962), who considers a married woman's time trade-off between housework and paid work. Nor is it quite the first to propose a household production function, in which purchased goods are converted into commodities like meals that generate utility. An earlier household production article he does not cite is Terence Gorman (1980), which was written in 1956 and widely circulated as a working paper for decades before it was finally published. However, what Becker does uniquely is to merge goods consumption with time use in the production of household utility. Previous models of labour supply considered consumption and leisure as distinct goods that separately provide utility. In contrast, Becker emphasizes that there are many different types of time use, just as there are many types of consumption goods, and that different types of time use and consumption goods combine in different ways to yield commodities, e.g., prepared meals, from which we get utility. He then draws a variety of important implications from the observation that various types of time and consumption combine into a single household objective function with a single overall budget constraint. In doing so, Becker (along with Mincer) created the foundational modeling framework for virtually all modern household level analyses of consumption and time use, in what was sometimes called the "New Home Economics". Becker's model is in some ways more general than those in common use today, in particular, he emphasizes that different types of time (e.g., weekends versus weekdays) should have different shadow prices within the household. This is in contrast to the vast majority of models today that associate a single observable wage rate to each individual's time.

The modern literature has generalized Becker in two main directions. A first strand of the literature considers unitary models which, like Becker's original time allocation theory, treat the household as maximizing a single utility function under a resource constraint. They are often dynamic and forward looking. A second strand of the literature follows a second Becker's paper [START_REF] Becker | A theory of social interactions[END_REF], where the allocation of resources within a household is determined by bargaining among household members, with outcomes that may therefore depend on the determinants of the bargaining power of each household member. We will not examine this second extension.

The paper is organized as follows. In Sect. 2, we recall some basic concepts and results in equilibrium problems. In Sect. 3, we show an existence result for our regularized bifunction. In Sect. 4, we present a generalized regularization algorithm for solving an equilibrium problem and study its convergence analysis. The method is performed in some academic examples in Sect. 5. An original application to Becker's household behavior theory is given in Sect. 6. Finally, some conclusions are presented.

Preliminaries

In this section, we recall some definitions and known results which will be used in the following sections. Usually, existence results for E P( f , K ) require either boundedness of the feasible set K or certain coercivity conditions. We recall the classical Ky Fan inequality result from [START_REF] Fan | A minimax inequality and applications[END_REF] in case of Minimax Problems.

Theorem 1 Assume that F satisfies the Conditions (P1)-(P3) and K is compact. Then S(F, K ) is nonempty.

Recently, many efforts were concentrated for obtaining coercivity conditions, which guarantee existence of solutions on unbounded feasible sets; see e.g. [START_REF] Iusem | On certain conditions for the existence of solutions of equilibrium problems[END_REF] and [START_REF] Burachik | On a generalized proximal point method for solving equilibrium problems in banach spaces[END_REF] and references therein. For such coercivity conditions it has been assumed beyond the first three basic assumptions on equilibrium problems, i.e., (P1)-(P3), one has been considered the following properties:

(P4) F(x, y) + F(y, x) ≤ 0 for all x, y ∈ K (monotone bifunction); (P5) For any sequence {z n } ⊂ K with lim n→∞ z n = ∞, there exists u ∈ K and n 0 ∈ N such that F(z n , u) ≤ 0, ∀ n ≥ n 0 .
Thus, under all these assumptions they obtained a well known result of existence of solution for the problem E P(F, K ); see (Iusem et al. 2009, Theorem 4.2). For sake of completeness, we present a version of the proof (adapted to the Euclidean space) which requires the following result proved by Ky Fan in 1961 in the context of convex feasibility problems; see (Fan 1961, Lemma 1).

Lemma 1 Let Y be a nonempty subset of R m and, for each y ∈ Y , consider a closed subset C(y) of R m . If the following two conditions hold: (C1) the convex hull of any finite subset {x 1 , . . . ,

x k } of Y is contained in ∪ k i=1 C(x i ); (C2) C(y) is compact for at least some y ∈ Y ; then ∩ y∈Y C(y) = ∅.
Theorem 2 Assume that F satisfies the Conditions (P1)-(P5). Then S(F, K ) is nonempty.

Proof Let n ∈ N be arbitrary. Define K n := {x ∈ K : x ≤ n} (since K n is nonempty for sufficiently large n, in what follows, for the sake of simplicity, we suppose without loss of generality that K n is nonempty for all n ∈ N, furthermore it is compact) and L F (n, y) := {x ∈ K n : F(y, x) ≤ 0} which is compact (closed from (P2) and bounded due to L F (n, y) ⊂ K n ). We intend to invoke Lemma 1 with Y = K n , C(y) = L F (n, y), and thus, we must check the validity of its hypotheses. Firstly, we verify (C1). For this, take x 1 , . . . , x k ∈ K n and λ 1 , . . . , λ k ∈ [0, 1] such that k i=1 λ i = 1, and denote by x = k i=1 λ i x i . We must verify that x belongs to ∪ k i=1 L F (n, x i ), i.e., that x ∈ K n and that F(x i , x) ≤ 0 for some i. Indeed,

x ∈ K n from convexity of K n and x ≤ k i=1 λ i x i ≤ n k i=1 λ i = n. Now, from (
P1) and (P3), we have 0 = F( x, x) ≤ k i=1 λ i F( x, x i ), so F( x, x i ) ≥ 0 for some i, thus by (P4) F(x i , x) ≤ 0. In the sequel, we verify (C2). Since C(y) = L F (n, y) is compact (as seen previously) and nonempty (since F(y, y) = 0 by (P1), implies y ∈ C(y)) for all y ∈ K n , and therefore (C2) holds. Thus, we are within the hypotheses of Lemma 1, and we conclude that ∩ y∈K n L F (n, y) is nonempty for each n ∈ N, so that for each n we may choose x n ∈ ∩ y∈K n L F (n, y). We distinguish two cases:

(i) There exists p ∈ N such that x p < p, or (ii) For all n ∈ N, x n = n . In this case, assumption (P5) ensures the existence of u ∈ K

and n 0 such that F(x n , u) ≤ 0 for all n ≥ n 0 . Take p ≥ n 0 such that u < p where F(x p , u) ≤ 0.

In both cases (i) or (ii), we will show that x p is a solution of E P(F, K ). Since x p ∈ ∩ y∈K p L F ( p, y), we have F(y, x p ) ≤ 0, for all y ∈ K p , by (P4) we get F(x p , y) ≥ 0, for all y ∈ K p . Thus, we only need to show that F(x p , w) ≥ 0, for all w ∈ K /K p . Let ȳ given by ȳ = x p (case (i)) or ȳ = u (case (ii)), so F(x p , ȳ) ≤ 0. Now, take any w ∈ K /K p , thus there exists t ∈ (0, 1) such that z t := (1t) ȳ + tw belongs to K p and F(x p , z t ) ≥ 0. By (P3), we have

0 ≤ F(x p , z t ) ≤ (1 -t)F(x p , ȳ) + t F(x p , w) ≤ t F(x p , w)
which implies that F(x p , w) ≥ 0, and hence, we conclude that F(x p , y) ≥ 0, for all y ∈ K , i.e., x p is a solution of E P(F, K ).

The following definition was introduced by [START_REF] Antipin | Iterative gradient prediction-type methods for computing fixed points of extremal mapping[END_REF] which encompasses a large class of functions that has interesting properties and can be used to make well-defined an equilibrium problem.

Definition 1 A mapping ϕ : K × K → R is said to be skew-symmetric if (A1) ϕ(x, x) -ϕ(x, y) -ϕ(y, x) + ϕ(y, y) ≥ 0, ∀x, y ∈ K .
If the inequality holds strictly, for all x, y ∈ K such that x = y, we say that ϕ is strictly skew-symmetric.

Skew-symmetric functions have some properties similar to monotonicity of the gradient and non-negativity of the second derivative of a convex function. For more properties and applications of skew symmetric functions we refer to [START_REF] Antipin | Iterative gradient prediction-type methods for computing fixed points of extremal mapping[END_REF].

We will consider the following assumptions on the mapping ϕ:

(A2) ϕ(x, •) : K -→ R is convex in the second argument, for all x ∈ K ; (A3) ϕ : K × K -→ R is continuous.
Example 1 Let f : R m → R and g : R m → R + be a convex functions. Define the bifunctions

ϕ i : R m × R m → R, i = 1, 2, given by ϕ 1 (x, y) = f (y) -f (x) and ϕ 2 (x, y) = g(x)g(y).
One can check that they satisfy assumptions (A1)-(A3). Indeed,

ϕ 1 (x, x) -ϕ 1 (x, y) -ϕ 1 (y, x) + ϕ 1 (y, y) = 0 and ϕ 2 (x, x) -ϕ 2 (x, y) -ϕ 2 (y, x) + ϕ 2 (y, y) = (g(x) -g(y)) 2 ≥ 0.
Conditions (A2) and (A3) directly follow from the convexity of f and g.

We recall that a set-valued operator T : R m ⇒ R m is called monotone (resp. strongly monotone with constant γ > 0) if

x -y, u -v ≥ 0, x -y, u -v ≥ γ ||x -y|| 2
for u ∈ T (x) and v ∈ T (y). A monotone operator is maximal if (considered as a graph) is it not strictly contained in any other monotone operator on R m . We denote the domain of T by dom (T ) and it is given by dom

(T ) = {x ∈ R m : T (x) = ∅}.
Example 2 Let T : R m → R m be a single-valued maximal monotone operator and define ϕ(x, y) = T (x), y . Then the condition (A1) is valid, i.e.,

ϕ(x, x) -ϕ(x, y) -ϕ(y, x) + ϕ(y, y) = T (x) -T (y), x -y ≥ 0, ∀x, y ∈ dom (T ).
Moreover, one can easily check that ϕ is convex in the second argument (condition (A2)) and continuous (condition (A3)).

Given x ∈ R n , we will denote by

ϒ(x) = ||x -arg min y∈K {F(x, y) + 1 2 ||y -x|| 2 }||.
The next proposition will be used in Sect. 5. It measures the quality of a candidate to solution of an (EP). It is a well known characterization of a solution of an equilibrium problem; see [START_REF] Iusem | On the maximal monotonicity of diagonal subdifferential operators[END_REF].

Proposition 1 A point x * ∈ S(F, K ) if and only if ϒ(x * ) = 0.

Existence and uniqueness of solution for GMEP

Let F be a bifunction, ϕ a regularization mapping and a constant λ > 0. We will consider the auxiliar bifunction F :

K × K → R defined by F(x, y) = F(x, y) + λ (ϕ(x, y) -ϕ(x, x)) . ( 9
)
Before present our results it is worth to mention some important examples of regularization existing in the literature which satisfy the assumptions made in the regularization ϕ for solving the equilibrium problem (1).

• [START_REF] Moudafi | Proximal point algorithm extended to equilibrium problems[END_REF], [START_REF] Moudafi | Proximal and dynamical approaches to equilibrium problems[END_REF] and [START_REF] Iusem | On the proximal point method for equilibrium problems in Hilbert spaces[END_REF], the authors considered the regularized auxiliar bifunction given by

F(x, y) = F(x, y) + λ x -x, y -x ,
where x ∈ K is fixed. Defining ϕ(x, y) = x -x, yx , one can check that ϕ is a skew-symmetric. Moreover, it satisfies the conditions (A2) and (A3). • [START_REF] Burachik | On a generalized proximal point method for solving equilibrium problems in banach spaces[END_REF], the following regularized bifunction F is considered

F(x, y) = F(x, y) + γ ∇φ(x) -∇φ( x), y -x ,
where φ is a Bregman function. Note that defining ϕ(x, y) = ∇φ(x) -∇φ( x), yx , we have that ϕ is skew-symmetric, and furthermore it satisfies the conditions (A2) and (A3).

• CruzNeto et al. ( 2019), the authors studied the following regularized equilibrium problems

F(x, y) = F(x, y) + γ (D φ (y, x) -D φ (x, x)),
where

D φ (x, y) = φ(x) -φ(y) -∇φ(y), x -y is the Bregman distance associated to the Bregman function φ. Considering ϕ(x, y) = D φ (y, x) -D φ (x, x)
we have that ϕ is a skew-symmetric mapping which satisfies the conditions (A2) and (A3). • Oliveira et al. ( 2013) considered the following Tikhonov-type regularization

F(x, y) = F(x, y) + λg(x, y),
where g : K × K → R is a strongly monotone bifunction which satisfies (A2), g(•, y) is upper semicontinuous and g(x, •) is lower semicontinuous, for all x, y ∈ K . Note that a strongly monotone bifunction is a (strongly) skew-symmetric mapping. It is worth to mention that we will consider both existence of solution and algorithmic approach for solving (EP) in a more general context than in [START_REF] Oliveira | A Tikhonov-type regularization for equilibrium problems in Hilbert spaces[END_REF].

From now on, we consider K ⊂ R m a closed and convex set, F : K × K → R a bifunction such that conditions (P1)-(P4) hold and ϕ be a generalized regularization mapping such that assumptions (A1)-(A3) hold.

As shown by [START_REF] Iusem | On certain conditions for the existence of solutions of equilibrium problems[END_REF] condition (P5) is a sufficient condition for the existence of solutions of (EP). Next, we consider a less restrictive condition than (P5) (here called (P6)) which was introduced by [START_REF] Burachik | On a generalized proximal point method for solving equilibrium problems in banach spaces[END_REF].

Theorem 3 Let x ∈ K be fixed. We have that F satisfies conditions (P1)-(P4). Moreover, if for every sequence {z n } ⊂ K such that lim n→∞ z n = ∞ implies (P6) lim inf n→∞ F( x, z n ) + λ ϕ(z n , z n ) -ϕ(z n , x) > 0, then F satisfies condition (P5).
Proof One can easily check that F satisfies (P1)-(P3) having in mind that K ⊂ int D, where D = dom ϕ. Now, we claim that F satisfies (P4). Indeed, we have

F(x, y) + F(y, x) = F(x, y) + F(y, x) +λ (ϕ(x, y) -ϕ(x, x) + ϕ(y, x) -ϕ(y, y)) ≤ 0,
where the inequality comes from the fact that F is monotone due to (P4), ϕ is skew-symmetric from (A1) and λ > 0. Thus, the first part of the assertion is proved. Next, we will show that F satisfies (P5) under the assumption (P6). Let {z n } ⊂ K be an arbitrary sequence such that lim n→∞ z n = ∞. From (9), we have

F(z n , x) = F(z n , x) + λ ϕ(z n , x) -ϕ(z n , z n ) .
Since F is monotone from (P4), the following inequality holds

F(z n , x) = F(z n , x) + λ ϕ(z n , x) -ϕ(z n , z n ) ≤ -F( x, z n ) + λ ϕ(z n , z n ) -ϕ(z n , x) . ( 10
)
From (P6), we have that there exists n 0 ∈ N such that

F(z n , x) + λ ϕ(z n , z n ) -ϕ( x, z n ) > 0,
for all n ≥ n 0 , which means from (10) that F(z n , x) < 0, for all n ≥ n 0 . This completes the proof.

Remark 1 One can easily check that if ϕ satisfies (A1), i.e., ϕ is skew-symmetric and x ∈ K , then the following condition (P6*) implies (P6), and hence, from Theorem 3 it implies (P5):

(P6 * ) lim inf n→∞ F( x, z n ) + λ ϕ( x, z n ) -ϕ( x, x) > 0.
Next, we provide a class of mapping ϕ in which condition (P6) holds, and hence, condition (P5) is verified. We consider the very important class of strongly skew-symmetric functions ϕ, i.e., it satisfies the following assumption:

(A4) ϕ(x, x) -ϕ(x, y) -ϕ(y, x) + ϕ(y, y) ≥ γ y -x 2 , for some γ > 0. Theorem 4 Assuming either (i) K is bounded, or (ii) ϕ is strongly skew-symmetric.
Then, F satisfies condition (P5).

Proof Using Theorem 3 it is enough to check that condition (P6) holds under (i) or (ii). If (i) is satisfied, i.e., if K is bounded, then one can easily check that condition (P6) holds since there is no {z n } ⊂ K such that ||z n || → ∞ as n → ∞. Hence, it is enough to prove that (ii) implies condition (P6). Take a sequence {z n } ⊂ K such that ||z n || → ∞ and x ∈ K fixed. From (A4), we have

F( x, z n )+λ(ϕ(z n , z n )-ϕ(z n , x)) ≥ F( x, z n )+λ(ϕ( x, z n )-ϕ( x, x))+γ z n -x 2 . (11) Since the mapping F(x, •) is convex for all x ∈ K , then for any u ∈ ∂ F( x, •)( x), we have F( x, z n ) ≥ F( x, x) + u, z n -x = u, z n -x , ( 12
)
taking into account that F( x, x) = 0. Again, since ϕ(x, •) is convex for all x ∈ K , for any v ∈ ∂ϕ( x, •)( x) and λ > 0, we obtain

λ(ϕ( x, z n ) -ϕ( x, x)) ≥ λ v, z n -x . ( 13
)
Adding inequalities ( 12) and ( 13), we have

F( x, z n ) + λ(ϕ( x, z n ) -ϕ( x, x)) ≥ u, z n -x + λ v, z n -x . ( 14
)
Using ( 14) in ( 11), we have

F( x, z n ) + λ(ϕ(z n , z n ) -ϕ(z n , x)) ≥ u, z n -x + λ v, z n -x + γ z n -x 2 , ≥ -w| z n -x + γ z n -x 2 (15)
where w = u +λv and applying the Cauchy-Schwartz inequality. Using the fact that ||z n || → ∞ in (15), then there exists n 0 ∈ N such that the right-hand side of ( 15) is positive for all n ≥ n 0 , and hence, we conclude (P6). Therefore, the desired result follows from Theorem 3.

Example 3 Let us consider ϕ(x, y) = T (x), y , where T : R m → R m is a strongly maximal monotone (single-valued) mapping with modulus τ > 0. One can prove that ϕ satisfies (A4) with γ = τ .

Example 4

The Tikhonov-type regularization analyzed in Oliveira et al. ( 2013) is given by ϕ(x, y) = g(x, y), where g : K ×K → R is a β-strongly monotone bifunction, i.e., g(x, y)+ g(y, x) ≤ -β||x -y|| 2 , for all x, y ∈ K . It generalizes the regularization considered in [START_REF] Iusem | On the proximal point method for equilibrium problems in Hilbert spaces[END_REF], [START_REF] Moudafi | Proximal point algorithm extended to equilibrium problems[END_REF] and [START_REF] Moudafi | Proximal and dynamical approaches to equilibrium problems[END_REF], where the regularization auxiliar bifunction is given by g(x, y) = x -x, y -x is strongly monotone with 0 < β ≤ 1.

In this case, one can easily check that ϕ satisfies (A4) with γ = β.

The next result establishes the existence and uniqueness of the solution of the GMEP (7) which can also be viewed as the regularized equilibrium problem E P( F, K ).

Theorem 5

The following assertions hold:

(i) If condition (P6) holds, then E P( F, K ) admits at least one solution; (ii) If ϕ is strictly skew-symmetric, then E P( F, K ) admits at most one solution.
Altogether, if condition (P6) holds and ϕ is strictly skew-symmetric, then E P( F, K ) has a unique solution.

Proof From Theorem 3, we have that F satisfies condition (P5). Thus, from Theorem 2, we obtain that E P( F, K ) has a solution. This proves the first assertion. To prove (ii), we assume that both x 1 and x 2 solve E P( F, K ). Hence,

0 ≤ F(x 1 , x 2 ) = F(x 1 , x 2 ) + λ(ϕ(x 1 , x 2 ) -ϕ(x 1 , x 1 )). ( 16
) 0 ≤ F(x 2 , x 1 ) = F(x 2 , x 1 ) + λ(ϕ(x 2 , x 1 ) -ϕ(x 2 , x 2 )). ( 17
)
Adding ( 16) with ( 17), we get

0 ≤ F(x 1 , x 2 ) + F(x 2 , x 1 ) -λ(ϕ(x 1 , x 1 ) -ϕ(x 2 , x 1 ) -ϕ(x 1 , x 2 ) + ϕ(x 2 , x 2 )).
Since λ > 0, F is monotone and ϕ is strictly skew-symmetric, then it follows from last inequality that x 1 = x 2 and the second assertion is proved. The last statement is a direct combination of (i) and (ii).

Remark 2 Note that item (ii) of the previous results holds if, in particular, ϕ is strongly skewsymmetric (condition (A4)). This is the case when ϕ is a Tikhonov-type regularization as given in [START_REF] Oliveira | A Tikhonov-type regularization for equilibrium problems in Hilbert spaces[END_REF]; see Example 4. Therefore, our last result generalizes (Oliveira et al. 2013, Theorem 3.2). This is an important generalization because there are strongly skew-symmetric functions which are not a bifunction (strongly monotone); see, for instance, Example 3.

Generalized regularization algorithm

Recall that we are assuming that F satisfies conditions (P1)-( P4) and ϕ verifies assumptions (A1)-(A3). In this section, we additionally assume that condition (P5) holds. As we prove in last section, there are some sufficient condition to guarantee condition (P5) is valid, for instance, if condition (P6) holds, or K is bounded, or ϕ is strongly skew-symmetric.

Next, we define and analyze the convergence of an inexact algorithm for solving E P(F, K ). Given two sequences of parameters {λ k } and {δ k } such that 0 ≤ λ k < λ, for some λ, and δ k ≥ 0. Let {x k } ⊂ K be a sequence such that x k is a solution of the inexact equilibrium problem

I E P(F k , K , δ k ), where F k : K × K → R is given by F k (x, y) = F(x, y) + λ k (ϕ(x, y) -ϕ(x, x)) . This means that x k ∈ K satisfies F(x k , y) + λ k ϕ(x k , y) -ϕ(x k , x k ) ≥ -δ k , ∀ y ∈ K . ( 18
)
In the sequel, we consider {x k } the sequence generated by ( 18). It is worth to mention that the well definition of {x k } follows from Theorem 5.

Theorem 6 Assume that lim →∞ λ k = 0 and lim →∞ δ k = 0.

(i) If the sequence {x k } has a cluster point, then it belongs to S(F, K ); (ii) Additionally, if ϕ satisfies condition (A4) and S(F, K ) is non-empty, then {x k } is bounded.

Proof Let x * be an accumulation point of {x k } and {x k j } be a subsequence converging to x * . From (18), we have

F k (x k , y) = F(x k , y) + λ k ϕ(x k , y) -ϕ(x k , x k ) ≥ -δ k , ∀ y ∈ K , ∀k ∈ N.
In particular the inequality above is valid for all k j . Thus, taking the limit with k j → ∞ and using that λ k → 0 and δ k → 0, we get F(x * , y) ≥ 0, for all y ∈ K . Now, let us prove the second assertion. Let x k be a solution of I E P(F k ; K , δ k ) and x * ∈ S(F, K ). Then,

F k (x k , x * ) = F(x k , x * ) + λ k (ϕ(x k , x * ) -ϕ(x k , x k )) + δ k ≥ 0
and, from (P4), we have that F(x k , x * ) ≤ 0, respectively. Combining last two inequalities, we have

λ k (ϕ(x k , x * ) -ϕ(x k , x k )) + δ k ≥ 0. ( 19
)
One has that λ k ≤ λ and δ k ≤ δ, for some positive real number λ, δ > 0, for all k ∈ N and hence, by ( 19), we have

ϕ(x k , x * ) -ϕ(x k , x k ) + δ λ ≥ 0. ( 20
)
Since ϕ is strongly skew-symmetric with modulus γ > 0, we obtain

ϕ(x * , x * ) -ϕ(x k , x * ) -ϕ(x * , x k ) + ϕ(x k , x k ) ≥ γ ||x k -x * || 2 . ( 21
)
Adding inequalities ( 20) and ( 21), we have

ϕ(x * , x * ) -ϕ(x * , x k ) ≥ γ ||x k -x * || 2 - δ λ . ( 22
)
By (A2), we have that ϕ(x * , •) is convex and this implies that

ϕ(x * , x k ) -ϕ(x * , x * ) ≥ v, x k -x * , ( 23
)
where v ∈ ∂ϕ(x * , x * ). Thus, adding inequalities ( 22) and ( 23), we obtain

γ ||x k -x * || 2 + v, x k -x * ≤ δ λ .
The assertion directly follows from last inequality.

Remark 3 In Oliveira et al. (2013, Theorem 3.3) is proved that if {x k } satisfying (18) with lim →∞ λ k = 0, δ k = 0, for all k ∈ N, and ϕ is a strongly monotone bifunction such that lim sup ||y||→∞ ϕ(x,y)

||x-y|| < +∞, then {x k } is bounded if, and only if, E P(F, K ) is non-empty. Our previous result extends this result and enables a broad class of regularization including the one proposed in [START_REF] Oliveira | A Tikhonov-type regularization for equilibrium problems in Hilbert spaces[END_REF] to solve equilibrium problems. Example 3 shows a class of regularization which fit with our set of assumption and does not verify the hypothesis in Oliveira et al. (2013, Theorem 3.3).

Numerical experiments

In this section, we illustrate our method for solving some academic problems considered in [START_REF] Santos | An inexact subgradient algorithm for equilibrium problems[END_REF] and Santos and Souza (2020) using different kind of regularizations. The algorithms have coded in MATLAB using the subroutine "quadprog" on a notebook 8 GB Intel Core i7 Dual-Core. For all tests, both inner and outer stop criteria is

x k, jx k, j+1 < , for some > 0. The subproblems (18) are solved applying Projected Subgradient Method (PSM) given by x k, j+1 = P C x k, jβ j γ j g j , with β j ≥ 0,

β j = +∞, β 2 j < +∞ with g j ∈ ∂ y F k (x k, j , x k, j
), γ j = max{1, g j }, where ∂ y F k stands to the subdifferential of F k with respect the variable y; see [START_REF] Santos | An inexact subgradient algorithm for equilibrium problems[END_REF].

It is worth to mention that regularization such as (3), ( 4) and ( 5) are well known and very popular for regularization methods in optimization. An important computational issue about these regularization is that when, for instance in equilibrium problems, the subproblem are solved with PSM which takes g j ∈ ∂ y F k (x k, j , x k, j ) one has that the regularization plays no role in the algorithm. This happens because using these regularizations we have

∂ y F k (x k, j , x k, j ) = ∂ y F(x k, j , x k, j ).
Next, we perform the proposed method comparing the performance of the regularization (3) with other regularizations which the previous mentioned issue does not happens, i.e., ∂ y F k (x k, j , x k, j ) = ∂ y F(x k, j , x k, j ). In the sequel, we consider the following regularization, for all k ∈ N: In Tables 1, 2 and 3, we report the number of iterates (k) until the method satisfies the stopping rule x k, jx k, j+1 < . It is worth to mention that for a general regularization it is not proved that if x k+1 = x k , then the method returns a solution. However, we use this classical stopping rule followed by the accuracy ϒ * = ϒ(x k ) in the last iterate x k showing how far from the solution the method stopped. The minimum, maximum and the median of the cpu time until the method stop is also reported. In Figs. 1 and2, we show the behaviour of method using three different regularizations for 10 randomly initial points for Example 5 and 6, respectively. We consider different choice for the sequence λ k setting constant for all k ∈ N. We take β k = 100/3k and γ k = max{3, g k } in the PSM.

I. ϕ(x, y) = λ k x -x k , y -x ; II. ϕ(x, y) = λ k T (x), y , where T (x) = ∇h(x) with h(x) = x 2 as in Example 2; III. ϕ(x, y) = λ k ( T (x), y + ψ(x) -ψ(y)), where T (x) = ∇h(x) with h(x) =
Example 5 Consider the 2-dimensional nonsmooth equilibrium problem defined by the bifunction F(x, y) = |y 1 | -|x 1 | + y 2 2x 2 2 and the constraint set given by K = x ∈ R 2 + ; x 1 + x 2 = 1 . One can check that F is monotone and the solution set is the single point

x * = ( 1 2 , 1 2 ).
Example 6 Consider the 2-dimensional variational inequality problem given by F(x) = (2x 1 + 8 3 x 2 -34, 2x 2 + 5 4 x 1 -24.25) and the constraint set defined by K = {x ∈ R 2 : 0 ≤ x i ≤ 10, i = 1, 2}. The solution set of this problem is a single point x * = (5, 9).

Example 7 Consider the equilibrium problem where the constraint set K is given by

K = x ∈ R 5 : 5 i=1 x i ≥ -1, -5 ≤ x i ≤ 5, i = 1, 2, . . . , 5
Fig. 1 The initial points, iterates of the sequence using three regularizations and the solution found Fig. 2 The initial points, iterates of the sequence using three regularizations and the solution found and the bifunction F is of the form F(x, y) = Px + Qy + q, yx , in which the matrices P, Q and the vector q are given by

P = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣
3.1 2 0 0 0 2 3.6 0 0 0 0 0 3.5 2 0 0 0 2 3.3 0 0 0 0 0 3

⎤ ⎥ ⎥ ⎥ ⎥ ⎦ ; Q = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣
1.6 1 0 0 0 1 1.6 0 0 0 0 0 1.5 1 0 0 0 1 1.5 0 0 0 0 0 2

⎤ ⎥ ⎥ ⎥ ⎥ ⎦ and q = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 1 -2 -1 2 -1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ .

Application to Becker's household behaviour theory

In this last section we give an application in behavioral sciences of our abstract regularization approach to solve equilibrium problems. We propose a dynamic version of the famous static Becker's "allocation of time problem" [START_REF] Becker | A theory of the allocation of time[END_REF], using a recent variational rationality approach of human dynamics [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](Soubeyran , 2019a, b), b).

Household behavior

Becker's allocation of time model. The static case.

Consider the Becker's household model and, for easy reading, let us adapt the notations to our framework. Let U (Z 1 , . . . , Z m ) be the utility function of a household over the quantities of final commodities (Z 1 , . . . , Z m ) ∈ R m + . This household produces at home these final goods. He uses the vector of inputs y j = (r j , t j ) to produce the quantity Z j = h j (y j ) of each final good j. Each r j is a bundle (quantity vector) of intermediate goods purchased at the vector of prices p j , and t j is a bundle of time use quantities. Becker defined t j as a vector also, to distinguish between, e.g., daytime from nighttime hours, or weekdays from weekends. Moreover he assigns a vector of wage rates w j to t j , thereby assuming that, e.g., the cost of a unit of time on weekends and weekdays would generally be different [START_REF] Chappori | A theory of the allocation of time[END_REF]. Rather than have one budget constraint for goods and another one for time, Becker supposes that the household can trade-off time for money, and so only faces the single budget constraint m i=1 ( p j r j + w j t j ) ≤ S, where S > 0 represents a full income. In this setting the household chooses a vector of resources y to maximize an 'earnings' function g(y) = U [h 1 (y 1 ), . . . , h m (y m )], where y = (y 1 , . . . , y m ) is a vector of resources (inputs and times) subject to his household's single budget constraint and to the production functions for each commodity. Then he solves the static resource allocation problem sup {g(y), y ∈ K } , where K = y ∈ X = R m + , q, y ≤ S , q = (q 1 , . . . , q m ) and q j = ( p j , w j ) > 0, j ∈ J = {1, 2, .., m} .

In this application, we are interested in a dynamic formulation of this famous model, using the recent variational rationality approach of human dynamics [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](Soubeyran , 2019a, b), b).

The variational rationality approach: an experience dependent formulation

Stay and change human dynamics

The (VR) variational rationality approach [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](Soubeyran , 2019a, b) , b) modelizes and unifies a lot of different models of stay and change human dynamics which appeared in behavioral sciences (economics, management sciences, psychology, sociology, political sciences, decision theory, game theory, artificial intelligence, etc.). Stays refer to static phases where, each period, an agent performs again and again the same bundle of activities, exploiting existing capabilities. This leads to the formation of temporary habits, routines, rules and norms, etc.... Changes represent transformation phases (including need's recognition, exploration, evaluation, learning and training change activities) where an agent stops doing some activities, continues doing others and starts doing new activities. These changes help to form and break habits and routines, etc.... This dynamical approach considers entities (an agent, an organization or several interacting agents), which are, at the beginning of the story, in an undesirable initial position. Being unable to reach immediately (one period) their final desired positions (desires), they try, each period, to improve their current position, to progressively approach and reach their desired ends (desires). The goal of this approach is to examine the transition problem: how such entities will accept to follow an acceptable and feasible transition before being able to reach their final desires; how they accept to overcome a lot of intermediate obstacles, difficulties and resistances to change; how a high initial motivation to change, a succession of not too high intermediate sacrifices and high enough intermediate satisfactions can sustain their motivation to change during the transition in order to persevere until reaching the final desired position. This (VR) approach admits a lot of variants, based on the same short list of general principles and concepts. The five main concepts refer to moves (changes and stays), worthwhile moves, worthwhile transitions made of a succession of worthwhile moves, variational traps, worthwhile to approach and reach but not worthwhile to leave, and desires not advantageous to leave.

Feasible activities

Bundles of activities. The variational rationality approach considers bundles of activities x ∈ X . They refer to different lists of elementary activities (tasks) which can be done within a period. In the present application relative to an allocation of activities problem [START_REF] Becker | A theory of the allocation of time[END_REF], x = (r , t) = (x 1 , . . . , x m ) ∈ X is a bundle of elementary activities x j = (r j , t j ), j ∈ J = {1, . . . , m} , where t = (t 1 , . . . , t m ) and r = (r 1 , . . . , r m ). In this context (r j , t j ) is the task x j = "spending the specific time t j ∈ R + to use the bundle of inputs r j ∈ j in order to produce the quantity Z j = h j (x j ) ∈ R + of the home final good j ∈ J = {1, . . . , m} ". That is, the home production function is Z = h(x).

Set of feasible bundles of activities The Becker's resource constraint. m j=1 ( p j r j + w j t j ) = α, x ≤ S defines a bounded convex polytope

K = {x ∈ X , α, x ≤ S} , where α = (α 1 , . . . , α j ., . . . , α m ) ∈ R m
++ with α j = ( p j , w j ) > 0.

Moves

Definition Consider two successive periods; the current period k + 1 and the previous period k. In this simplified version of the variational rationality approach a current move m = (x, y) goes from x = x k = having done the bundle of activities x in the previous period k to y = x k+1 = doing the bundle of activities y in the current period k + 1. This move is a change if y = x and a stay if y = x. In general, a move is m = (x, ω, y) where ω ∈ modelizes a transition stage, that is a preparation stage where an agent chooses and becomes able to do the bundle of activities y within the current period.

Extensive and intensive moves Soubeyran (2019a, b) make the important distinction between extensive and intensive moves. In this paper, this distinction will play a major role for the definition of resistance to move (see later). When doing a move, an agent, each period, either stops, continues and starts doing some activities (extensive move), or he performs, each period, the same list of activities, but more or less of each (intensive move). In the first case we move, i) for some j ∈ J from x j > 0 to y j = 0, for others j from x j = 0 to y j > 0 and for the last j we have x j = y j > 0. In the second case all x j > 0 and y j > 0 remain strictly positive. When we consider moves, we start to consider a dynamic version of Becker's allocation of time problem. Then, we depart from Becker's model which is static. To match with the mathematical formulation of equilibrium problems our dynamic version of Becker's model will only treat the case of intensive moves.

Worthwhile balances and worthwhile moves

An experience dependent variational rationality structure. The (VR) approach starts with a general definition of a worthwhile change: a change is worthwhile if motivation to change rather than to stay is "high enough" with respect to resistance to change rather than to stay. This definition allows a lot of variants, given the different aspects of motivation to move (more than one hundred theories/aspects of motivations exist in Psychology), the different aspects of resistance to move and different formulations of the sentence "being high enough". To cover all these different aspects [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](Soubeyran , 2019a, b, b) defined a general variational rationality (VR) structure which modelizes successively activities, moves, utility and costs of moving, advantages and inconveniences to move, motivation and resistance to move, worthwhile balances and finally worthwhile moves. Two special cases emerge where utility and costs of moving do not depend of the whole move m = (x, ω, y), but depend only of the end y of this move, or of the beginning x and the end y of this move, but not of the transition ω. First, the simplest case where utility g(y) ∈ R and costs C(y) ∈ R + depend only of doing a bundle of activities y. Second, the intermediate case where utility g s (y) and costs C s (y) depend of the internal state and the external environment s of the agent, including, among other things, different reference points (for example the status quo x and aspiration levels x * ), experience, etc. To match with the mathematical part of this paper, this application considers a simplified intermediate case (which is still quite complicated), where (i) a move m = (x, y) does not specify the preparation phase, that is, the transition ω which goes from x = having done the bundle of activities x in the previous period to being able to do the bundle of activities y in the current period ; (ii) the internal state of the agent and the state of his external environment is limited to his experience s = e ; (iii) experience e depends only of what the agent have done in the previous period, that is, e = x. This is the Markov case; (iv) motivation and resistance to move are identified with advantages and inconveniences to move. This defines a linear VR structure; (v) costs of moving which are the sum of two costs, costs of being able to do and costs to do a bundle of activities y, refer only to costs to do. This is the case when changes are intensive (see above).

For the justifications of all these simplifications see Soubeyran (2019a, b).

Formulation Then, within this simplified VR structure:

1. g x (y) ∈ R represents the utility of doing a bundle of activities y in the current period, after having done a bundle of activities x in the previous period; 2. C x (y) ∈ R is the cost of being able to do a bundle of activities y in the current period, after having done a bundle of activities x in the previous period; 3. C x (y) ∈ R is the cost of doing a bundle of activities y in the current period, after having done a bundle of activities x in the previous period;

4. C(x, y) = C x (y) + C x (y) is the cost of moving from x to y; 5. A x (x, y) = g x (y)g x (x) defines advantages to move (change rather than stay); 6. L x (x, y) = -A x (x, y) represents losses to move; 7. I x (x, y) = C(x, y) -C(x, x) defines inconveniences to move (change rather than stay); 8. M x (x, y) = U x A x (x, y) ∈ R + is the utility U x A x of advantages to move A x = A x (x, y) ≥ 0. It defines motivation to change rather than to stay; 9. R x (x, y) = D x I x (x, y) ∈ R + is the dis-utility D x I x of inconveniences to move I x = I x (x, y) ≥ 0. It defines resistance to change rather than to stay; 10. B x,ξ (x, y) = M x (x, y) -ξ R x (x, y), ξ > 0 represents a worthwhile balance between motivation and resistance to move; 11. Then, a move m = (x, y) is worthwhile if B x,ξ (x, y) ≥ 0.

We suppose that:

(a) costs C x (y) of being able to move from x to y are zero for all intensive moves m = (x,y).

That is, C x (y) = 0 for all y ∈ K such that the move m = (x,y) is intensive; (b) the values of advantages and inconveniences to move are linear. That is, U x A x (x, y) = A x (x, y) and R x (x, y) = D x I x (x, y) = I x (x, y) for all x, y ∈ X .

Then, worthwhile balances become

B x,ξ (x, y) = A x (x, y) -ξ I x (x, y) = -L x (x, y) + ξ I x (x, y) .

Ends: desires and traps

The variational rationality approach [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](Soubeyran , 2019a, b defines successively the following aspects of an end position. The bundle of activities x * is:

• a desire if it is not advantageous to move away from it. That is, if

A x * (x * , y) ≤ 0, ∀y ∈ K .
The bundle of activities x * is,

• a weak stationary trap if it is not worthwhile to move away from it. That is, if B x * ,ξ (x * , y) ≤ 0, for all y ∈ K . • a strong stationary trap if it is strictly not worthwhile to move away from it. That is, if B x * ,ξ (x * , y) < 0, for all y ∈ K , y = x * . • an almost (inexact) stationary trap if it is not enough worthwhile to move away from it.

That is, if B x * ,ξ (x * , y) ≤ δ, for all y ∈ K and some δ > 0.

The VR approach defines also variational traps which are worthwhile to reach from an initial position, but not worthwhile to leave. In the present paper, we will not consider these variational traps because the generalized regularization algorithm of this paper does not tell something about them.

Transitions as successions

After having defined worthwhile moves, desires and traps, the VR approach considers different human dynamics. Here are some.

Definition A stay and change human dynamic is a succession of moves m k = (x k , x k+1 ), k = 0, 1, . . . , where k + 1 is the current period and k is the previous period. This transition can be a succession of worthwhile moves, or a succession of stationary or almost stationary traps, or a succession of variational traps, etc. To fit with the mathematical part of the paper we will consider a succession of almost stationary traps x 0 , . . . , x k , x k+1 , . . . such that, for each k, B x k ,ξ k+1 (x k , y) ≤ δ k with δ k > 0 and for all y ∈ K . This means that

B x k ,ξ k+1 (x k , y) = A x k (x k , y) -ξ k+1 I x k (x k , y) = -L x k (x k , y) + ξ k+1 I x k (x k , y) ≤ δ k , that is, L x k (x k , y) + ξ k+1 C(x k , y) -C(x k , x k ) ≥ -δ k , ∀y ∈ K .
In this context, each period k + 1, an agent moves from a previous trap to a new one. To be able to do that, he must break the old trap. A way to do that is to choose a new worthwhile ratio ξ k+1 = ξ k .

Definition An aspiration point of a human dynamic x k is a bundle of activities x * ∈ X such that it exists k

0 ∈ N such that A x k (x k , x * ) ≥ 0, or B x k ,ξ k+1 (x k , x * ) ≥ 0, for all k ≥ k 0 .
This means that, after a succession of moves, an agent wants or finds worthwhile to approach x * .

Dynamic Markov allocation problems with resistance to move

We are now in a good position to show how, very surprisingly, our abstract regularization method can modelize a dynamic formulation of the famous Becker's household's resource allocation problem.

A static experience dependent resource allocation model

We start by giving a generalized (but still static) version of the static Becker's model (1965) given at the beginning of this section. We suppose that, (a) the household utility function g x (y) is experience dependent. That is, the utility of consuming the bundle of final goods y in the current period depends of how much of all these final goods the consumer consumed in the previous period, i.e., x. For example a linear utility function with experience dependent weights is g x (y) = α(x), y , where

α(x) = (α 1 (x), . . . , α m (x)) ∈ R m + . (b)
(y) = β(x) y 2 ,β(x) ∈ R + or C x (y) = β(x), y , where β(x) = (β 1 (x),...,β l (x)) ∈ R l + .
In this setting the static maximization program of the household is sup g x (y) -ξC x (y), y ∈ K , ξ>0.

Equilibrium and regularized equilibrium problems as desires and traps problems

Let us show how, in the context of the Becker's time allocation problem [START_REF] Becker | A theory of the allocation of time[END_REF], the variational rationality approach helps to identify equilibria to desires and generalized regularized equilibria to stationary traps.

Losses of moving They are the opposite of advantages to move, i.e., F(x, y) = L(x, y) = -A x (x, y) = g x (x)g x (y), while advantages to move are A x (x, y) = g x (y)g x (x). Then, when moving from the status quo position x k to y in the current period k + 1, losses of moving are

F k (x k , y) = -A x k (x k , y) = g x k (x) -g x k (y).
Hypothesis (P2) and (P3) suppose that the experience dependent utility function ψ(•, •) :

(x, y) ∈ X -→ ψ(x, y) = g x (y) ∈ R is, respectively
• lower semicontinuous with respect to the first variable y, for all y ∈ K ;

• concave and upper semicontinuous in the second variable y, for all x ∈ K . Hypothesis (P1) means that advantages to stay are zero, that is,

A x (x, x) = g x (x) -g x (x) = 0.
Hypothesis (P4) supposes that if advantages to move from x to y are positive, then, there is no advantages to move from y to x.

Hypothesis (P5) supposes that any dynamic which goes to infinity have an aspiration point. This is a position that, after some time, an agent wants or finds worthwhile to approach. Hence, the merit of such unbounded dynamics is to do not wander.

Generalized regularization terms

They represent home production costs C x (y) = C(x, y) = ϕ(x, y). Then, the difference ϕ(x, y) -ϕ(x, x) refers to inconveniences to move. That is, ϕ(x, y) -ϕ(x, x) = I x (x, y) = C(x, y) -C(x, x).

Hypothesis (A1) supposes that, if inconveniences to move from x to y are positive, then, inconveniences to move from y to x are negative, that is, I x (x, y) + I y (y, x) ≤ 0, for all x, y ∈ K . This is true if C x (y) -C x (x) + C y (x) -C y (y) ≤ 0, for all x, y ∈ K .

Hypothesis (A2) and (A3) suppose that experience dependent costs of production ϕ(•, •) : (x, y) ∈ X -→ ϕ(x, y) = C x (y) ∈ R are convex in the second argument y (the usual case of decreasing returns) and are continuous (a natural hypothesis).

Hypothesis (A4) means that I x (x, y)+ I y (y, x) ≤ -γ yx 2 , γ > 0, for all x, y ∈ K . That is, if inconveniences to move from x to y are positive, then, inconveniences to move from y to x are strongly negative.

Example 1 considers strictly convex costs ϕ(x, y) = C x (y) = y 2 while example 2 considers the linear case ϕ(x, y) = C x (y) = T (x), y with experience dependent weights T (x) = β(x). These costs satisfy hypothesis (A1), (A2) and (A3). A regularized loss F λ (x k , y) = f (x k , y) + λ k ϕ(x k , y) -ϕ(x k , x k ) refers to the worthwhile balance B x k ,λ k+1 (x k , y). Equilibria and regularized equilibria as desires and traps At this stage of the presentation, it is evident that equilibria represent desires, while regularized equilibria refer to stationary traps.

Regularized losses

The dynamic household problem: results

The mathematical part of the paper gives three different kinds of results. In term of the variational rationality approach, Theorems 3 and 4 give sufficient conditions for the existence of aspiration points, Theorem 5 gives sufficient conditions for the existence of desires, and the proof of Theorem 6 helps to show when a succession of stationary traps is an improving worthwhile to move dynamic, made of a succession of worthwhile moves going from a stationary trap to the next one. A striking result being that such worthwhile successions of stationary traps modelize the famous "Unfreeze, change, refreeze" Lewin's behavioral change model Lewin (1947aLewin ( , b, 1951) ) which is at the origin of a huge literature on "change management" and behavioral interventions in management sciences and health theory (more than 80 different models and variants). More precisely, here are our results in the context of behavioral sciences.

1. Theorems 3 and 4 give sufficient conditions which guarantee, for each unbounded succession of moves, the existence of aspirations points "worthwhile to approach" after doing a sufficient amount of moves. These conditions require that worthwhile balances satisfy conditions (P1)-( P4) and (P5). Condition (P5) is satisfied if, either condition (P6) works, or K is bounded or costs to do are strongly skew-symmetric. K is bounded is a weak hypothesis. It means that, each period, the household have limited resources. 2. A behavioral interpretation of (P6 * ) is the following. This hypothesis means that, given

x, for any unbounded human dynamic {z n }, there exists a period n 0 such that, after this period, it is not worthwhile to move away from x to each successive {z n } . This means that x is a kind of local stationary trap for this dynamic. 3. Theorem 5 shows the existence and unicity of traps if advantages to move satisfy assumptions (P1)-(P4), costs to do satisfy assumptions (A1)-(A4) and condition (P6) holds. 4. Theorem 6 shows the existence of desires as accumulation points of a succession of almost worthwhile moves if condition (P5) is satisfied.

Conclusion

We have proposed a generalized regularization method for solving equilibrium problems. At each subproblem, the proposed method solves a generalized mixed equilibrium problem. Under some mild assumptions we have proved the well-definition and the convergence of the method in the sense that every accumulation point of the sequence generated by the method is a solution of the equilibrium problem even if the subproblem are solved inexactly. The proposed method retrieves some existing ones in the literature on equilibrium problems enlarging the application of this kind of regularization method. An original application to Becker's household behavior theory has given using the variational rationality approach of human dynamics as well as some numerical experiments.

  x 2 and ψ(t) = log t as inOliveira et al. (2013, Remark 3.2).

  They are F λ (x, y) = F(x, y) + λ [ϕ(x, y) -ϕ(x, x)] . They represent the opposite of worthwhile balances B x,λ (x, y) = [-F(x, y)] -λI x (x, y) = A x (x, y) -λI x (x, y).

Table 1

 1 Example 5: 10 times with random initial points in [-10, 10] × [-10 × 10], λ k = 1 and = 10 -4

	Regularization Iterations (k) Min CPU time (s) Max CPU time (s) Median CPU time (s) Solution found
	I	4	0.2432	0.5086	0.3411	(0.5, 0.5)
	II	3	0.0806	0.0910	0.0836	(0.5, 0.5)
	III	2	0.0889	0.1052	0.0977	(0.5, 0.5)

Table 2

 2 Example 6: 10 times with random initial points in [-10, 10]×[-10 × 10], λ k = 0.001 and = 10 -3

	Regularization Iterations (k) Min CPU time (s) Max CPU time (s) Median cpu time (s) Solution found
	I	14.2	0.4337	4.3946	2.7958	(5.0042, 8.9971)
	II	14	3.5532	3.9570	3.6818	(5.0063, 8.9954)
	III	15	2.7884	2.9132	2.8244	(5.0077, 8.9941)
	Table 3 Example 7: the initial point x 0 = (1, 3, 1, 1, 2) and = 10 -3 as in Santos and Scheimberg (2011)
	and λ k = 0.001
	Regularization		Iterations (k)		CPU time (s)	ϒ(x k )
	I		2		0.4015	4.6e-04
	II		2		0.4103	9.1e-04
	III		2		0.4026	1.1e-03

  the household chain of production does not produce directly final goods from market goods and labor. In a first upstream stage market goods and labor allow to produce at home intermediate goods. In a second downstream stage intermediate goods, ingredients and labor allow to produce at home final goods. The production of intermediate home goods adds home costs of production C x (y) ∈ R + which are experience dependent.

The set of resource constraints K includes now different kind of times spend to produce intermediate goods. Examples of experience dependent production costs are C x


Ontological and spatial hybridities of Man-nature relationship: An application to walker in the forests of the Montagne

Keywords: Relation homme-nature, spatialité, ontologies, espaces naturels, expérience Human-nature relationship, spatiality, ontologies, natural spaces, experience

Hybridités ontologiques et spatiales de la relation Homme-nature : une application aux marcheurs en forêt du Parc naturel régional de la Montagne de Reims Résumé Alors que les espaces naturels comme la forêt constituent la toile de fond de nombreuses pratiques et expériences de consommation, peu de recherches sont conduites sur la manière dont ces espaces sont perçus et appréhendés par leurs usagers. Notre recherche propose une exploration des représentations de la forêt du parc naturel régional de la Montagne de Reims croisant les ontologies de Philippe Descola (2005) et la géographie critique de l'espace d'Henri Lefebvre (1974). Nous déclinons la manière dont la forêt, comme espace « conçu » et espace « perçu », entre en résonance avec un régime naturaliste dominant, laissant place à des infiltrations analogiques secondaires. Cette contribution enrichit la compréhension des relations homme-nature, des attentes en matière de naturalité ou d'aménagement et de contrôle de l'espace.

Alors que l'approche des espaces urbanisés -destinations touristiques, centres commerciaux, musées, villes intelligentes, etc. -a fait l'objet de nombreux travaux, peu est dit sur les espaces naturels et la manière dont les individus les appréhendent. Quelques travaux ont été menés sur la temporalité de l'expérience vécue dans les parcs urbains [START_REF] Mceachern | Producing and Consuming Public Space: A 'Rhythmanalysis' of the Urban Park[END_REF] ou sur la suspension du temps permise par des épisodes de retour à la nature [START_REF] Belk | The mountain man myth: a contemporary consuming fantasy[END_REF]. D'autres mettent en avant le caractère extraordinaire des activités extrêmes pratiquées dans la nature [START_REF] Arnould | River magic: extraordinary experience and the extended service encounter[END_REF]Celsi, Rose et Leigh, 1993), et la dimension existentielle du corps-à-corps avec les éléments lorsque la vie moderne coupe les individus de toute sensation d'eux-mêmes [START_REF] Scott | Selling Pain to the Saturated Self[END_REF]. D'autres enfin [START_REF] Canniford | Purifying practices: how consumers assemble romantic experiences of nature[END_REF] mettent en lumière le caractère construit -ou hybride -des espaces « naturels », perçus comme tels grâce à un travail de purification des consommateurs [START_REF] Latour | Nous n'avons jamais été modernes[END_REF]. Bien que ces perspectives offrent chacune un ensemble d'enseignements utiles sur des formes de consommation individuelle et collective de la nature, aucune n'en explore l'ontologie sousjacente, implicitement fondée sur une vision classique de l'homme, maître et possesseur d'une nature qui le domine et qu'il cherche à maîtriser. Notre approche propose d'éclairer la relation Homme-Nature à partir du cadre ontologique de Philippe [START_REF] Descola | Par-delà nature et culture[END_REF] et de la géographique critique d'Henri [START_REF] Lefebvre | La production de l'espace[END_REF]. Alors que l'approche par les ontologies s'intéresse aux modes d'identification et de relation entre humains et non-humains, la conception politique de l'espace articule les principes selon lesquels celui-ci est conçu et les modalités par lesquelles il est approprié. Le croisement fructueux de ces deux cadres permet de mettre en lumière la manière dont les usagers d'un massif forestier perçoivent et appréhendent cet espace. Ce cadre est mis à l'épreuve au travers d'une étude qualitative menée auprès de 35 usagers (promeneurs, randonneurs, VTTistes, chasseurs, etc.) du parc naturel régional de la Montagne de Reims. Les résultats mettent au jour deux types d'ontologies, naturalistes et analogiques, caractérisant à la fois l'espace « conçu » et l'espace perçu [START_REF] Lefebvre | La production de l'espace[END_REF]. Alors que la première traduit le côté domestiqué, contrôlé et fonctionnel de la forêt, la seconde traduit son insertion dans un réseau de correspondances qui lient l'individu au monde et cet espace à d'autres espaces auxquels il est relié.

CONCEPTIONS DE LA FORÊT

La fréquentation des forêts constitue un objet de recherche multidisciplinaire et hybride, à la frontière des loisirs de plein air, de l'écotourisme et des enjeux de santé des populations [START_REF] Bell | Introduction[END_REF]. Sur le plan marketing, la forêt, objet « naturel » par excellence, fait l'objet de peu de réflexions et d'investigation en comparaison des sites urbains et commerciaux, alors même que le développement des espaces forestiers constitue un enjeu de développement non négligeable pour les territoires qu'ils participent à revitaliser [START_REF] Pröbstl | Challenges for outdoor recreation and nature based tourism[END_REF]. Dans les approches de l'expérience en Consumer Culture Theory, la forêt, de même que d'autres espaces naturels, ne sont pas le centre de l'attention. Cette dernière, en effet, porte sur le caractère extraordinaire [START_REF] Arnould | River magic: extraordinary experience and the extended service encounter[END_REF], authentique [START_REF] Belk | The mountain man myth: a contemporary consuming fantasy[END_REF] ou extrême [START_REF] Celsi | An Exploration of High-Risk Consumption through Skydiving[END_REF][START_REF] Scott | Selling Pain to the Saturated Self[END_REF] des activités qui s'y déroulent. Les recherches sur le tourisme naturel font quant à elles ressortir la complexité de gestion des espaces forestiers du fait de la multiplicité des parties prenantes impliquées dans leur fréquentation, leur exploitation ou leur protection [START_REF] Virapongse | A socialecological systems approach for environmental management[END_REF]. Ces espaces posent en effet des questions de conflits d'usages dus à leur multifonctionnalité [START_REF] Pröbstl | Management of Recreation and Nature Based Tourism in European Forests[END_REF], entre exploitation du bois et activités de loisirs [START_REF] Harshaw | Using the recreation opportunity spectrum to evaluate the temporal impacts of timber harvesting on outdoor recreation settings[END_REF], accès aux publics et protection de la biodiversité [START_REF] Pickering | Impacts of recreation and tourism on plant biodiversity and vegetation in protected areas in Australia[END_REF], et coexistence de populations aux motivations variées, sinon divergentes [START_REF] Bell | Introduction[END_REF][START_REF] Kearsley | Balancing Tourism and Wilderness Qualities in New Zealand's Native Forests[END_REF]. Pour ce qui concerne les publics, Fredman, Wall-Reinius et Grundén (2012) décrivent les forêts comme des environnements de service (servicescapes), dont le caractère « naturel » et « accessible » constitue les deux principales dimensions perçues. Alors que la naturalité fait référence à des environnements non transformés par l'homme, la seconde évoque le travail d'aménagement facilitant l'appropriation et la consommation de ces espaces. Cette tension entre naturalité d'une part et accessibilité d'autre part fait écho à l'approche spatiale de [START_REF] Lefebvre | La production de l'espace[END_REF] qui envisage les espaces comme « conçus » (pensés, agencés, équipés) et « perçus » (appropriés, utilisés, parcourus). L'approche des ontologies de Descola (2005) permet quant à elle d'examiner quelle forme de continuité les usagers de la forêt envisagent entre eux et les non-humains qui la peuplent, à la fois sur le plan de la physicalité et de l'intériorité. Peu abordée dans la littérature, cette approche permet d'approfondir ce recouvre la naturalité et quelle position l'homme y occupe, en surplomb ou en égalité. [START_REF] Descola | Par-delà nature et culture[END_REF] montre en effet que quatre ontologies, croisant les continuités/discontinuités des intériorités/physicalités entre les existants, sont repérables dans le monde. Dans le naturalisme, les humains sont les seuls à posséder un esprit, une conscience (discontinuité de l'intériorité), tout en étant liés aux non-humains par des lois physiques, chimiques et biologiques (continuité de la physicalité). Par-delà la dualité homme/nature qui est propre au monde moderne occidental, l'animisme apparaît comme l'ontologie symétrique du naturalisme. La plupart des existants (humains, animaux, plantes et certains objets) sont réputés posséder une intériorité similaire à celle des hommes (continuité de l'intériorité), mais ils se distinguent par une discontinuité de leur physicalité qui leur confère des caractères et une manière spécifiques de voir le monde. Dans l'analogisme, tous les éléments se différencient les uns des autres sur le plan ontologique (discontinuité de l'intériorité et de la physicalité), ce qui conduit, pour construire une intelligibilité du monde, à rechercher des correspondances stables entre ces éléments. Dans le totémisme, des ensembles particuliers d'humains partagent des propriétés physiques et morales avec des non-humains (continuité de l'intériorité et de la physicalité), répartis au sein d'une même classe d'existants dont un type (animal, plante) constitue le représentant (le totem). Comme le montre [START_REF] Descola | Par-delà nature et culture[END_REF], les ontologies sont reliées à des aires culturelles spécifiques, et à des modes de relation qui peuvent être univoques entre des termes non équivalents -production, protection et transmission -ou potentiellement réversibles entre des termes équivalents -échange, prédation et don.

CONTEXTE ET METHODE

L'étude a été menée auprès d'un échantillon de convenance de 35 usagers du parc naturel régional de la Montagne de Reims (PNRMR) dont les caractéristiques sont présentées en annexe 1. Il s'agit d'un échantillon de convenance dont la distribution montre une proportion plus forte d'hommes que de femmes, plutôt jeunes et dont la moitié se situe en-dessous du salaire médian (1 798€ mensuel). D'autres critères de variance ont été introduits comme la composition familiale, le lieu et le type d'habitation, et l'ancienneté de fréquentation du massif forestier, sur lesquels la répartition se révèle très équilibrée. Les répondants ont été interrogés à leur domicile et tous les entretiens ont été retranscrits. Ils portent sur des éléments factuels de fréquentation -avec qui ? Quand ? Avec quelle fréquence ? Par quels moyens d'accès ? Dans quel(s) but(s) ? -avant d'aborder les représentations et perceptions des forêts du PNRMR. Des comparaisons ont été fréquemment introduites par les répondants qui n'étaient pas natifs de la région, mais habitués à fréquenter d'autres forêts, comme celles des Ardennes, de l'Argonne, de l'Aisne, mais aussi d'autres régions de France et d'autres pays (Vénézuela). Les entretiens ont été analysés et codés manuellement de manière inductive, à la recherche de noyaux de sens significatifs [START_REF] Glaser | The Discovery of Grounded Theory[END_REF]. Les catégories émergentes ont été confrontées aux cadres théoriques susceptibles de les éclairer. Les aller-retour entre théories et terrain ont permis de faire ressortir des thèmes qui résonnent à la fois avec la conception/production de l'espace de [START_REF] Lefebvre | La production de l'espace[END_REF] et avec les modes d'identification et de relation propres aux ontologies de Descola (2005).

RESULTATS : LA FORÊT, UN HYBRIDE SPATIAL ET ONTOLOGIQUE

Espaces produits, régulés, domestiqués

Les discours des répondants témoignent de leur perception que la forêt est un espace produit (ou « conçu » au sens de [START_REF] Lefebvre | La production de l'espace[END_REF], dont les finalités sont à la fois d'extraire des ressources -« Des fois on est surpris, on traverse des bois, des forêts, quand on voit tout ce qui est coupé on se demande s'il y a un contrôle de fait ou pas » (Jules, 71 ans) -mais aussi d'offrir aux promeneurs des lieux de déambulation sécurisés -« Quand vous avez des enfants ou quand vous ne vous ne connaissez pas et que vous ne maîtrisez pas obligatoirement le déplacement dans la forêt, c'est est un lieu complètement balisé qui permet de la pratique pour le plus grand nombre » (Fabrice, 59 ans). Les aménagements cités (parkings, tables de pique-nique, balisages, poubelles, etc.) illustrent les nombreuses adaptations des espaces naturels à la fréquentation des publics, mais aussi la canalisation de ces derniers auxquels certains accès sont interdits : « Aujourd'hui, on voit bien que la forêt, il y a des chemins qui sont autorisés et puis 90% des chemins deviennent interdits » (Fabrice, 59 ans). L'exemple des Faux de Verzy en particulier -des hêtres tortillards très connus dans la région dont Verzy est la principale réserve mondialerevient fréquemment illustrer les limites posées à la fréquentation des marcheurs : « Avant, il y a un siècle, tu pouvais y aller, il y avait une vie formidable là-bas, les gens ils se retrouvaient, t'avais des guinguettes, ce n'était pas fermé. Maintenant, c'est fermé parce que sinon les Faux, ils prennent trop cher et du coup tu es obligé de fermer » (Manon, 38 ans). La protection de certains espaces, forme typique de l'ontologie naturaliste, consiste alors à réguler les flux et à soustraire la nature à l'égard de laquelle s'instaure une relation dissymétrique du fait « d'une domination non réversible de celui qui l'exerce sur celui qui en bénéficie » (Descola, 2005, p.555) : « Je comprends, hein, qu'on ait envie de protéger ces arbres et éviter qu'ils se fassent piétinés par les visiteurs… je pense que c'est fait exprès pour éviter que les gens marchent dessus et c'est vraiment compréhensible » (Sabrina, 29 ans) L'ontologie naturaliste est également très présente lorsque les répondants expriment un besoin de se retrouver dans une nature domestiquée et débarrassée de certaines espèces : « Il y a une chose que je n'aime pas quand même, c'est les mouches, et quand ça passe très proche de ton oreille. Bon c'est normal, c'est la forêt hein ? Mais je n'aime pas trop » (Victor, 24 ans) ; « Maintenant, les tiques, je les aime pas trop, donc dès que je vois que ce n'est pas entretenu, je n'y vais pas. Il faut entretenir la nature » (Léna, 23 ans). Si la rencontre avec certains animaux sauvages (cerfs, chevreuils) est souvent espérée, la peur du loup ou du sanglier demeure tenace : « Je pense que j'aurais un peu peur de voir un loup en forêt. Après on dit toujours que le loup a peur de l'homme. Mais imaginons si le loup a faim, s'il est en meute, je pense que je ne survivrai pas, parce qu'il va m'attaquer, forcément… Se faire charger par un sanglier, hein, il y a un peu de risques, oui. Ce n'est pas une expérience que j'aimerais tenter » (Léna, 23 ans). De la même manière, la domestication de la nature sauvage se traduit par une préférence pour des espaces aménagés qui tendent à calquer la forêt sur le modèle du parc : « C'est une forêt normale, on va dire. Et je n'ai pas trop aimé parce qu'il avait aussi beaucoup de gadoue, c'est à dire que ce n'est pas très propre comme dans l'autre où il y a des chemins. Là c'était plus sauvage, c'était vraiment très sauvage et peut-être que c'est ça que je n'ai pas trop aimé » (Victor, 24 ans). La forêt comme espace conçu fait toutefois l'objet de formes d'appropriation qui font ressortir la dimension spatiale et fonctionnelle de ce qu'elle peut offrir, en-deçà de la naturalité recherchée (Redman et al., 2012).

Espaces pratiqués, appropriés, perçus

La forêt est pour de nombreux répondants un réservoir d'activités aussi diverses que la randonnée, la balade, le VTT, la marche nordique, le pique-nique, la cueillette (champignons, muguet), la photographie, le géocaching, la chasse ou tout simplement « pour fêter un anniversaire, fêter un truc » (Sabrina, 29 ans). C'est un réservoir de ressources -faune et flore -qui fait ressortir les relations au monde dominantes dans l'ontologie naturaliste c'est-à-dire l'exploitation -« Je sais que tu peux acheter des forêts, même toi, à titre privé, qui sont soit exploitées pour leur bois, pour les meubles, pour de la papeterie, soit tu peux avoir des droits pour les chasseurs, c'est à dire qu'ils viennent chasser sur ton terrain qui te payent donc ça. C'est de la monétisation dans le vent » (Frédéric, 36 ans)avec comme contrepartie la protection -« Pour protéger ce qui va être animaux, bah de temps en temps on va récupérer quelques animaux spéciaux d'une espèce et puis on va les reproduire entre eux » (Juliette, 22 ans). En même temps, cette appropriation révèle aussi le plaisir d'aller prélever ce que la nature produit -« Je sais qu'il y a des endroits où tu peux récupérer des champignons, récupérer du muguet » (Romain, 39 ans) ; « Je pratique la chasse à tir et je suis suiveur de la chasse à courre. Oui, ça me plaît. C'est quelque chose qui est inscrit dans notre patrimoine et c'est une culture » (Antoine, 22 ans). Ces activités qui, dont certaines pourraient être pratiquées dans les parcs en ville, marquent toutefois une forme de contraste marqué avec les espaces urbains : « C'est plus sympa d'être réellement dans des grands espaces verts, il y a un côté moins artificiel, en plus c'est plus grand, donc on est plus tranquille et c'est plus reposant. La biodiversité n'est pas du tout la même que dans un parc en ville. On peut trouver des traces de passages d'animaux, des petits sentiers un peu cachés et des empreintes de l'histoire laissés à l'abandon » (Lou, 25 ans). S'approprier la forêt, c'est aussi pour certains contourner le guidage, le balisage et les interdictions de l'espace « conçu » afin d'échapper au caractère policé de l'espace urbain et restaurer un droit d'accès au monde sauvage : « Quand on s'écarte un peu, il n'y a plus ces infrastructures et c'est sûr qu'il y a des coins qui sont encore plus beaux à voir sans les infrastructures. Alors, ce qui est bien, c'est qu'on y croise des biches avec leurs petits » (Patrice, 67 ans). Ce que d'autres viennent s'approprier dans l'espace « perçu » [START_REF] Lefebvre | La production de l'espace[END_REF], c'est, au contraire des répondants décrits plus haut, ce contact espéré avec la sauvagerie : « On aime voir les animaux sauvages, on n'en voit pas souvent, mais voilà, c'est cool quoi. On a l'impression de partir en exploration » (Manon, 38 ans). Au-delà, aller en forêt, c'est aussi rechercher une émotion esthétique -« L'envie de voir de belles choses, des arbres en particulier » (Hugo, 21 ans) ; « Se retrouver un petit peu face à face à la nature avec des paysages quand même qui sont assez jolis à regarder » (Marine, 26 ans) -, mais aussi un élan spirituel -« J'ai trouvé incroyable la hauteur des arbres. Et ça, c'est une nouvelle sensation pour moi. Il y a un côté mystique, je dirais un côté, mystérieux à la fois, parce qu'il y a une verticalité aussi dans les arbres, tu vois l'immensité, tu as quelque chose qui élève » (Victor, 24 ans). Si le contact avec les animaux ne s'exprime pas dans une forme de continuité des âmes et ne reflète donc pas une ontologie animiste, le besoin de reconnexion au monde naturel illustre assez bien, en revanche, l'ontologie analogique que nous développons à la suite.

Infiltrations analogiques dans l'ontologie naturaliste

L'ontologie analogique est caractérisée par le réseau de correspondances qui reconstitue et reconnecte ce que la discontinuité des intériorités et des physicalités a produit comme disjonction. Les répondants établissent ainsi des relations entre la nature et la santé, celle de la terre résonant avec celle de du corps et de l'esprit : « On sait que les gens qui sont en prison, ceux qui ont vu du côté forêt, sont moins malades que ceux qui ont pas vu ce côté forêt » (Fabrice, 59 ans) ; « La couleur verte, c'est apaisant. Ça apporte de la sérénité, du calme. Ça incite à la méditation, mais de manière éveillée, à absorber tout ce qu'on peut en image, en sensations, en vibrations et j'allais dire en zen attitude » (Laurence, 50 ans). Pour cette répondante, l'appropriation de la forêt débouche sur des pratiques visant à rééquilibrer ses énergies : « Je reste au moins 3h par visite. Je suis très attirée par ce qu'on appelle les bains de forêt. J'ai lu énormément de livres là-dessus, sur la sylvothérapie, donc ce sont des choses que j'aime faire parce que je trouve que la nature est extrêmement apaisante et me permet de me ressourcer après mon travail à l'hôpital ». Cette homologie entre l'être et la nature devrait donc conduire les humains à modérer leurs usages intensifs des espaces forestiers comme le propose Léna, 23 ans : « Pour moi la nature, il ne faut pas y faire trop d'activités, comment dirais-je, il faut laisser la nature comme elle est et lui rendre visite de temps en temps pour revenir à notre vraie nature ». De même, les connections énergétiques entre la nature et les humains rejaillissent dans la conscientisation de la pollution des forêts, ce qui conduit certains répondants à stigmatiser la présence de déchets : « Bon faut quand même faire attention quoi ! On n'est pas là pour dégueulasser la forêt. Faut faire quand même attention, c'est un bien qui appartient à tout le monde, on ne peut pas se permettre de polluer inutilement. La pollution, tout ça, c'était des sujets très loin, très loin pour nous. Et maintenant, en fait, on sait très bien qu'il y a le réchauffement climatique et que pour moi, tout ce qui est écologie, environnemental, c'est des sujets auxquels on devrait faire plus attention avant qu'il n'y ait plus rien » (Céline, 47 ans). Comme cette répondante le précise : « Pour nous, pour notre survie, pour la survie de l'espèce humaine aussi, à partir de ce moment, tu respectes les forêts. C'est notre poumon. On ne peut pas vivre sans. Sans cela, donc je trouve ça génial qu'il y ait des gens qui s'en occupent et qui apprennent aux nouvelles générations à les respecter ». Cette double correspondance entre la forêt et le poumon humain et dans une chaîne de dépendances intergénérationnelle est précisément ce qui caractérise l'analogisme. Nous résumons nos résultats dans la figure 1.

Figure 1 .

 1 Figure 1. La forêt comme hybride spatial et ontologique


F Nivole

C Brousseau

S Avrillon

D Lemur

L Bertel

Radio direction finding applied to DVB-T network for vehicular mobile reception

To improve the quality of the mobile reception of DVB-T (Digital Video Broadcasting -Terrestrial) signal, the knowledge of the propagation channel characteristics is necessary. In this aim, this paper presents sounding methods and results for the estimation of Direction of Arrival (DoA) of DVB-T signals in mobile receiving configuration. The mobile passive sounder is presented, including post-processing and radio direction finding tools. Then, results in different kinds of environment (rural area, motorway, low density and high density town centres) are given and discussed.

I. INTRODUCTION

The aim of the CAVITE project [START_REF] Brousseau | Vectorial characterization of DVB-T propagation channel -Application to radiomobile communications: the "CAVITE" Project[END_REF] is to improve the reception of DVB-T signal in mobile conditions (car, train …) characterized by important propagation effects (Doppler, delay spread …). The first step of this project is to evaluate the spatial and time propagation channel characteristics, to ensure that diversity exists and that it could be exploited to increase the quality of received images on vehicular board. DVB-T system uses COFDM transmission. This modulation is suitable to high numerical data rate transmission but is very sensitive to Doppler frequency shift, noise and fading effects. One way to improve the DVB-T reception for mobile application is to use heterogeneous antenna array [START_REF] Brousseau | Vectorial characterization of DVB-T propagation channel -Application to radiomobile communications: the "CAVITE" Project[END_REF]. The choice of these antennas is critical and it is necessary to have a good knowledge of propagation channel effect. In particular, the estimation of the Direction of Arrival (DoA) allows to define the features of the antennas and their location on the vehicle, and the characteristics of diversity receiver.

II. MULTI-ANTENNA MOBILE SOUNDER AND SIGNAL

PROCESSING

The challenge for channel sounding is to get information from broadcasted DVB-T received signals with an omnidirectional angular span, under fading condition, horizontal polarization and antennas mounted on a car roof.

A 4-inputs antenna coherent receiver, including a radio frequency (RF) front-end and baseband signal processing, has been developed in IETR laboratory.

A. RF front-end

As shown on Fig. Omnidirectional antennas are constituted with pairs of printed halo antennas and are set as a circular array mounted on car roof [START_REF] Nivole | Comparison of antenna performances for COFDM DVB-T system -Application to channel sounding[END_REF].

B. Baseband signal processing

Estimations of directions of arrival are computed using impulse responses from the four coherent receiving channels. These channel impulse responses are obtained from the structure of DVB-T frame (see Fig. 2) using baseband signal processing [START_REF] Combelles | A receiver architecture conforming to the OFDM based digital video broadcasting standard for terrestrial transmission (DVB-T)[END_REF] that includes:

-OFDM coarse-synchronization, phase offset correction and fine synchronization, -Static and scattered pilot recovery for channel frequency estimation and signal equalization, -Estimation of channel impulse response from frequency channel interpolated information. A photo of the receiving and acquisition system is given in Fig. 3.

C. DoA estimation methods

Two different methods Capon [START_REF] Combelles | A receiver architecture conforming to the OFDM based digital video broadcasting standard for terrestrial transmission (DVB-T)[END_REF] and MUSIC [START_REF] Schmidt | Multiple emitter location and signal parameter estimation[END_REF] are used to estimate directions of arrival from measured impulse responses.

1) Capon method

The Capon method algorithm uses the covariance matrix of channel impulse response samples X:

T N n XX n X n X N R ) ( ) ( 1 1 ∑ = = ( 1
)
Where n is the sample number, N, the number of samples, and (.) T , the transposition.

Then, the Capon spectrum is given by:

= ( 2
)
Where Az is the azimuth angle, El, the elevation angle, and a, the steering vector of the antenna array.

2) MUSIC high resolution algorithm

This algorithm uses an Eigen-decomposition of the covariance matrix R xx given in equation ( 1). The aim is to separate the samples in two orthogonal subspaces. The first is the noise subspace and the second is the signal one.

After the Eigen-decomposition, the number of sources NSE is evaluated by most important Eigen values. Then, a pseudo spectrum PSSP is determined using the normalized steering vector b(Az, El) of antenna array.

It is given by:

∑ + = = NC NSE k T k El Az b v Az PSSP 1 ) , ( .
1 ) ( ( 3
)
Where NC is the number of sensors, k, the sensor number, v k T , the Eigen-vector, and b = a/(NC) 1/2 , the normalized steering vector.

The maximum as a function of azimuth and elevation angles gives the DoA of DVB-T propagation paths.

III. FIELD TESTS

D. Measurement setup

For the measurements, a car has been equipped with the 4 inputs diversity receiver and the antennas have been mounted on the vehicle in a roof box. Figure 4 shows the embedded mobile system inside the car. Figure 5 gives the reference azimuth and elevation regarding the vehicle orientation. A GPS receiver records the vehicle routes during measurements.

Field tests have been performed in four different environments, three in Brittany (France): motorway, rural area, low density town centre, and one in Paris (France): high density town centre.

1) Brittany measurement setup

In Brittany, the receiver is set up to receive signals from the Rennes/St Pern transmitter. Figure 6 shows the routes, the transmitter location, and the direction of vehicle along these routes. In these three cases, direction of vehicle in relation to transmitter has very little variation along each route. Speed limit is set to 110 km/h in motorway area, 90 km/h in rural area, and 50 km/h in low density town center.

2) Paris measurement setup

In Paris, the receiver is set up to receive signals from the Eiffel tower transmitter. Figure 7 shows the routes, the transmitter location, and the direction of vehicle along these routes. Speed limit is 50 km/h in Paris.

A. Brittany measurement results

For the three measurements realized in Brittany, results are quite similar. Figures 10,11, and 12, show that angles of arrival are limited to a reduced angular sector, typically less than 60° in azimuth plane and less than 40° in elevation plane. Azimuth angle is around 230° during motorway measurements and 90° during rural and low density town centre measurements. These angles correspond with the direction of Rennes/St Pern transmitter so that the vehicle is in Line of Sight (LoS) configuration for the three measurements.

In the three measurements realized in Brittany, Capon and MUSIC methods give very similar results. This is probably due to LoS configuration that provides high signal to noise ratio and permits to easily separate noise and signal orthogonal subspaces in MUSIC algorithm.

B. Paris measurement results

In Paris, DoA estimations results are very different from Brittany results. Azimuth angular spread is wider, from 60° to 150° and elevation angular spread can be higher than 60°.

For field test 1, the first path (Fig. 13) is spread between 60° and 120° in azimuth plane and between 0° to 60° in elevation plane. Capon and MUSIC give similar results in this case. The second path (Fig. 14) has a wider angular spread in azimuth, with two most important directions around 220° and around 300°.

For field test 2, the first path (Fig. 15) has two most important angular directions in azimuth plane spread around 60° and 120° using Capon method and spread around 45° and 130° using MUSIC algorithm. In elevation plane, a wide spread is also observed from 0° to 60°. The second path (Fig. 16) has a wide angular spread in azimuth from 230° to 360° using Capon method but a smaller spread in elevation plane with angles less than 20°.

In the measurements realized in Paris, Capon and MUSIC methods do not generally give the same results. This is probably due to the high number of multipaths that reduces signal to noise ratio and does not permit to easily separate noise and signal orthogonal subspaces in MUSIC algorithm.

C. Comments on measurement results

Measurements in Brittany and in Paris give very different DoA estimation results. In Brittany, the environments chosen for these tests are free of reflexion and/or diffraction mechanism because of the absence of high buildings or obstacles. In these LoS configurations, angular spreads of DoA are very low. In Paris, the presence of high buildings and the very high density of the town introduce multi-paths with high angular spreads. However, in most of high spread DoA estimations (Fig. 14 and Fig. 15), two main directions can be observed. This is probably due to urban-canyon phenomena in which waves can propagate only in very few directions which correspond to buildings alignment.

In high density environments like town centres, the high angular diversity is suitable with receivers using EGC (Equal Gain Combining) or MRC (Maximum Ratio Combining) with omni-directional antennas or wide-sectorial antennas.

In other environments where angular diversity is lower, omni-directional reception can be improved by using beamforming process that can be used to have better gain in one direction and to focus the wave energy. Other solution is to use several high gain directional antennas which have to be distributed to cover the 360° angular span, with a switched antenna diversity algorithm. In this case, multi-receiver system does not need to use all synthesizers at the same time. The consumption will be reduced but a regular scan to get quality of signal has to be done to choose the antenna which has to be used with its respective receiver.

V. CONCLUSIONS A channel sounder for DVB-T network has been presented. It is constituted with four coherent receivers which permit to exploit DVB-T broadcasted signals in order to estimate directions of arrival. RF front-end, acquisition system, and baseband signal processing have been entirely developed in IETR laboratory. Direction of Arrival (DoA) estimations have been obtained from channel impulse responses using two different methods: Capon method and MUSIC algorithm.

DoA estimations have been performed during several field tests in four different environments: motorway, rural area, low density town centre, and high density town centre (Paris). Results of DoA estimations in Paris show that angular spread is important, both in azimuth and elevation plane, and that urban-canyon phenomena can reduce this angular spread around two main directions. In the other environments, the vehicle is in LoS configuration with only one low angular spread path in the direction of the transmitter.

This radio direction finding applied to DVB-T signal in vehicular condition permits to validate that two types of reception principles would be optimal: an antenna combining receiver with omnidirectional or wide-sectorial antennas in high density environments, and beamforming or switched antenna receiver with narrow-sectorial antennas in other environments.

Fig. 1

 1 Fig. 1 Receiving RF front-end architecture used for the DVB-T channel sounder.

Fig. 2

 2 Fig. 2 DVB-T frame structure with static and scattered pilots positions as a function of frequency and symbol time.

Fig. 3

 3 Fig. 3 Receiving and acquisition system

Fig. 4 :

 4 Fig. 4: Photo of the embedded system inside the car.

Fig. 5 :

 5 Fig. 5: Car representation in azimuth and elevation plan.

Fig. 6 :Fig. 7 :Fig. 9 :

 679 Fig. 6: Route of mobile measurements recorded by the GPS receiver in Brittany (north of Rennes)

Fig. 10 :

 10 Fig. 10: Azimuth and elevation angle estimations in motorway area.

Fig. 11 :

 11 Fig. 11: Azimuth angle estimations in rural area.

Fig. 12 :

 12 Fig. 12: Azimuth angle estimations in low density town centre.

Fig. 14 :

 14 Fig. 14: Azimuth angle estimations in Paris (field test 1), for the second path.

Fig. 16 :

 16 Fig. 16: Azimuth angle estimations in Paris (field test 2), for the second path.

ACKNOWLEDGMENT

The authors thank the council of Brittany -France (Conseil Régional de Bretagne) and the FNADT (Fonds National d'Aménagement et de Développement du Territoire) for their supports to this project.