Antoine Soubeyran

Faculty ,  Aix-Marseille Université , Faculté d'économie et de gestion (FEG)

Econometrics, Finance and mathematical methods
Status
Emeritus professor
Research domain(s)
Behavioral and experimental economics, Industrial organization
Thesis
1975, Aix-Marseille Université
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Contact
antoine.soubeyran[at]gmail.com
+33(0)4 13 94 20 28
Address

Maison de l'économie et de la gestion d'Aix
424 chemin du viaduc, CS80429
13097 Aix-en-Provence Cedex 2

Abstract In this article, we obtain an extension of the Ekeland variational principle in quasi-uniform spaces. Since the Ekeland variational principle is a type of perturbed optimization problem, the perturbations do not need to satisfy the triangle property to obtain results. We also give some equivalent results of our main results. Moreover, we present a new version of the Ekeland variational principle and its equivalent results, in the setting of quasi-gage spaces. Finally, we establish the Ekeland variational principle in a (metric) modular space as an application of our results.
Keywords Ekeland&#039, s variational principle, Quasi-uniform space, Quasi-gauge space, Modular space, Fixed point
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Abstract In this paper, we examine the quasi-equilibrium problem from a variational rationality perspective. To this end, we first study the convergence of the proximal point method proposed by Bento et al. [Ann. Oper. Res. 316 (2022), 1301-1318] in the more general context of quasi-equilibrium problems using a Bregman distance. Thus, we provide an application of the method through a recent behavioral perspective, more precisely, the variational rationality approach of staying and changing human dynamics, and the important example of climbing the goal ladder in goal pursuit theory. An illustrative simulation demonstrates that Bregman distances improve the computational performance of the method compared to the Euclidean distance.
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Abstract Local proximal point algorithms with quasi distances to find critical points (or minimizer points in the convex case) of functions in finite dimensional Riemannian manifolds are introduced. We prove that bounded sequences of the algorithm generated by proper bounded from below, lower semicontinuous and locally Lipschitz functions have accumulation points which are critical points (minimizer points in the convex case). Moreover, for KurdykaLojasiewicz functions, the sequence globally converges to a critical point. We applied the algorithm to a behavioral traveler’s problem where an individual tries to satisfy locally his needs and desires by moving from one city to the next, with costs to move playing a major role.
Keywords Local search, Proximal algorithms, Riemannian manifolds, The behavioral traveler’s problem
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Abstract This paper has two parts. In the mathematical part, we present two inexact versions of the proximal point method for solving quasi-equilibrium problems (QEP) in Hilbert spaces. Under mild assumptions, we prove that the methods find a solution to the quasi-equilibrium problem with an approximated computation of each iteration or using a perturbation of the regularized bifunction. In the behavioral part, we justify the choice of the new perturbation, with the help of the main example that drives quasi-equilibrium problems: the Cournot duopoly model, which founded game theory. This requires to exhibit a new QEP reformulation of the Cournot model that will appear more intuitive and rigorous. It leads directly to the formulation of our perturbation function. Some numerical experiments show the performance of the proposed methods.
Keywords Quasi-equilibrium problem, Proximal point method, Inexact version, Monotone bifunction, Cournot duopoly, Moving constraints
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Abstract We present an inexact proximal point algorithm using quasi distances to solve a minimization problem in the Euclidean space. This algorithm is motivated by the proximal methods introduced by Attouch et al., section 4, (Math Program Ser A, 137: 91–129, 2013) and Solodov and Svaiter (Set Valued Anal 7:323–345, 1999). In contrast, in this paper we consider quasi distances, arbitrary (non necessary smooth) objective functions, scalar errors in each objective regularized approximation and vectorial errors on the residual of the regularized critical point, that is, we have an error on the optimality condition of the proximal subproblem at the new point. We obtain, under a coercivity assumption of the objective function, that all accumulation points of the sequence generated by the algorithm are critical points (minimizer points in the convex case) of the minimization problem. As an application we consider a human location problem: How to travel around the world and prepare the trip of a lifetime.
Keywords Proximal point methods, Inexact algorithms, Coercivity, Quasi distances, Variational rationality, Traveler problem
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Abstract This paper has two parts. The mathematical part provides generalized versions of the robust Ekeland variational principle in terms of set-valued EVP with variable preferences, uncertain parameters and changing weights given to vectorial perturbation functions. The behavioural part that motivates our findings models the formation and stability of a partnership in a changing, uncertain and complex environment in the context of the variational rationality approach of stop, continue and go human dynamics. Our generalizations allow us to consider two very important psychological effects relative to ego depletion and goal gradient hypothesis.
Keywords Variational rationality, Variable preferences, Ego depletion, Goal gradient hypothesis, Robust Ekeland variational principle
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Abstract We establish general versions of the Ekeland variational principle (EVP), where we include two perturbation bifunctions to discuss and obtain better perturbations for obtaining three improved versions of the principle. Here, unlike the usual studies and applications of the EVP, which aim at exact minimizers via a limiting process, our versions provide good-enough approximate minimizers aiming at applications in particular situations. For the presentation of applications chosen in this paper, the underlying space is a partial quasi-metric one. To prove the aforementioned versions, we need a new proof technique. The novelties of the results are in both theoretical and application aspects. In particular, for applications, using our versions of the EVP together with new concepts of Ekeland points and stop and go dynamics, we study in detail human dynamics in terms of a psychological traveler problem, a typical model in behavioral sciences.
Keywords Stop and go dynamics, Variational trap, Worthwhile move, Partial quasi-metric space, Ekeland points, Better perturbation, Two perturbation bifunctions, Ekeland variational principle
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Abstract In this paper, we introduce a new proximal algorithm for equilibrium problems on a genuine Hadamard manifold, using a new regularization term. We first extend recent existence results by considering pseudomonotone bifunctions and a weaker sufficient condition than the coercivity assumption. Then, we consider the convergence of this proximal-like algorithm which can be applied to genuinely Hadamard manifolds and not only to specific ones, as in the recent literature. A striking point is that our new regularization term have a clear interpretation in a recent “variational rationality” approach of human behavior. It represents the resistance to change aspects of such human dynamics driven by motivation to change aspects. This allows us to give an application to the theories of desires, showing how an agent must escape to a succession of temporary traps to be able to reach, at the end, his desires.
Keywords Desires, Worthwhile changes, Trap, Hadamard manifold, Equilibrium problem, Proximal algorithms
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Abstract In this paper, we consider an abstract regularized method with a skew-symmetric mapping as regularization for solving equilibrium problems. The regularized equilibrium problem can be viewed as a generalized mixed equilibrium problem and some existence and uniqueness results are analyzed in order to study the convergence properties of the algorithm. The proposed method retrieves some existing one in the literature on equilibrium problems. We provide some numerical tests to illustrate the performance of the method. We also propose an original application to Becker’s household behavior theory using the variational rationality approach of human dynamics.
Keywords Equilibrium problem, Variational rationality, Desires, Traps, Household behavior, Resource allocation problems
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Abstract In this paper we introduce a definition of approximate Pareto efficient solution as well as a necessary condition for such solutions in the multiobjective setting on Riemannian manifolds. We also propose an inexact proximal point method for nonsmooth multiobjective optimization in the Riemannian context by using the notion of approximate solution. The main convergence result ensures that each cluster point (if any) of any sequence generated by the method is a Pareto critical point. Furthermore, when the problem is convex on a Hadamard manifold, full convergence of the method for a weak Pareto efficient solution is obtained. As an application, we show how a Pareto critical point can be reached as a limit of traps in the context of the variational rationality approach of stay and change human dynamics.
Keywords Multiobjective proximal method, Riemannian manifold, Approximate solution, Variational rationality, Worthwhile moveTrap
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Abstract In this paper, in the context of quasi-metric spaces, we obtain two set-valued versions of the Ekeland variational-type principle by means of lower and upper set less relations, for the case where the perturbations need not satisfy the triangle inequality. An application in terms of migration problems and quality of life is given.
Keywords Variational rationality, Quasi-metric space, Trap, Set-valued mapping, Lower and upper set less relations
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Abstract We first give a pre-order principle whose form is very general. Combining the pre-order principle and generalized Gerstewitz functions, we establish a general equilibrium version of set-valued Ekeland variational principle (denoted by EVP), where the objective function is a set-valued bimap defined on the product of quasi-metric spaces and taking values in a quasi-ordered linear space, and the perturbation consists of a subset of the ordering cone multiplied by the quasi-metric. From this, we obtain a number of new results which essentially improve the related results. Particularly, the earlier lower boundedness condition has been weakened. Finally, we apply the new EVPs to Psychology.
Keywords Gerstewitz function, Pre-order principle, Quasi-metric space, Set-valued perturbation, Equilibrium version of Ekeland variational principle
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Abstract This paper has two aspects. Mathematically, in the context of global optimization, it provides the existence of an optimum of a perturbed optimization problem that generalizes the celebrated Ekeland variational principle and equivalent formulations (Caristi, Takahashi), whenever the perturbations need not satisfy the triangle inequality. Behaviorally, it is a continuation of the recent variational rationality approach of stay (stop) and change (go) human dynamics. It gives sufficient conditions for the existence of traps in a changing environment. In this way it emphasizes even more the striking correspondence between variational analysis in mathematics and variational rationality in psychology and behavioral sciences.
Keywords Traps, Changing environment, Variational rationality, The Ekeland variational principle, Quasi-metric space
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Abstract By using a pre-order principle in [Qiu JH. A pre-order principle and set-valued Ekeland variational principle. J Math Anal Appl. 2014;419:904–937], we establish a general equilibrium version of set-valued Ekeland variational principle (denoted by EVP), where the objective function is a set-valued bimap defined on the product of left-complete quasi-metric spaces and taking values in a quasi-ordered linear space, and the perturbation consists of a cone-convex subset of the ordering cone multiplied by the quasi-metric. Moreover, we obtain an equilibrium EVP, where the perturbation contains a σ-convex subset and the quasi-metric. From the above two general EVPs, we deduce several interesting corollaries, which extend and improve the related known results. Several examples show that the obtained set-valued EVPs are new. Finally, applying the above EVPs to organizational behavior sciences, we obtain some interesting results on organizational change and development with leadership. In particular, we show that the existence of robust organizational traps.
Keywords Robust trap problem, Caristi&#039, s fixed point theorem, Pre-order principle, Quasi-metric space, Equilibrium version of Ekeland variational principle
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Abstract We consider a contracting relationship where the agent's effort induces monetary costs, and limits on the agent's resource restrict his capability to exert effort. We show that, the principal finds it best to offer a sharing contract while providing the agent with an up-front financial transfer only when the monetary cost is neither too low nor too high. Thus, unlike in the limited liability literature, the principal might find it optimal to fund the agent. Moreover, both incentives and the amount of funding are non-monotonic functions of the monetary cost. These results suggest that an increase in the interest rate may affect the form of contracts differently , depending on the initial level of the former. Using the analysis, we provide and discuss several predictions and policy implications.
Keywords Funding, Wealth constraint, Moral hazard, Contract theory
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Abstract In this paper, we extend the general descent method proposed by Attouch, Bolte and Svaiter [Math. Program. 137 (2013), 91-129] to deal with possible asymmetric like-distances. Using a w-distance as regularization term our results guarantee the convergence of bounded sequences, under the assumption that the objective function satisfies the Kurdyka-Łojasiewicz inequality. In particular, it improves some existing works on proximal point methods with quasi-distance as regularization term because we prove convergence of bounded sequences without any additional assumption on the w-distance unlike it have been done with quasi-distances. The last section gives an application relative to the emergence of habits after a succession of worthwhile moves which balance motivation and resistance to move.
Keywords Variational rationality, Habits, W-distance, Kurdyka-Łojasiewicz inequality, Descent methods
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Abstract The purpose of this paper is twofold. First, we examine convergence properties of an inexact proximal point method with a quasi distance as a regularization term in order to find a critical point (in the sense of Toland) of a DC function (difference of two convex functions). Global convergence of the sequence and some convergence rates are obtained with additional assumptions. Second, as an application and its inspiration, we study in a dynamic setting, the very important and difficult problem of the limit of the firm and the time it takes to reach it (maturation time), when increasing returns matter in the short run. Both the formalization of the critical size of the firm in term of a recent variational rationality approach of human dynamics and the speed of convergence results are new in Behavioral Sciences.
Keywords Variational rationality, Proximal point method, Kurdyka–Łojasiewicz inequality, DC function, Limit of the firm
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Abstract We consider the constrained multi-objective optimization problem of finding Pareto critical points of difference of convex functions. The new approach proposed by Bento et al. (SIAM J Optim 28:1104–1120, 2018) to study the convergence of the proximal point method is applied. Our method minimizes at each iteration a convex approximation instead of the (non-convex) objective function constrained to a possibly non-convex set which assures the vector improving process. The motivation comes from the famous Group Dynamic problem in Behavioral Sciences where, at each step, a group of (possible badly informed) agents tries to increase his joint payoff, in order to be able to increase the payoff of each of them. In this way, at each step, this ascent process guarantees the stability of the group. Some encouraging preliminary numerical results are reported.
Keywords Behavioral sciences, Variational rationality, DC function, Proximal point method, Multi-objective programming
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Abstract This paper is devoted to new versions of Ekeland’s variational principle in set optimization with domination structure, where set optimization is an extension of vector optimization from vector-valued functions to set-valued maps using Kuroiwa’s set-less relations to compare one entire image set with another whole image set, and where domination structure is an extension of ordering cone in vector optimization; it assigns each element of the image space to its own domination set. We use Gerstewitz’s nonlinear scalarization function to convert a set-valued map into an extended real-valued function and the idea of the proof of Dancs-Hegedüs-Medvegyev’s fixed-point theorem. Our setting is applicable to dynamic processes of changing jobs in which the cost function does not satisfy the symmetry axiom of metrics and the class of set-valued maps acting from a quasimetric space into a real linear space. The obtained result is new even in simpler settings.
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Abstract This chapter considers potential games, where agents play, each period, Nash worthwhile moves in alternation, such that their unilateral motivation to change rather than to stay, other players being supposed to stay, are high enough with respect to their resistance to change rather than to stay. This defines a generalized proximal alternating linearized algorithm, where resistance to change plays a major role, perturbation terms of alternating proximal algorithms being seen as the disutilities of net costs of moving.
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Abstract This paper concerns applications of variational analysis to some local aspects of behavioral science modeling by developing an effective variational rationality approach to these and related issues. Our main attention is paid to local stationary traps, which reflect such local equilibrium and the like positions in behavioral science models that are not worthwhile to quit. We establish constructive linear optimistic evaluations of local stationary traps by using generalized differential tools of variational analysis that involve subgradients and normals for nonsmooth and nonconvex objects as well as variational and extremal principles.
Keywords Variational rationality, Applications to behavioral sciences, Variational and extremal principles, Normals, Sub-gradients, Worthwhile moves, Optimization, Variational analysis
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Abstract We introduce and examine an inexact multi-objective proximal method with a proximal distance as the perturbation term. Our algorithm utilizes a local search descent process that eventually reaches a weak Pareto optimum of a multi-objective function, whose components are the maxima of continuously differentiable functions. Our algorithm gives a new formulation and resolution of the following important distributive justice problem in the context of group dynamics: In each period, if a group creates a cake, the problem is, for each member, to get a high enough share of this cake; if this is not possible, then it is better to quit, breaking the stability of the group.
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Abstract This paper studies the constrained multiobjective optimization problem of finding Pareto critical points of vector-valued functions. The proximal point method considered by Bonnel, Iusem, and Svaiter [SIAM J. Optim., 15 (2005), pp. 953--970] is extended to locally Lipschitz functions in the finite dimensional multiobjective setting. To this end, a new (scalarization-free) approach for convergence analysis of the method is proposed where the first-order optimality condition of the scalarized problem is replaced by a necessary condition for weak Pareto points of a multiobjective problem. As a consequence, this has allowed us to consider the method without any assumption of convexity over the constraint sets that determine the vectorial improvement steps. This is very important for applications; for example, to extend to a dynamic setting the famous compromise problem in management sciences and game theory.
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Abstract Using a pre-order principle in [Qiu JH. A pre-order principle and set-valued Ekeland variational principle. J Math Anal Appl. 2014;419:904–937], we establish a general equilibrium version of set-valued Ekeland variational principle (denoted by EVP), where the objective bimap is defined on the product of left-complete quasi-metric spaces and taking values in a quasi-order linear space, and the perturbation consists of the quasi-metric and a positive vector . Here, the ordering is only to be -closed, which is strictly weaker than to be topologically closed. From the general equilibrium version, we deduce a number of particular equilibrium versions of EVP with set-valued bimaps or with vector-valued bimap. As applications of the equilibrium versions of EVP, we present several interesting results on equilibrium problems, vector optimization and fixed point theory in the setting of quasi-metric spaces. These results extend and improve the related known results. Using the obtained EVPs, we further study the existence and the robustness of traps in Behavioural Sciences.
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Abstract We analyze a sequential decision making process, in which at each step the decision is made in two stages. In the first stage a partially optimal action is chosen, which allows the decision maker to learn how to improve it under the new environment. We show how inertia (cost of changing) may lead the process to converge to a routine where no further changes are made. We illustrate our scheme with some economic models.
Keywords Convergence MSC2010 91B06, Costs to change, Sequential decision making
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Abstract This paper has a twofold focus. The mathematical aspect of the paper shows that new and existing quasimetric and weak r-distance versions of Ekeland’s variational principle are equivalent in the sense that one implies the other, and so are their corresponding fixed-point results. The practical aspect of the paper, using a recent variational rationality approach of human behaviour, offers a model of organizational change, where generalized distances model inertia in terms of resistance to change. The formation and breaking of routines relative to hiring and firing workers will be used to illustrate the obtained results.
Keywords Variational rationality, R-distance, W-distance, Quasimetric, Caristi’s fixed-point theorem, Ekeland’s variational principle
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Abstract In this paper, driven by Behavioral applications to human dynamics, we consider the characterization of completeness in pseudo-quasimetric spaces in term of a generalization of Ekeland’s variational principle in such spaces, and provide examples illustrating significant improvements to some previously obtained results, even in complete metric spaces. At the behavioral level, we show that the completeness of a space is equivalent to the existence of traps, rather easy to reach (in a worthwhile way), but difficult (not worthwhile to) to leave. We first establish new forward and backward versions of Ekeland’s variational principle for the class of strict-decreasingly forward (resp. backward)-lsc functions in pseudo-quasimetric spaces. We do not require that the space under consideration either be complete or to enjoy the limit uniqueness property since, in a pseudo-quasimetric space, the collections of forward-limits and backward ones of a sequence, in general, are not singletons.
Keywords Pseudo-quasimetric, Forward-completeness, Backward-completeness, Variational principle, Group dynamics, Existence of trap
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Abstract In this paper, driven by applications in Behavioral Sciences, wherein the speed of convergence matters considerably, we compare the speed of convergence of two descent methods for functions that satisfy the well-known Kurdyka–Lojasiewicz property in a quasi-metric space. This includes the extensions to a quasi-metric space of both the primal and dual descent methods. While the primal descent method requires the current step to be more or less half of the size of the previous step, the dual approach considers more or less half of the previous decrease in the objective function to be minimized. We provide applications to the famous “Tension systems approach” in Psychology.
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Abstract A proximal linearized algorithm for minimizing difference of two convex functions is proposed. If the sequence generated by the algorithm is bounded it is proved that every cluster point is a critical point of the function under consideration, even if the auxiliary minimizations are performed inexactly at each iteration. Linear convergence of the sequence is established under suitable additional assumptions.
Keywords Economie quantitative
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Abstract In this paper, we generalize Ekeland’s variational principle in the new context of cone pseudo-quasimetric spaces. We propose this extension for applications to group dynamics in behavioral sciences. In this setting, a cone pseudo-quasimetric helps to model, in a crude way, multidimensional aspects of resistance to change for a group, where each component represents resistance to change of one agent in the group. At the behavioral level, our new version of Ekeland’s variational principle shows how a group, forming and breaking routines each period by balancing between motivations and resistances to change of all members, can improve step by step their payoffs to end in a trap worthwhile to approach and reach, but not worthwhile to leave.
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Abstract We consider a bilevel problem involving two monotone equilibrium bifunctions and we show that this problem can be solved by a proximal point method with generalized proximal distances. We propose a framework for the convergence analysis of the sequence generated by the algorithm. This class of problems is very interesting because it covers mathematical programs and optimization problems under equilibrium constraints. As an application, we consider the problem of the stability and change dynamics of a leader-follower relationship in a hierarchical organization.
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Abstract The paper is devoted to variational analysis of set-valued mappings acting from quasimetric spaces into topological spaces with variable ordering structures. Besides the mathematical novelty, our...
Keywords Economie quantitative
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Abstract Multiobjective optimization has a significant number of real-life applications. For this reason, in this paper we consider the problem of finding Pareto critical points for unconstrained multiobjective problems and present a trust-region method to solve it. Under certain assumptions, which are derived in a very natural way from assumptions used to establish convergence results of the scalar trust-region method, we prove that our trust-region method generates a sequence which converges in the Pareto critical way. This means that our generalized marginal function, which generalizes the norm of the gradient for the multiobjective case, converges to zero. In the last section of this paper, we give an application to satisficing processes in Behavioral Sciences. Multiobjective trust-region methods appear to be remarkable specimens of much more abstract satisficing processes, based on “variational rationality” concepts. One of their important merits is to allow for efficient computations. This is a striking result in Behavioral Sciences.
Keywords Worthwhile change, Variational rationality, Unconstrained multiobjective problem, Trust-region methods, Satisficing process, Pareto critical point
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Abstract In this paper, we present a proximal point algorithm for multicriteria optimization, by assuming an iterative process which uses a variable scalarization function. With respect to the convergence analysis, firstly we show that, for any sequence generated from our algorithm, each accumulation point is a Pareto critical point for the multiobjective function. A more significant novelty here is that our paper gets full convergence for quasi-convex functions. In the convex or pseudo-convex cases, we prove convergence to a weak Pareto optimal point. Another contribution is to consider a variant of our algorithm, obtaining the iterative step through an unconstrained subproblem. Then, we show that any sequence generated by this new algorithm attains a Pareto optimal point after a finite number of iterations under the assumption that the weak Pareto optimal set is weak sharp for the multiobjective problem.
Keywords Weak sharp, Quasi-convexity, Proximal, Multicriteria optimization, Fejér convergence
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Abstract In this article, we consider a knowledge accumulation problem within an organization that cannot prevent the worker from quitting and using the knowledge outside the organization. We show that knowledge accumulation is delayed: The fraction of working time allocated to knowledge creation is highest at the early career stage, falls gradually, then rises again, before falling finally toward zero. We determine the effect of a change in the severity of the enforcement problem (or the specificity of knowledge). We also discuss the form of the optimal life'cycle wage profiles, the role of the initial knowledge level, and the role of discounting.
Keywords Economie quantitative
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Abstract In this paper, we consider minimization problems with constraints. We show that, if the set of constraints is a Finslerian manifold of non-positive flag curvature, and the objective function is differentiable and satisfies the Kurdyka-Lojasiewicz property, then the proximal point method can be naturally extended to solve this class of problems. We prove that the sequence generated by our method is well defined and converges to a critical point. We show how tools of Finslerian geometry, specifically non-symmetrical metrics, can be used to solve non-convex constrained problems in Euclidean spaces. As an application, we give one result regarding decision-making speed and costs related to change.
Keywords Proximal algorithms, Non-convex optimization, Kurdyka-Lojasiewicz inequality, Finslerian manifolds
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Abstract In this paper we study an inexact steepest descent method for multicriteria optimization whose step-size comes with Armijo's rule. We show that this method is well-defined. Moreover, by assuming the quasi-convexity of the multicriteria function, we prove full convergence of any generated sequence to a Pareto critical point. As an application, we offer a model for the Psychology's self regulation problem, using a recent variational rationality approach.
Keywords Steepest descent, Self regulation, Quasi-convexity, Multiple objective programming
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Abstract This paper develops some mathematical models arising in psychology and some other areas of behavioral sciences that are formalized via general preferences with variable ordering structures. Our considerations are based on the recent variational rationality approach, which unifies numerous theories in different branches of behavioral sciences using, in particular, worthwhile change and stay dynamics and variational traps. In the mathematical framework of this approach, we derive a new variational principle, which can be viewed as an extension of the Ekeland variational principle to the case of set-valued mappings on quasimetric spaces with cone-valued ordering variable structures. Such a general setting is proved to be appropriate for broad applications to the functioning of goal systems in psychology, which are developed in the paper. In this way, we give a certain answer to the following striking question: in the world, where all things change (preferences, motivations, resistances, etc.), where goal systems drive a lot of entwined course pursuits between means and ends, what can stay fixed for a while? The obtained mathematical results and new insights open the door to developing powerful models of adaptive behavior, which strongly depart from pure static general equilibrium models of the Walrasian type, which are typical in economics.
Keywords Economie quantitative
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Abstract We consider minimization problems with constraints. We show that if the set of constraints is a Riemannian manifold of non positive curvature and the objective function is lower semicontinuous and satisfies the Kurdyka-Lojasiewicz property, then the alternating proximal algorithm in Euclidean space is naturally extended to solve that class of problems. We prove that the sequence generated by our algorithm is well defined and converges to an inertial Nash equilibrium under mild assumptions about the objective function. As an application, we give a welcome result on the difficult problem of "learning how to play Nash" (convergence, convergence in finite time, speed of convergence, constraints in action spaces in the context of "alternating potential games" with inertia).
Keywords Riemannian manifold, Proximal algorithm, Nash equilibrium, Learning in games, Kurdyka-Lojasiewicz property, Inertia, Finite time, Convergence, Alternation
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Abstract We consider variable preference relations, also called reference dependent preference relations, which are typical in the study of dynamic models in economic theories. We introduce a new concept of weak consistency, a generalization of acyclicity, as an immediate regret condition for variable preferences. The main result to establish is on an existence criterion for maximal elements of a space equipped with a weakly consistent variable preference relation. It is expressed via a preference completeness condition which is equivalent to existence of aspiration points. As applications, we show that a number of results known in the recent literature on maximum principles on a space with or without topological structure can be obtained from the unifying approach of this paper. Habit formation and state functions are also discussed in the framework of variable preference relations.
Keywords Variable preference relation, State function, State, Maximal point, Improving path, Habit formation, Function
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Abstract Acceptable moves for the "worthwhile-to-move" incremental principle are such that "advantages-to-move" are higher than some fraction of "costs-to-move". When combined with optimization, this principle gives raise to adaptive local search proximal algorithms. Convergence results are given in two distinctive cases, namely low local costs-to-move and high local costs-to-move. In this last case, one obtains a dynamic cognitive approach to Ekeland's -variational principle. Introduction of costs-to-move in the algorithms yields robustness and stability properties.
Keywords Costs-to-move, Decision dynamics, Exploration process, Friction, Inertia, Local optimization, Local search algorithms, Proximal algorithms, Worthwhile-to-move incremental process
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Abstract We study the convergence properties of an alternating proximal minimization algorithm for nonconvex structured functions of the type: L( x, y) = f( x )+Q( x , y) +g (y) , where f and g are proper lower semicontinuous functions, defined on Euclidean spaces, and Q is a smooth function that couples the variables x and y. The algorithm can be viewed as a proximal regularization of the usual Gauss-Seidel method to minimize L. We work in a nonconvex setting, just assuming that the function L satisfies the Kurdyka-Łojasiewicz inequality. An entire section illustrates the relevancy of such an assumption by giving examples ranging from semialgebraic geometry to "metrically regular" problems. Our main result can be stated as follows: If L has the Kurdyka-Łojasiewicz property, then each bounded sequence generated by the algorithm converges to a critical point of L. This result is completed by the study of the convergence rate of the algorithm, which depends on the geometrical properties of the function L around its critical points. When specialized to to f , g indicator functions, the algorithm is an alternating projection mehod (a variant of von Neumann's) that converges for a wide class of sets including semialgebraic and tame sets, transverse smooth manifolds or sets with "regular" intersection. To illustrate our results with concrete problems, we provide a convergent proximal reweighted algorithm for compressive sensing and an application to rank reduction problems.
Keywords Gradient systems, Sparse reconstruction, Finite convergence time, Alternating minimization algorithms, O-minimal structures, Convergence rate, Tame optimization, Alternating projections algorithms, Proximal algorithms, Nonconvex optimization, Kurdyka-Łojasiewicz inequality
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Abstract This paper studies the properties of joint-venture relationship between a technologically advanced multinational firm and a local firm operating in a developing economy where the ability to enforce contracts is weak. We formulate a dynamic model of principal-agent relationship in which at any point of time the local firm can quit without legal penalties. An early breakup may be prevented if the multinational designs a suitable scheme in which both the pace and aggregate amount of technology transfer deviate from the first-best, and a suitable flow of side payments to encourage the local firm to stay longer.
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Abstract We try to understand why firms producing goods by means of complementary components do not merge, especially in industries in which investments in component-based knowledge matters. As Audretsch, we state that these activities are developed by "individuals" who do their best to appropriate the return from their knowledge and whose effort is non-contractible. The organization of the industry into firms is identified to a partition of the set of individuals. In this context, we prove that an organization in which each individual hold his own firms is both stable with respect to unilateral deviation and optimal in the line of the property right approach. If the returns are high enough, this structure is even the only one which shares both properties.
Keywords Lateral disintegration, Dual Cournot competition, Property rights, Incomplete contracts, Composite goods
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Abstract In this paper, we study monopolistic pricing behaviors within a two-way network. In this symbiotic production system, independent decision centers carry out an activity which concurs to the production of different system goods. The players are assumed to know the whole network. Due to this rationality, they try to capture a share of the profit of the firms who sell the system goods to the consumers. These double marginalization behaviors are studied within very general networks. Conditions with ensure existence and uniqueness are discussed. We even provided a complete characterization of an equilibrium. Potential applications are also discussed
Keywords Non-cooperative games, Two-way networks, Monopolistic behaviors
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Abstract As physics provides the equations of motion of a body, this paper formulates, for the first time, at the conceptual and mathematical levels, the inequations of motion of an individual seeking to meet his needs and quasi needs in an adaptive (not myopic) way. Successful (failed) dynamics perform a succession of moves, which are, at once, satisficing and worthwhile (free from too many sacrifices), or not. They approach or reach desires (fall in traps). They balance the desired speed of approach to a desired end (a distal promotion goal) with the size of the required immediate sacrifices to go fast (a proximal prevention goal). Therefore, each period, need/quasi need satisfaction success requires enough self control to be able to make, in the long run, sufficient progress in need/quasi need satisfaction without enduring, in the short run, too big sacrifices. A simple example (lose or gain weight) shows that the size of successful moves must be not too small and not too long. A second paper will solve this problem, using variational principles and inexact optimizing algorithms in mathematics. This strong multidisciplinary perspective refers to a recent mathematical model to psychology: the variational rationality theory of human life stay and change dynamics.
Keywords Need satisfaction, Speed of progress, Sacrifices, DYNAMICAL SYSTEM, Variational rationality
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Abstract This paper provides a general and formalized theory of self-regulation success and failures as an application of the recent Variational rationality approach of stay and change human dynamics (Soubeyran, 2009, 2010, 2021.a,b,c,d). For concreteness purposes, it starts with an example in psychology: how to gain or to lose weight ? It ends with a general, conceptual, dynamical and computable formulation of self-regulation and goal pursuit in the context of variational principles and adaptive optimizing algorithms in mathematics.
Keywords Variational rationality, Self regulation, Variational principles, Adaptive optimizing algorithms
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