The self regulation problem as an inexact steepest descent method for multicriteria optimizationJournal articleGlaydston Carvalho Bento, Joao Xavier Neto, Paulo Roberto Oliveira and Antoine Soubeyran, European Journal of Operational Research, Volume 235, Issue 3, pp. 494-502, 2014

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.

Proximal Point Method on Finslerian Manifolds and the "Effort-Accuracy" Trade-offJournal articleJoao Xavier Cru Neto, Paulo Roberto Oliveira, Jr Soares and Antoine Soubeyran, Journal of Optimization Theory and Applications, Volume 162, Issue 3, pp. 873-891, 2014

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.

A Proximal Point-Type Method for Multicriteria OptimizationJournal articleGlaydston Carvalho Bento, J.X. Cruz Neto and Antoine Soubeyran, Set-Valued and Variational Analysis, Volume 22, Issue 3, pp. 557-573, 2014

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.

Variational Analysis in Psychological ModelingJournal articleTruong Q. Bao, Boris S. Mordukhovich and Antoine Soubeyran, Journal of Optimization Theory and Applications, Volume 164, Issue 1, pp. 290-315, 2014

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.

A Trust-Region Method for Unconstrained Multiobjective Problems with Applications in Satisficing ProcessesJournal articleKelyD.V. Villacorta, Paulo R. Oliveira and Antoine Soubeyran, Journal of Optimization Theory and Applications, Volume 160, Issue 3, pp. 865-889, 2014

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.

Variable preference relations: Existence of maximal elementsJournal articleDinh The Luc and Antoine Soubeyran, Journal of Mathematical Economics, Volume 49, Issue 4, pp. 251-262, 2013

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.

Learning how to Play Nash, Potential Games and Alternating Minimization Method for Structured Nonconvex Problems on Riemannian ManifoldsJournal articleJoao Xavier Cru Neto, Paulo Roberto Oliveira, Soares Jr A. Pedro and Antoine Soubeyran, Journal of Convex Analysis, Volume 20, Issue 2, pp. 395-438, 2013

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).

Water Shortages and ConflictJournal articleAntoine Soubeyran and Agnès Tomini, Revue d'économie politique, Volume 122, Issue 2, pp. 279-297, 2012

This study analyzes the risk of a conflict between countries sharing freshwater. While some scholars claim that water-based conflicts can never occur, this analysis identifies a negotiation interval whose size depends on water availability and asymmetry in productive ability between countries. This interval is assimilated to the probability-toconflict which is decreasing with its size. We show that the risk of conflict increases with scarcer water resources but, as well, the higher the asymmetry level, the higher is the probability-to-conflict. Whenever this heterogeneity is extremely large, there is no opportunity for cooperation. Then, given the existence of this negotiation interval, we turn to the Nashbargaining solution to highlight the optimal water allocation. We show that the amount of water allocated to a country is decreasing as long its productive ability increases.

A proximal algorithm with quasi distance. Application to habit's formationJournal articleAntoine Soubeyran, Felipe Garcia Moreno and Paulo Roberto Oliveira, Optimization, Volume 61, Issue 12, pp. 1383-1403, 2012
Maximal Elements Under Reference-Dependent Preferences with Applications to Behavioral Traps and GamesJournal articleAntoine Soubeyran, Fabián Flores Bazán and Dinh The Luc, Journal of Optimization Theory and Applications, Volume 155, Issue 3, pp. 883-901, 2012