Equilibrium versions of variational principles in quasi-metric spaces and the robust trap problemJournal articleJing-Hui Qiu, He Fei and Antoine Soubeyran, Optimization, Volume 67, Issue 1, pp. 25-53, 2018

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.

Inexact Multi-Objective Local Search Proximal Algorithms: Application to Group Dynamic and Distributive Justice ProblemsJournal articleGlaydston de Carvalh Bento, Orizon Pereira Ferreira, Antoine Soubeyran and Valdinês Leite de S. Júnior, Journal of Optimization Theory and Applications, Volume 177, Issue 1, pp. 181-200, 2018

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.

Variational principles, completeness and the existence of traps in behavioral sciencesJournal articleTruong Q. Bao, S. Cobzaş and Antoine Soubeyran, Annals of Operations Research, Volume 269, Issue 1, pp. 53-79, 2018

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.

Generalized Proximal Distances for Bilevel Equilibrium ProblemsJournal articleGlaydston Carvalho Bento, J.X. Cruz Neto, J. Lopes, P. Soares, Jr and Antoine Soubeyran, SIAM Journal on Optimization, Volume 26, Issue 1, pp. 810-830, 2016

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.

Variational Analysis in Cone Pseudo-Quasimetric Spaces and Applications to Group DynamicsJournal articleTruong Q. Bao and Antoine Soubeyran, Journal of Optimization Theory and Applications, Volume 170, Issue 2, pp. 458-475, 2016

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.

Global convergence of a proximal linearized algorithm for difference of convex functionsJournal articleJoão Carlos O. Souza, Paulo Roberto Oliveira and Antoine Soubeyran, Optimization Letters, Volume 10, Issue 7, pp. 1529-1539, 2016

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.

Dual Descent Methods as Tension Reduction SystemsJournal articleGlaydston de Carvalh Bento, João Xavier da Neto, Antoine Soubeyran and Valdinês Leite de S. Júnior, Journal of Optimization Theory and Applications, Volume 171, Issue 1, pp. 209-227, 2016

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.

Convergence in a sequential two stages decision making processJournal articleJuan-Enrique Martinez-Legaz and Antoine Soubeyran, Bulletin of the Iranian Mathematical Society, Volume 42, Issue 7, pp. 25-29, 2016

We analyze a sequential decision making process, in which at each stepthe decision is made in two stages. In the rst stage a partially optimalaction is chosen, which allows the decision maker to learn how to improveit under the new environment. We show how inertia (cost of changing)may lead the process to converge to a routine where no further changesare made. We illustrate our scheme with some economic models.

Variational principles with generalized distances and the modelization of organizational changeJournal articleTruong Q. Bao, Phan Q. Khanh and Antoine Soubeyran, Optimization, Volume 65, Issue 12, pp. 2049-2066, 2016

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.

Minimal points, variational principles, and variable preferences in set optimizationJournal articleTruong Q. Bao, Boris S. Mordukhovich and Antoine Soubeyran, Journal of Nonlinear and Convex Analysis, Volume 16, Issue 8, pp. 1511-1537, 2015

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...