Inexact Multi-Objective Local Search Proximal Algorithms: Application to Group Dynamic and Distributive Justice Problems

  • Glaydston de Carvalho Bento
  • Orizon Pereira Ferreira
  • Antoine Soubeyran
  • Valdinês Leite de Sousa Júnior


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.


Multi-objective Inexact proximal Group dynamic Distributive justice Behavioral sciences Variational rationality 

Mathematics Subject Classification

90C29 90C30 49M30 



The work was supported by CAPES, CNPq, MathAmSud (CAPES) 88881.117595/2016-01 and the ANR GREEN-Econ research project (ANR-16-CE03-0005).


  1. 1.
    Mordukhovich, B.: Variational Analysis and Generalized Differentiation. I. Basic Theory. Grundlehren der Mathematischen Wissenschaften, vol. 330. Springer, Berlin (2006)Google Scholar
  2. 2.
    Bonnel, H., Iusem, A.N., Svaiter, B.F.: Proximal methods in vector optimization. SIAM J. Optim. 15(4), 953–970 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bento, G.C., Ferreira, O.P., Sousa Junior, V.L.: Proximal point method for a special class of nonconvex multiobjective optimization functions. Optim. Lett. 12(2), 311–320 (2018)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Aarts, E., Lenstra, K.: Local Search. Princeton University Press, Princeton (2003)zbMATHGoogle Scholar
  5. 5.
    Attouch, H., Soubeyran, A.: Local search proximal algorithms as decision dynamics with costs to move. Set Valued Var. Anal. 19(1), 157–177 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Soubeyran, A.: Variational rationality, a theory of individual stability and change: worthwhile and ambidextry behaviors. GREQAM, Aix-Marseille University, France (2009, Pre-print)Google Scholar
  7. 7.
    Soubeyran, A.: Variational rationality and the “unsatisfied man”: routines and the course pursuit between aspirations, beliefs. GREQAM, Aix-Marseille University, France (2010, Pre-print)Google Scholar
  8. 8.
    Soubeyran, A: Variational rationality. Worthwhile stay and change approach-avoidance human dynamics ending in traps. GREQAM, Aix-Marseille University, France (2016, Pre-print)Google Scholar
  9. 9.
    Hiriart-Urruty, J.B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms, Published in Two Volumes. Springer, Berlin (1993)zbMATHGoogle Scholar
  10. 10.
    Clarke, F.H.: Optimization and Nonsmooth Analysis Volume 5 of Classics in Applied Mathematics, vol. 2. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1990)CrossRefGoogle Scholar
  11. 11.
    Auslender, A., Teboulle, M.: Interior gradient and proximal methods for convex and conic optimization. SIAM J. Optim. 16(3), 697–725 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Burachik, R., Dutta, J.: Inexact proximal point methods for variational inequality Problems. SIAM J. Optim. 20(5), 2653–2653 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Custódio, A.L., Madeira, J.F.A., Vaz, A.I.F., Vicente, L.N.: Direct multisearch for multiobjective optimization. SIAM J. Optim. 21(3), 1109–1140 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Chen, Z., Huang, X.X., Yang, X.Q.: Generalized proximal point algorithms for multi-objective optimization problems. Appl. Anal. 90(6), 935–949 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Polyak, B.T.: Introduction to Optimization. Optimization Software, New York (1987)zbMATHGoogle Scholar
  16. 16.
    Rockafellar, R.T., Wets, R.: Variational Analysis. Grundlehren der Mathematischen Wissenschafte. Springer, New York (1998)CrossRefzbMATHGoogle Scholar
  17. 17.
    Rockafellar, R.T.: Convex Analysis. Princeton Mathematical Series, vol. 28. Princeton University Press, Princeton (1970)CrossRefzbMATHGoogle Scholar
  18. 18.
    Arrow, K.J.: Social Choice and Individual Values. Yale University Press, New Haven (1951)zbMATHGoogle Scholar
  19. 19.
    Rawls, J.: A Theory of Justice. Harvard University Press, Cambridge (1971)Google Scholar
  20. 20.
    Lewin, K.: A Dynamic Theory of Personality. McGraw-Hill, New York (1935)Google Scholar
  21. 21.
    Lewis, K.: Field theory in Social Science. Harper, New York (1951)Google Scholar
  22. 22.
    Townsend, J.T., Busemeyer, J.R.: Approach-avoidance: Return to dynamic decision behavior. The Tulane Flowerree Symposia on Cognition, Psychology Press, In: Current Issues in Cognitive Processes (2014)Google Scholar
  23. 23.
    Elliot, A.J.: The hierarchical model of approach-avoidance motivation. Motiv. Emot. 30(2), 111–116 (2006)CrossRefGoogle Scholar
  24. 24.
    Bento, G.C., Cruz Neto, J.X., Soubeyran, A., Sousa Junior, V.L.: Dual descent methods as tension reduction systems. J. Optim. Theory Appl. 171(1), 209–277 (2016)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Glaydston de Carvalho Bento
    • 1
  • Orizon Pereira Ferreira
    • 1
  • Antoine Soubeyran
    • 2
  • Valdinês Leite de Sousa Júnior
    • 1
  1. 1.IMEUniversidade Federal de GoiásGoiâniaBrazil
  2. 2.CNRS & EHESS, Aix-Marseille School of EconomicsAix-Marseille UniversityMarseilleFrance

Personalised recommendations