Skip to main content

Advertisement

SpringerLink
Log in

Generalized Inexact Proximal Algorithms: Routine’s Formation with Resistance to Change, Following Worthwhile Changes

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

This paper shows how, in a quasi-metric space, an inexact proximal algorithm with a generalized perturbation term appears to be a nice tool for Behavioral Sciences (Psychology, Economics, Management, Game theory,...). More precisely, the new perturbation term represents an index of resistance to change, defined as a “curved enough” function of the quasi-distance between two successive iterates. Using this behavioral point of view, the present paper shows how such a generalized inexact proximal algorithm can modelize the formation of habits and routines in a striking way. This idea comes from a recent “variational rationality approach” of human behavior which links a lot of different theories of stability (habits, routines, equilibrium, traps,...) and changes (creations, innovations, learning and destructions,...) in Behavioral Sciences and a lot of concepts and algorithms in variational analysis.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Soubeyran, A.: Variational rationality, a theory of individual stability and change: worthwhile and ambidextry behaviors. GREQAM, Aix Marseillle University (2009, pre-print)

  2. Soubeyran, A.: Variational rationality and the “unsatisfied man”: routines and the course pursuit between aspirations, capabilities and beliefs. GREQAM, Aix Marseillle University (2010, preprint)

  3. Attouch, H., Soubeyran, A.: Local search proximal algorithms as decision dynamics with costs to move. Set-Valued Var. Anal. 19(1), 157–177 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  4. Flores-Bazán, F., Luc, D., Soubeyran, A.: Maximal elements under reference-dependent preferences with applications to behavioral traps and games. J. Optim. Theory Appl. 155, 883–901 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  5. Attouch, H., Redont, P., Soubeyran, A.: A new class of alternating proximal minimization algorithms with costs-to-move. SIAM J. Optim. 18, 1061–1081 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  6. Cruz Neto, J.X., Oliveira, P.R., Soares Jr., P.A., Soubeyran, A.: Learning how to play Nash and alternating minimization method for structured nonconvex problems on Riemannian manifolds. J. Convex Anal. 20, 395–438 (2013)

  7. Bento, G.C., Soubeyran, A.: A generalized inexact proximal point method for nonsmooth functions that satisfies Kurdyka Lojasiewicz inequality. Set-Valued Var. Anal. (2015). doi:10.1007/s11228-015-0319-6

  8. Kahneman, D., Tversky, A.: Prospect theory: an analysis of decision under risk. Econometrica 47(2), 263–291 (1979)

    Article  MATH  Google Scholar 

  9. Tversky, A., Kahneman, D.: Loss aversion in riskless choice: a reference dependent model. Q. J. Econ. 106(4), 1039–1061 (1991)

    Article  Google Scholar 

  10. Attouch, H., Bolte, J.: On the convergence of the proximal algorithm for nonsmooth functions involving analytic features. Math. Programming, Ser. B 116(1—-2), 5–16 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  11. Attouch, H., Bolte, J., Svaiter, B.F.: Convergence of descent methods for semi-algebric and tame problems: proximal algorithms, forward–backward splitting, and regularized Gauss–Seidel methods. Math. Program. Ser. A 137, 91–129 (2012)

    Article  MathSciNet  Google Scholar 

  12. Moreno, F.G., Oliveira, P.R., Soubeyran, A.: A proximal algorithm with quasi distance. Application to Habit’s formation. Optimization 61(12), 1383–1403 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  13. Bento, G.C., Soubeyran, A.: Some Comparisons Between the Variational Rationality, Habitual Domain, and DMCS Approaches. arXiv:submit/1077264 (2014)

  14. Yu, P.L.: Forming Winning Strategies, An Integrated Theory of Habitual Domains. Springer, Berlin (1990)

    Book  Google Scholar 

  15. Larbani, M., Yu, P.L.: Decision making and optimization in changeable spaces, a new paradigm. J. Optim. Theory Appl. 155(3), 727–761 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  16. Simon, H.: A behavioral model of rational choice. Q. J. Econ 69, 99–118 (1955)

    Article  Google Scholar 

  17. Van Long, N., Soubeyran, A., Soubeyran, R.: Knowledge accumulation within an organization. Int. Econ. Rev. 55(4), 1089–1128 (2014)

  18. Kremer, M.: The O-ring theory of economic development. Q. J. Econ. 108, 551–575 (1993)

    Article  Google Scholar 

  19. Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  20. Moreau, J.: Proximité et dualité dans un espace hilbertien. (French) Bull. Soc. Math. 93, 273–299 (1965)

    MATH  MathSciNet  Google Scholar 

  21. Martinet, B.: Régularisation, d’inéquations variationelles par approximations successives. (French) Rev. Française Informat. Recherche Opérationnelle 4(Ser. R–3), 154–158 (1970)

    MATH  MathSciNet  Google Scholar 

  22. Rockafellar, R.T., Wets, R.: Variational Analysis, 317 of Grundlehren der Mathematischen Wissenschafte. Springer, Berlin (1998)

    Google Scholar 

  23. Zaslavski, A.J.: Inexact proximal point methods in metric spaces. Set-Valued Anal. 19, 589–608 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  24. Lojasiewicz, S.: Une propriété topologique des sous-ensembles analytiques réels. Les Équations aux Dérivées Partielles, Éditions du centre National de la Recherche Scientifique, 117, 87–89 (1963)

  25. Kurdyka, K.: On gradients of functions definable in o-minimal structures. Ann. Inst. Fourier 48, 769–783 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  26. Attouch, H., Redont, P., Bolte, J., Soubeyran, A.: Proximal alternating minimization and projection methods for nonconvex problems. An approach based on the Kurdyka–Lojasiewicz inequality. Math. Oper. Res. 35(2), 438–457 (2010)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

The work was supported by CAPES, CAPES-MES-CUBA 226/2012, FAPEG 201210267000909–05/2012 and CNPq Grants 458479/2014-4, 471815/2012-8, 303732/2011-3, 236938/2012-6.

Author information

Authors and Affiliations

  1. Universidade Federal de Goiás, Goiânia, Brazil

    G. C. Bento

  2. CNRS & EHESS, Aix-Marseille University (Aix-Marseille School of Economics), Marseille, France

    A. Soubeyran

Authors
  1. G. C. Bento
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Soubeyran
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to G. C. Bento.

Additional information

Communicated by Po-Lung Yu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bento, G.C., Soubeyran, A. Generalized Inexact Proximal Algorithms: Routine’s Formation with Resistance to Change, Following Worthwhile Changes. J Optim Theory Appl 166, 172–187 (2015). https://doi.org/10.1007/s10957-015-0711-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-015-0711-2

Keywords

Mathematics Subject Classification

Search

Navigation