Skip to main content

Advertisement

Springer Nature Link
Log in

Proximal Point Method on Finslerian Manifolds and the “Effort–Accuracy” Trade-off

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

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.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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. Payne, J.W., Bettman, J.R., Johnson, E.J.: The Adaptive Decision Maker. Cambridge University Press, Cambridge (1993)

    Book  Google Scholar 

  2. Rinkenauer, G., Osman, A., Ulrich, R., Müller-Gethmann, H., Mattes, S.: On the locus of speed–accuracy trade-off in reaction time: inferences from the lateralized readiness potential. J. Exp. Psychol. Gen. 133(2), 261–282 (2004)

    Article  Google Scholar 

  3. Holland, J.H.: Adaptation in Natural and Artificial Systems. University of Michigan Press, Ann Arbor (1975)

    Google Scholar 

  4. March, J.G.: Exploration and exploitation in organizational learning. Organ. Sci. 2(1), 71–87 (1991)

    Article  MathSciNet  Google Scholar 

  5. Levitt, B., March, J.G.: Organizational learning. Annu. Rev. Sociol. 14, 319–338 (1988)

    Article  Google Scholar 

  6. Cohen, W.M., Levinthal, D.A.: Absorptive capacity: a new perspective on learning and innovation. Adm. Sci. Q. 35(1), 128–152 (1990)

    Article  Google Scholar 

  7. Levinthal, D.A., March, J.G.: The myopia of learning. Strateg. Manag. J. 14, 95–112 (1993)

    Article  Google Scholar 

  8. Fu, W.T., Gray, W.D.: Suboptimal tradeoffs in information seeking. Cogn. Psychol. 52(3), 195–242 (2006)

    Article  Google Scholar 

  9. Fu, W.T.: A Rational-Ecological Approach to the Exploration/Exploitation Trade-Offs: Bounded Rationality and Suboptimal Performance. Oxford University Press, Oxford (2007)

    Google Scholar 

  10. Busemeyer, J.R., Townsend, J.T.: Decision field theory: a dynamic cognition approach to decision making. Psychol. Rev. 100(3), 432–459 (1993)

    Article  Google Scholar 

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

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

  13. Attouch, H., Soubeyran, A.: Inertia and reactivity in decision making as cognitive variational inequalities. J. Convex Anal. 13(2), 207–224 (2006)

    MATH  MathSciNet  Google Scholar 

  14. 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 

  15. Matsumoto, M.: A slope of a mountain is a Finsler surface with respect to a time measure. J. Math. Kyoto Univ. 29(1), 17–25 (1989)

    MATH  MathSciNet  Google Scholar 

  16. Bao, D., Chern, S.S., Shen, Z.: An Introduction to Riemann–Finsler Geometry. Graduate Texts in Mathematics, vol. 200. Springer, Berlin (2000)

    Book  MATH  Google Scholar 

  17. Kristály, A., Morosanu, G., Róth, Á.: Optimal Placement of a Deposit between Markets: Riemann–Finsler Geometrical Approach. J. Optim. Theory Appl. 139(2), 263–276 (2008)

    Article  MathSciNet  Google Scholar 

  18. Ferreira, O.P., Oliveira, P.R.: Proximal point algorithm on Riemannian manifold. Optimization 51(2), 257–270 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  19. Li, C., Lopez, G., Martin-Marquez, V.: Monotone vector fields and the proximal point algorithm on Hadamard manifolds. J. Lond. Math. Soc. 79(2), 663–683 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  20. Papa Quiroz, E.A., Oliveira, P.R.: Proximal point methods for quasiconvex and convex functions with Bregman distances on Hadamard manifolds. J. Convex Anal. 16(1), 49–69 (2009)

    MATH  MathSciNet  Google Scholar 

  21. Bento, G.C., Ferreira, O.P., Oliveira, P.R.: Local convergence of the proximal point method for a special class of nonconvex functions on Hadamard manifolds. Nonlinear Anal. 73, 564–572 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  22. Kurdyka, K.: On gradients of functions definable in o-minimal structures. Ann. Inst. Fourier 48(3), 769–783 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  23. Bolte, J., Daniilidis, A., Lewis, A.: The Lojasiewicz inequality for nonsmooth subanalytic functions with applications to subgradient dynamical systems. SIAM J. Optim. 17(4), 1205–1223 (2006)

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  25. Bolte, J., Daniilidis, A., Lewis, A., Shiota, M.: Clarke subgradients of stratifiable functions. SIAM J. Optim. 18(2), 556–572 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  26. Lageman, C.: Convergence of gradient-like dynamical systems and optimization algorithms. Dissertation zur Erlangung des naturwissenschaftlichen Doktorgrades der Bayerischen Julius-Maximilians-Universitat Wurzburg (2007)

  27. Shen, Z.: Lectures on Finsler Geometry. World Scientific, Singapore (2001)

    Book  MATH  Google Scholar 

  28. van den Dries, L., Miller, C.: Geometric categories and o-minimal structures. Duke Math. J. 84(2), 497–540 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  29. Bierstone, E., Milman, P.D.: Semianalytic and subanalytic sets. Publ. Math. Paris 67, 5–42 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  30. van den Dries, L.: O-minimal structures and real analytic geometry. In: Current Developments in Mathematics, Cambridge, MA, 1998, pp. 105–152. Int. Press, Somerville (1999)

    Google Scholar 

  31. Attouch, H., Bolte, J., Redont, P., 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 

  32. Cruz Neto, J.X., Oliveira, P.R., Soares, P.A. Jr, Soubeyran, A.: Learning how to Play Nash, Potential Games and Alternating Minimization Method for Structured Nonconvex Problems on Riemannian Manifolds. J. Convex Anal. 20(2), 395–438 (2013)

    MATH  MathSciNet  Google Scholar 

  33. Moreno, F., Oliveira, P., Soubeyran, A.: A proximal algorithm with quasi distance. Application to habit’s formation. Optimization 61(2), 1383–1403 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  34. Simon, H.A.: A behavioral model of rational choice. Q. J. Econ. 69, 99–188 (1955)

    Article  Google Scholar 

  35. Simon, H.A.: Rationality in psychology and economics. Part 2: The behavioral foundations of economic theory. J. Bus. 59(4), S209–S224 (1986)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. CCN, DM, Universidade Federal do Piauí, Terezina, PI, 64049-550, Brazil

    J. X. Cruz Neto

  2. COPPE-Sistemas, Universidade Federal do Rio de Janeiro, Rio de Janeiro, RJ, 21945-970, Brazil

    P. R. Oliveira

  3. CCN, DM, Universidade Etadual do Piauí, Terezina, PI, 64002-150, Brazil

    P. A. Soares Jr.

  4. Aix-Marseille School of Economics, Aix-Marseille University, CNRS & EHESS, Marseille, France

    A. Soubeyran

Authors
  1. J. X. Cruz Neto
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. P. R. Oliveira
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. P. A. Soares Jr.
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. A. Soubeyran
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. X. Cruz Neto.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cruz Neto, J.X., Oliveira, P.R., Soares, P.A. et al. Proximal Point Method on Finslerian Manifolds and the “Effort–Accuracy” Trade-off. J Optim Theory Appl 162, 873–891 (2014). https://doi.org/10.1007/s10957-013-0483-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-013-0483-5

Keywords