Set-Valued and Variational Analysis

, Volume 23, Issue 3, pp 501–517 | Cite as

A Generalized Inexact Proximal Point Method for Nonsmooth Functions that Satisfies Kurdyka Lojasiewicz Inequality

  • G. C. Bento
  • A. Soubeyran


In this paper, following the ideas presented in Attouch et al. Math. Program. Ser. A, 137: 91–129, (2013), we present an inexact version of the proximal point method for nonsmooth functions, whose regularization is given by a generalized perturbation term. More precisely, the new perturbation term is defined as a “curved enough” function of the quasi distance between two successive iterates, that appears to be a nice tool for Behavioral Sciences (Psychology, Economics, Management, Game theory, … ). Our convergence analysis is an extension, of the analysis due to Attouch and Bolte Math. Program. Ser. B, 116: 5–16, (2009) and, more generally, to Moreno et al. Optimization, 61:1383–1403, (2011), to an inexact setting of the proximal method which is more suitable from the point of view of applications. In a dynamic setting, (Bento and Soubeyran (2014)) present a striking application on the famous Nobel Prize (Kahneman and Tversky. Econometrica 47(2), 263–291 (1979); Tversky and Kahneman. Q. J. Econ. 106(4), 1039–1061 (38))“loss aversion effect” in Psychology and Management. This application shows how the strength of resistance to change can impact the speed of formation of a habituation/routinization process.


Nonconvex optimization Kurdyka-Lojasiewicz inequality Inexact proximal algorithms Speed of convergence 

Mathematics Subject Classification (2010)

49J52 49M37 65K10 90C30 91E10 


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  1. 1.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    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 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Attouch, H., Redont, P., Soubeyran, A.: A new class of alternating proximal minimization algorithms with costs-to-move. SIAM J. Optimiz. 18, 1061–1081 (2007)MathSciNetCrossRefzbMATHGoogle 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.
    Bento, C.G., Cruz Neto, J.X., Oliveira, P.R. Convergence of inexact descent methods for nonconvex optimization on Riemannian manifolds (2011). arXiv:1103.4828
  7. 7.
    Bento, G.C., Soubeyran, A.: Generalized inexact proximal algorithms: habit’s/ routine’s formation with resistance to change, following worthwhile changes. J. Optim. Theory Appl. (2015). doi: 10.1007/s10957-015-0711-2
  8. 8.
    Bento, G.C., Soubeyran, A.: A generalized inexact proximal point method for nonsmooth functions that satisfies Kurdyka Lojasiewicz inequality (2014).
  9. 9.
    Bolte, J., Daniilidis, J.A., Lewis, A.: The Lojasiewicz inequality for nonsmooth subanalytic functions with applications to subgradient dynamical systems. SIAM J. Optim. 17(4), 1205–1223 (2006)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Bolte, J., Daniilidis, A., Ley, O., Mazet, L.: Characterizations of Lojasiewicz inequalities: Subgradient flows, talweg, convexity. Trans. Amer. Math. Soc. 362, 3319–3363 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Bolte, J., Daniilidis, J.A., Lewis, A., Shiota, M.: Clarke subgradients of stratifiable functions. SIAM J. Optim. 18(2), 556–572 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Bot, R.I., Csetnek, E.R.: An inertial Tsengs type proximal algorithm for nonsmooth and nonconvex optimization problems. arXiv:1406.0724v1 (2014)
  13. 13.
    Cruz Neto, J.X., Oliveira, P.R., Soares, P.A., Soubeyran, A.: Learning how to play Nash, potential games and alternating minimization method for structured nonconvex problems on Riemannian manifolds. J. Convex Anal. 20, 395–438 (2013)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Combettes, P.L., Pennanen, T.: Proximal methods for cohypomonotone operators. SIAM J. Control. Optim. 43, 731–742 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    van den Dries, L., Miller, C.: Geometric categories and o-minimal structures. Duke Math. J. 84, 497–540 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Feldman, M.: Organizational routines as a source of continuous change. Organ. Sci. 11(6), 611–629 (2000)CrossRefGoogle Scholar
  17. 17.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Frankel, P., Garrigos, G., Peypouquet, J.: Splitting methods with variable metric for KL functions. arXiv:1405.1357 (2014)
  19. 19.
    Fukushima, M., Mine, H.: A generalized proximal point algorithm for certain nonconvex minimization problems. Int. J. Syst. Sci. 12, 989–1000 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Gárciga-Otero, R., Iusem, A.N.: Proximal methods in reflexive Banach spaces without monotonicity. J. Math. Anal. Appl. 330(1), 433–450 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Iusem, A.N., Penannen, T., Svaiter, B.F.: Inexact variants of the proximal point algorithm without monotonicity. SIAM J. Optim. 13(4), 1080–1097 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Kahneman, D., Tversky, A.: Prospect theory: An analysis of decision under risk. Econometrica 47(2), 263–291 (1979)CrossRefzbMATHGoogle Scholar
  23. 23.
    Kaplan, A., Tichatschke, R.: Proximal point methods and nonconvex optimization. J. Global Optim 13(4), 389–406 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Kurdyka, K.: On gradients of functions definable in o-minimal structures. Ann. Inst. Fourier 48, 769–783 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Kurdyka, K., Mostowski, T., Parusinski, A.: Proof of the gradient conjecture of R. Thom. Ann. Math 152, 763–792 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Lageman, C.: Convergence of gradient-like dynamical systems and optimization algorithms, Ph.D., Thesis, Universität Wü rzburg (2007)Google Scholar
  27. 27.
    Lojasiewicz, S.: Une propriété topologique des sous-ensembles analytiques réels. Les Équations aux Dérivées Partielles, Éditions du centre National de la Recherche Scientifique. 87–89 (1963)Google Scholar
  28. 28.
    Martinet, B.: Régularisation, d’inéquations variationelles par approximations successives. (French) Rev. Française Informat. Recherche Opérationnelle 4(Ser. R-3), 154–158 (1970)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Mordukhovich, B.: Variational analysis and generalized differentiation I: basic theory (Grundlehren der mathematischen Wissenschaften (2010)Google Scholar
  30. 30.
    Moreau, J.: Proximité et dualité dans un espace hilbertien. (French). Bull. Soc. Math 93, 273–299 (1965)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Moreno, F.G., Oliveira, P.R., Soubeyran, A.: A proximal algorithm with quasi distance. Application to habit’s formation. Optimization 61(12), 1383–1403 (2011)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Pennanen, T.: Local convergence of the proximal point algorithm and multiplier methods without monotonicity. Math. Oper. Res. 27, 170–191 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control. Optim. 14, 877–898 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Rockafellar, R.T., Wets, R.: Variational Analysis, 317 of Grundlehren der Mathematischen Wissenschafte. Springer, Berlin (1998)Google Scholar
  35. 35.
    Solodov, M.V., Svaiter, B.F.: A hybrid approximate extragradient - proximal algorithm using the enlargement of a maximal monotone operator. Set-Valued Var Anal 7, 323–345 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Solodov, M.V., Svaiter, B.F.: Forcing strong convergence of proximal point iterations in a Hilbert space. Math. Program. Ser. A 87, 189–202 (2000)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Spingarn, J.E.: Submonotone mappings and the proximal point algorithm. Numer. Funct. Anal. Optim. 4(2), 123–150 (1981)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Tversky, A., Kahneman, D.: Loss aversion in riskless choice: a reference dependent model. Q. J. Econ. 106(4), 1039–1061 (1991)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Universidade Federal de GoiásGoiâniaBrazil
  2. 2.Aix-Marseille University (Aix-Marseille School of Economics)CNRS and EHESSMarseilleFrance

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