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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
Article

Abstract

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.

Keywords

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

Mathematics Subject Classification (2010)

49J52 49M37 65K10 90C30 91E10 

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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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