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 nonsmoth 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 a extension, of the analysis due to Attouch and Bolte (Math. Program. Ser. B, 116: 5-16, 2009) or, 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. We give, in a dynamic setting, a striking application to the famous Nobel Prize Kahneman and Tversky~\cite{Tversky1979}, Tversky and Kahneman~\cite{Tversky1991} “loss aversion effect” in Psychology and Management. This application shows how the strength of resistance to change can impact the speed of formation of an habituation/routinization process.
Citation
Federal University of Goias April2014