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

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, aEuro broken vertical bar ). 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.