Generalized Inexact Proximal Algorithms: Routine’s Formation with Resistance to Change, Following Worthwhile Changes

Abstract

This paper shows how, in a quasi metric space, an inexact proximal algorithm with a generalized perturbation term appears to be a nice tool for Behavioral Sciences (Psychology, Economics, Management, Game theory,. . . ). More precisely, the new perturbation term represents an index of resistance to change, defined as a "curved enough" function of the quasi distance between two successive iterates.Using this behavioral point of view, the present paper shows how such a generalized inexact proximal algorithm can modelize the formation of habits and routines in a striking way. This idea comes from a recent "variational rationality approach" of human behavior which links a lot of different theories of stability (habits, routines, equilibrium, traps,. . . ) and changes (creations, innovations, learning and destructions,. . . ) in Behavioral Sciences and a lot of concepts and algorithms in Variational Analysis. In this variational context, the perturbation term represents a specific instance of the very general concept of resistance to change, which is the disutility of some inconvenients to change. Central to the analysis are the original variational concepts of "worthwhile changes" and "marginal worthwhile stays". At the behavioral level, this paper advocates that proximal algorithms are well suited to modelize the emergence of habituation/routinized human behaviors. We show when, and at which speed, a "worthwhile to change" process converges to a behavioral trap.keywords. Nonconvex optimizationKurdyka-Lojasiewicz inequalityinexact proximal algorithmshabitsroutinesworthwhile changes IntroductionThe main message of this paper is that, using the behavioral context of a recent "Variational rationality" approach of worthwhile stay and change dynamics proposed by Soubeyran [1, 2], a generalized proximal algorithm can modelize fairly well an habituation process as described in Psychology for an agent, or a routinization process, in Management Sciences, for an organization. This opens the door to a new vision of proximal algorithms. They are not only very nice mathematical tools in optimization theory, with striking computational aspects. They can also be nice tools to modelize the dynamics of human behaviors.Theories of stability and change consider successions of stays and changes. Stays refer to habits, routines, equilibrium, traps, rules, conventions,. . . . Changes represent creations, destructions, learning processes, innovations, attitudes as well as beliefs formation and revision, self regulation problems, including goal setting, goal striving and goal revision, the formation and break of habits and routines,. . . . In the interdisciplinary context which characterizes all these theories in Behavioral Sciences, the "Variational rationality approach" (see [1,2]), shows how to modelize the course of human activities as a succession of worthwhile temporary

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

“…The question becomes: when a critical point is a variational trap. This point has been examined in Bento and Soubeyran [7]. In the case of weak resistance to change (the present paper) a striking result for applications in Behavioral Sciences has been to show how the strength of resistance to change can impact the speed of convergence of a habituation/routinization process; see [8].…”

“…[7,8] have shown that generalized inexact proximal algorithms which use, at each period, a sufficient descent condition and a marginal stopping rule, represent a specific but leading prototype for the (VR) approach which allows to modelize, among many potential other applications, the formation and break of habits and routines. To save space, let us consider, in few terms, the example of hiring, firing and repeated employment routines, see [7], which represents a leading and concrete example in the vast literature on routine formation and break; see Feldman [16]. Consider an entrepreneur (leader) of an innovative firm who, at each period, can choose to hire and fire different kinds and numbers of skilled and specialized workers {1, 2, .…”

“…Inconvenients to change represent the difference I (x, y) = C(x, y) − C(x, x) ≥ 0 between costs C(x, y) to be able to change from profile x to profile y and costs C(x, x) to be able to stay with the same profile x used in the last period. See [7] for the precise modelization of costs C(x, y) to hire and to fire workers, as a quasi distance C(x, y) = q(x, y) between profiles of workers. Motivation and resistance to change functions are M(x, y) = U (A(x, y)) = [g(y) − g(x)] μ and R(x, y) = D (I (x, y)) = q(x, y) ν , μ,ν > 0, where the utility and disutility functions are U (A) = A μ and D (I ) = I ν .…”

“…These traps can represent the end of a succession of worthwhile temporary stays and changes, where, at each period, the agent can prefer to stay for a while and then prefers to change rather than to stay again, and, at the end, he prefers to stay than to change. This has been done in a second paper, Bento and Soubeyran [7]. The authors show how such a generalized inexact proximal algorithm can help to modelize, in Psychology, habit formation for an agent and, in Management Sciences, routine formation and knowledge management for an organization.…”

The Proximal Point Method for Locally Lipschitz Functions in Multiobjective Optimization with Application to the Compromise Problem

“…In the general case C(x, y) = C(y, x), and C(x, x) 0 is possible. Hence, future research will examine the case where the Euclidean norm in (8) is replaced by a "like-distance" as done for instance in Bento and Soubeyran [3] and Moreno, Oliveira, and Soubeyran [43] for scalar-valued functions. This is better adapted to applications in behavioral sciences.…”