A proximal algorithm with quasi distance. Application to habit's formation

We consider a proximal algorithm with quasi distance applied to nonconvex and nonsmooth functions involving analytic properties for a minimization problem. We show the behavioural importance of this proximal point model for the formation of habit in Decision and Making Sciences. The convergence of the sequence generated by our algorithm to a critical-limit point is guaranteed under standard conditions of coercivity and by using the Kurdyka–Łojasiewicz inequality. We present a definition of habit that contemplates that kind of convergence.

[1]  Kevin M. Murphy,et al.  A Theory of Rational Addiction , 1988, Journal of Political Economy.

[2]  K. Kurdyka.  On gradients of functions definable in o-minimal structures , 1998 .

[3]  Aleksandar Stojmirovic.  Quasi-metric spaces with measure , 2003 .

[4]  Jeffrey C. Fuhrer.  Habit Formation in Consumption and Its Implications for Monetary Policy Models , 2000 .

[5]  Bastian Goldlücke,et al.  Variational Analysis , 2014, Computer Vision, A Reference Guide.

[6]  B. Mordukhovich,et al.  Nonsmooth sequential analysis in Asplund spaces , 1996 .

[7]  H. Attouch,et al.  Local Search Proximal Algorithms as Decision Dynamics with Costs to Move , 2011 .

[8]  B. Martinet.  Brève communication. Régularisation d'inéquations variationnelles par approximations successives , 1970 .

[9]  Hans-Peter A. Künzi,et al.  Partial quasi-metrics , 2006, Theor. Comput. Sci..

[10]  B. Martinet,et al.  R'egularisation d''in'equations variationnelles par approximations successives , 1970 .

[11]  Antoine Soubeyran,et al.  Existence of solutions in variational relation problems without convexity , 2010 .

[12]  S. Łojasiewicz.  Ensembles semi-analytiques , 1965 .

[13]  Hédy Attouch,et al.  Proximal Alternating Minimization and Projection Methods for Nonconvex Problems: An Approach Based on the Kurdyka-Lojasiewicz Inequality , 2008, Math. Oper. Res..

[14]  K. Kiwiel.  Proximal Minimization Methods with Generalized Bregman Functions , 1997 .

[15]  J. Bolte,et al.  Characterizations of Lojasiewicz inequalities and applications , 2008, 0802.0826.

[16]  Adrian S. Lewis,et al.  Clarke Subgradients of Stratifiable Functions , 2006, SIAM J. Optim..

[17]  A. Tversky,et al.  Prospect theory: an analysis of decision under risk — Source link , 2007 .

[18]  Salvador Romaguera,et al.  Applications of utility functions defined on quasi-metric spaces , 2003 .

[19]  Alexander Kaplan,et al.  Proximal Point Methods and Nonconvex Optimization , 1998, J. Glob. Optim..

[20]  J. van Leeuwen,et al.  Theoretical Computer Science , 2003, Lecture Notes in Computer Science.

[21]  Robert E. Mahony,et al.  Convergence of the Iterates of Descent Methods for Analytic Cost Functions , 2005, SIAM J. Optim..

[22]  H. Simon.  Method and Appraisal in Economics: From substantive to procedural rationality , 1976 .

[23]  A. Tversky,et al.  Prospect theory: analysis of decision under risk , 1979 .

[25]  Vasco Brattka,et al.  Recursive quasi-metric spaces , 2003, Theor. Comput. Sci..

[26]  Heinz H. Bauschke,et al.  Joint minimization with alternating Bregman proximity operators , 2005 .

[27]  Giangiacomo Gerla,et al.  Quasi-metric spaces and point-free geometry , 2006, Mathematical Structures in Computer Science.

[28]  Gary S. Becker,et al.  Habits, Addictions, and Traditions , 1992 .

[29]  H. Simon,et al.  From substantive to procedural rationality , 1976 .

[30]  H. Attouch,et al.  Worthwhile-to-move behaviors as temporary satisficing without too many sacrificing processes , 2009, 0905.1238.

[31]  Hédy Attouch,et al.  On the convergence of the proximal algorithm for nonsmooth functions involving analytic features , 2008, Math. Program..

[32]  J. Bolte,et al.  Characterizations of Lojasiewicz inequalities: Subgradient flows, talweg, convexity , 2009 .

[33]  R. Rockafellar.  Monotone Operators and the Proximal Point Algorithm , 1976 .

[34]  Talia N. Lerner.  Habits , 1902, The Hospital.