Dual Descent Methods as Tension Reduction Systems

In this paper, driven by applications in Behavioral Sciences, wherein the speed of convergence matters considerably, we compare the speed of convergence of two descent methods for functions that satisfy the well-known Kurdyka–Lojasiewicz property in a quasi-metric space. This includes the extensions to a quasi-metric space of both the primal and dual descent methods. While the primal descent method requires the current step to be more or less half of the size of the previous step, the dual approach considers more or less half of the previous decrease in the objective function to be minimized. We provide applications to the famous “Tension systems approach” in Psychology.

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

[2]  Boris S. Mordukhovich,et al.  Variational Analysis in Psychological Modeling , 2015, J. Optim. Theory Appl..

[3]  Antoine Soubeyran,et al.  A proximal algorithm with quasi distance. Application to habit's formation , 2012 .

[4]  M. Smith Field Theory in Social Science: Selected Theoretical Papers. , 1951 .

[5]  K. Kurdyka,et al.  Proof of the gradient conjecture of R. Thom , 1999, math/9906212.

[6]  Antoine Soubeyran,et al.  Generalized inexact proximal algorithms : habit ’ s formation with resistance to change , following worthwhile changes , 2014 .

[7]  G. Bouza Allende,et al.  A Steepest Descent-Like Method for Variable Order Vector Optimization Problems , 2014 .

[8]  Jérôme Malick,et al.  Descentwise inexact proximal algorithms for smooth optimization , 2012, Comput. Optim. Appl..

[9]  S. Hosseini Convergence of nonsmooth descent methods via Kurdyka-Lojasiewicz inequality on Riemannian manifolds , 2017 .

[10]  Benar Fux Svaiter,et al.  Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward–backward splitting, and regularized Gauss–Seidel methods , 2013, Math. Program..

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

[12]  Adrian S. Lewis,et al.  The [barred L]ojasiewicz Inequality for Nonsmooth Subanalytic Functions with Applications to Subgradient Dynamical Systems , 2006, SIAM J. Optim..

[13]  A. Soubeyran Variational rationality and the unsatised man: routines and the course pursuit between aspirations, capabilities and beliefs , 2018 .

[14]  Albert Bandura,et al.  Failures in self-regulation: Energy depletion or selective disengagement? , 1996 .

[15]  Charles S. Carver,et al.  On the structure of behavioral self-regulation. , 2000 .

[16]  B. Mordukhovich Variational analysis and generalized differentiation , 2006 .

[17]  Gabriele Oettingen,et al.  Strategies of setting and implementing goals: Mental contrasting and implementation intentions , 2010 .

[18]  G. C. Bento,et al.  A Generalized Inexact Proximal Point Method for Nonsmooth Functions that Satisfies Kurdyka Lojasiewicz Inequality , 2015 .

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

[20]  Chong Li,et al.  Mathematics 4-1-2009 Weak Sharp Minima on Riemannian Manifolds , 2014 .

[21]  Boris S. Mordukhovich,et al.  Minimal points, variational principles, and variable preferences in set optimization , 2015 .

[23]  João X. da Cruz Neto,et al.  A New Approach to the Proximal Point Method: Convergence on General Riemannian Manifolds , 2016, J. Optim. Theory Appl..

[24]  John de Wit,et al.  Self-regulation in health behavior , 2006 .

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

[26]  L. Dries,et al.  Geometric categories and o-minimal structures , 1996 .

[27]  G. C. Bento,et al.  Finite termination of the proximal point method for convex functions on Hadamard manifolds , 2012, 1205.4763.

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