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Journal of Convex Analysis 13 (2006), No. 2, 207--224
Copyright Heldermann Verlag 2006

Inertia and Reactivity in Decision Making as Cognitive Variational Inequalities

Hedy Attouch
I3M UMR CNRS 5149, Université Montpellier II, Place Eugène Batallion, 34095 Montpellier, France

Antoine Soubeyran
GREQAM, Université Aix-Marseille, Centre de la Vielle Charité, 13236 Marseille, France

We modelize a Decision Process of an agent where Inertia and Reactivity Aspects help to converge to Stable Routines. A decision is a move. We consider an agent who can only take incremental decisions, moving step by step on an unknown landscape, due to limited knowledge of his environment, resources, efforts, energy, money or time constraints. The agent explores around to be able to compare incremental advantages and costs of moving (changing). The agent reaches a stable routine, stops moving, and prefers to stay there, when advantages to move are lower than costs to move. Inertia is modelized by costs to move quickly (Reactivity is the learned ability to move quickly in a cheap way). We apply our "Worthwhile to Move Approach" (Attouch-Soubeyran, 2005, mimeo) to build "Cognitive Versions of Variational Inequalities" for "Second Order Dynamical Gradients Systems with Inertia" ("HBF" differential equations). In this model, we have mainly insisted on costs to move. Advantages to move (the other side of the balance which drives a "worthwhile to move decision") are detailed in Attouch-Soubeyran (2005). To save space the annex concentrates only on costs to move.

Keywords: Inertia and Reactivity, worthwhile to move process, costs to move, improving with cost of improving quickly, second order dynamical gradient process, differential inclusions, viscous and dry friction.

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