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Sur des systèmes dynamiques dissipatifs de type gradient. Applications en Optimisation.

Abstract : This work is intended to introduce and to study new gradient-like dynamical systems.
The dissipative aspects of those kind of dynamics stand at the crossroads
of many field in Analysis : Optimization, Mechanics, PDE.

A first part of the work is devoted to the construction of mappings that are able
to control gradient (or subdifferntial vector fields). They are called barrier-operators.
One of the main motivation is to derive interior descent methods. The abstract
framework that is proposed allow to recover many dynamics : projected
gradient, Riemannian methods, continuous Newton method, Lotka-Volterra based
dynamics ... Within such a setting, we may evoke several results concerning strong
viability, well-posedness, and global convergence.
Keeping in mind the fact that "good" trajectories are those that remain
in the feasible set : it is natural to pay a particular attention to hessian Riemannian
structure induced by Legendre functions. Those Riemannian manifolds enjoys
many properties, and may be characterized as "the metrics that are the more
appropriate to solve, a certain class of variational inequalities" . Another
interesting aspects, is that those type of structure has a sense in Hilbert Spaces :
it corresponds to some well-known subdifferntial formulation of some
parabolic equations arising in Thermodynamic.

The second part oif the thesis is devoted to the study of second-order in time
gradient method. It is shown that the use of Hessian-driven damping yields a
nice class of dynamical systems.
A first interest of those methods, is to give rise to non-descent methods with
convergent trajectories. Indeed, if one follows some minimization purposes,
it may be interesting to avoid local minima in order to attain a global minimum.
Document type :
Theses
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https://tel.archives-ouvertes.fr/tel-00002568
Contributor : Jérôme Bolte <>
Submitted on : Thursday, March 20, 2003 - 5:10:23 PM
Last modification on : Thursday, January 11, 2018 - 6:12:20 AM
Long-term archiving on: : Tuesday, September 11, 2012 - 8:05:45 PM

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  • HAL Id : tel-00002568, version 1

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Jérôme Bolte. Sur des systèmes dynamiques dissipatifs de type gradient. Applications en Optimisation.. Mathématiques [math]. Université Montpellier II - Sciences et Techniques du Languedoc, 2003. Français. ⟨tel-00002568⟩

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