The steepest descent dynamical system with control. Applications to constrained minimization

Free access article

Issue		ESAIM: COCV Volume 10, Number 2, April 2004


Page(s)		243 - 258
DOI		10.1051/cocv:2004005

ESAIM: COCV, April 2004, Vol. 10, pp. 243-258
DOI: 10.1051/cocv:2004005

The steepest descent dynamical system with control. Applications to constrained minimization

Alexandre Cabot

Laboratoire LACO, Faculté des Sciences, 123 avenue Albert Thomas, 87060 Limoges Cedex, France; alexandre.cabot@unilim.fr.

(Received February 17, 2003.)

Abstract
Let H be a real Hilbert space, $\Phi_1: H\to \xR$ a convex function of class ${\mathcal C}^1$ that we wish to minimize under the convex constraint S. A classical approach consists in following the trajectories of the generalized steepest descent system (cf. Brézis [CITE]) applied to the non-smooth function $\Phi_1+\delta_S$ . Following Antipin [1], it is also possible to use a continuous gradient-projection system. We propose here an alternative method as follows: given a smooth convex function $\Phi_0: H\to \xR$ whose critical points coincide with S and a control parameter $\varepsilon:\xR_+\to \xR_+$ tending to zero, we consider the "Steepest Descent and Control" system

$\begin{displaymath}(SDC) \qquad \dot{x}(t)+\nabla \Phi_0(x(t))+\varepsilon(t)\, \nabla \Phi_1(x(t))=0,\end{displaymath}$

where the control $\varepsilon$ satisfies $\int_0^{+\infty} \varepsilon(t)\, {\rm d}t =+\infty$ . This last condition ensures that $\varepsilon$ "slowly" tends to 0. When H is finite dimensional, we then prove that $d(x(t), {\rm argmin}\kern 0.12em_S \Phi_1) \to 0 \quad (t\to +\infty),$ and we give sufficient conditions under which $x(t) \to \bar{x}\in \,{\rm argmin}\kern 0.12em_S \Phi_1$ . We end the paper by numerical experiments allowing to compare the (SDC) system with the other systems already mentioned.

Mathematics Subject Classification. 34A12, 34D05, 34G20, 34H05, 37N40.

Key words: Dissipative dynamical system, steepest descent method, constrained optimization, convex minimization, asymptotic behaviour, non-linear oscillator.

© EDP Sciences, SMAI 2004

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