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

Cabot, Alexandre. "The steepest descent dynamical system with control. Applications to constrained minimization." ESAIM: Control, Optimisation and Calculus of Variations 10.2 (2004): 243-258. <http://eudml.org/doc/244608>.

@article{Cabot2004,
abstract = {Let $H$ be a real Hilbert space, $\Phi _1: H\rightarrow \mathbb \{R\}$ 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 [5]) 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 $\Phi _0: H\rightarrow \mathbb \{R\}$ whose critical points coincide with $S$ and a control parameter $\varepsilon :\mathbb \{R\}_+\rightarrow \mathbb \{R\}_+$ tending to zero, we consider the “Steepest Descent and Control” system\[(SDC) \qquad \dot\{x\}(t)+\nabla \Phi \_0(x(t))+\varepsilon (t)\, \nabla \Phi \_1(x(t))=0,\]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 $d(x(t), \operatorname\{argmin\}_S \Phi _1) \rightarrow 0 \quad (t\rightarrow +\infty ),$ and we give sufficient conditions under which $x(t) \rightarrow \bar\{x\}\in \,\operatorname\{argmin\}_S \Phi _1$. We end the paper by numerical experiments allowing to compare the $(SDC)$ system with the other systems already mentioned.},
author = {Cabot, Alexandre},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {dissipative dynamical system; steepest descent method; constrained optimization; convex minimization; asymptotic behaviour; non-linear oscillator},
language = {eng},
number = {2},
pages = {243-258},
publisher = {EDP-Sciences},
title = {The steepest descent dynamical system with control. Applications to constrained minimization},
url = {http://eudml.org/doc/244608},
volume = {10},
year = {2004},
}

TY - JOUR
AU - Cabot, Alexandre
TI - The steepest descent dynamical system with control. Applications to constrained minimization
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2004
PB - EDP-Sciences
VL - 10
IS - 2
SP - 243
EP - 258
AB - Let $H$ be a real Hilbert space, $\Phi _1: H\rightarrow \mathbb {R}$ 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 [5]) 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 $\Phi _0: H\rightarrow \mathbb {R}$ whose critical points coincide with $S$ and a control parameter $\varepsilon :\mathbb {R}_+\rightarrow \mathbb {R}_+$ tending to zero, we consider the “Steepest Descent and Control” system\[(SDC) \qquad \dot{x}(t)+\nabla \Phi _0(x(t))+\varepsilon (t)\, \nabla \Phi _1(x(t))=0,\]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 $d(x(t), \operatorname{argmin}_S \Phi _1) \rightarrow 0 \quad (t\rightarrow +\infty ),$ and we give sufficient conditions under which $x(t) \rightarrow \bar{x}\in \,\operatorname{argmin}_S \Phi _1$. We end the paper by numerical experiments allowing to compare the $(SDC)$ system with the other systems already mentioned.
LA - eng
KW - dissipative dynamical system; steepest descent method; constrained optimization; convex minimization; asymptotic behaviour; non-linear oscillator
UR - http://eudml.org/doc/244608
ER -