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

• Volume: 10, Issue: 2, page 243-258
• ISSN: 1292-8119

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## Abstract

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Let H be a real Hilbert space, ${\Phi }_{1}:H\to$ a convex function of class ${𝒞}^{1}$ 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 ${\Phi }_{0}:H\to$ whose critical points coincide with S and a control parameter $\epsilon {:}_{+}{\to }_{+}$ tending to zero, we consider the “Steepest Descent and Control” system $\left(SDC\right)\phantom{\rule{2.0em}{0ex}}\stackrel{˙}{x}\left(t\right)+\nabla {\Phi }_{0}\left(x\left(t\right)\right)+\epsilon \left(t\right)\phantom{\rule{0.166667em}{0ex}}\nabla {\Phi }_{1}\left(x\left(t\right)\right)=0,$ where the control ε satisfies ${\int }_{0}^{+\infty }\epsilon \left(t\right)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}t=+\infty$. This last condition ensures that ε “slowly” tends to 0. When H is finite dimensional, we then prove that $d\left(x\left(t\right),\mathrm{argmin}{\phantom{\rule{1.19995pt}{0ex}}}_{S}{\Phi }_{1}\right)\to 0\phantom{\rule{1.0em}{0ex}}\left(t\to +\infty \right),$ and we give sufficient conditions under which $x\left(t\right)\to \overline{x}\in \phantom{\rule{0.166667em}{0ex}}\mathrm{argmin}{\phantom{\rule{1.19995pt}{0ex}}}_{S}{\Phi }_{1}$. We end the paper by numerical experiments allowing to compare the (SDC) system with the other systems already mentioned.

## How to cite

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Cabot, Alexandre. "The steepest descent dynamical system with control. Applications to constrained minimization." ESAIM: Control, Optimisation and Calculus of Variations 10.2 (2010): 243-258. <http://eudml.org/doc/90728>.

@article{Cabot2010,
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 $(SDC) \qquad \dot\{x\}(t)+\nabla \Phi\_0(x(t))+\varepsilon(t)\, \nabla \Phi\_1(x(t))=0,$ where the control ε satisfies $\int_0^\{+\infty\} \varepsilon(t)\, \{\rm d\}t =+\infty$. This last condition ensures that ε “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. },
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.; dissipative dynamical system; non-linear oscillator},
language = {eng},
month = {3},
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/90728},
volume = {10},
year = {2010},
}

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
DA - 2010/3//
PB - EDP Sciences
VL - 10
IS - 2
SP - 243
EP - 258
AB - 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 $(SDC) \qquad \dot{x}(t)+\nabla \Phi_0(x(t))+\varepsilon(t)\, \nabla \Phi_1(x(t))=0,$ where the control ε satisfies $\int_0^{+\infty} \varepsilon(t)\, {\rm d}t =+\infty$. This last condition ensures that ε “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.
LA - eng
KW - Dissipative dynamical system; steepest descent method; constrained optimization; convex minimization; asymptotic behaviour; non-linear oscillator.; dissipative dynamical system; non-linear oscillator
UR - http://eudml.org/doc/90728
ER -

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