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

Alexandre Cabot

ESAIM: Control, Optimisation and Calculus of Variations (2010)

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

Abstract

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Let H be a real Hilbert space, Φ 1 : H → 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  Φ 1 + δ 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  Φ 0 : H → whose critical points coincide with S and a control parameter ε : + → + tending to zero, we consider the “Steepest Descent and Control” system ( S D C ) x ˙ ( t ) + ∇ Φ 0 ( x ( t ) ) + ε ( t ) ∇ Φ 1 ( x ( t ) ) = 0 , where the control ε satisfies ∫ 0 + ∞ ε ( t ) d t = + ∞ . This last condition ensures that ε “slowly” tends to 0. When H is finite dimensional, we then prove that d ( x ( t ) , argmin S Φ 1 ) → 0 ( t → + ∞ ) , and we give sufficient conditions under which  x ( t ) → x ¯ ∈ argmin S Φ 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 -

References

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  9. W. Hirsch and S. Smale, Differential equations, dynamical systems and linear algebra. Academic Press, New York (1974).  
  10. J.P. Lasalle and S. Lefschetz, Stability by Lyapounov's Direct Method with Applications. Academic Press, New York (1961).  
  11. Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Amer. Math. Soc.73 (1967) 591-597.  
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