The steepest descent dynamical system with control. Applications to constrained minimization
ESAIM: Control, Optimisation and Calculus of Variations, Tome 10 (2004) no. 2, pp. 243-258.

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 [5]) 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

(SDC)x ˙(t)+Φ 0 (x(t))+ε(t)Φ 1 (x(t))=0,
where the control ε satisfies 0 + ε(t)dt=+. 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.

DOI : 10.1051/cocv:2004005
Classification : 34A12, 34D05, 34G20, 34H05, 37N40
Mots clés : dissipative dynamical system, steepest descent method, constrained optimization, convex minimization, asymptotic behaviour, non-linear oscillator
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     title = {The steepest descent dynamical system with control. {Applications} to constrained minimization},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {243--258},
     publisher = {EDP-Sciences},
     volume = {10},
     number = {2},
     year = {2004},
     doi = {10.1051/cocv:2004005},
     mrnumber = {2083486},
     zbl = {1072.49004},
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     url = {http://www.numdam.org/articles/10.1051/cocv:2004005/}
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Cabot, Alexandre. The steepest descent dynamical system with control. Applications to constrained minimization. ESAIM: Control, Optimisation and Calculus of Variations, Tome 10 (2004) no. 2, pp. 243-258. doi : 10.1051/cocv:2004005. http://www.numdam.org/articles/10.1051/cocv:2004005/

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