Issue |
ESAIM: COCV
Volume 10, Number 2, April 2004
|
|
---|---|---|
Page(s) | 243 - 258 | |
DOI | https://doi.org/10.1051/cocv:2004005 | |
Published online | 15 March 2004 |
The steepest descent dynamical system with control. Applications to constrained minimization
Laboratoire LACO, Faculté des Sciences, 123 avenue Albert Thomas,
87060 Limoges Cedex, France; alexandre.cabot@unilim.fr.
Received:
17
February
2003
Let H be a real Hilbert space, a
convex function of class
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
. 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
whose critical points coincide
with S
and a control parameter
tending to zero,
we consider the “Steepest Descent and Control” system
where
the control ε satisfies
. This last condition ensures that ε “slowly” tends
to 0. When H is finite dimensional, we then prove that
and we give sufficient conditions under which
.
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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