The steepest descent dynamical system with control. Applications to constrained minimization
Laboratoire LACO, Faculté des Sciences, 123 avenue Albert Thomas,
87060 Limoges Cedex, France; firstname.lastname@example.org.
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 , 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