Continuous Gradient Projection Method in Hilbert Spaces

Jérôme Bolte

Journal of Optimization Theory and Applications, 2003, vol. 119, issue 2, No 2, 235-259

Abstract: Abstract This paper is concerned with the asymptotic analysis of the trajectories of some dynamical systems built upon the gradient projection method in Hilbert spaces. For a convex function with locally Lipschitz gradient, it is proved that the orbits converge weakly to a constrained minimizer whenever it exists. This result remains valid even if the initial condition is chosen out of the feasible set and it can be extended in some sense to quasiconvex functions. An asymptotic control result, involving a Tykhonov-like regularization, shows that the orbits can be forced to converge strongly toward a well-specified minimizer. In the finite-dimensional framework, we study the differential inclusion obtained by replacing the classical gradient by the subdifferential of a continuous convex function. We prove the existence of a solution whose asymptotic properties are the same as in the smooth case.

Keywords: Gradient projection methods; dissipative dynamical systems in optimization; differential inclusions; asymptotic control; Lyapunov functions (search for similar items in EconPapers)
Date: 2003
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Citations: View citations in EconPapers (7)

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DOI: 10.1023/B:JOTA.0000005445.21095.02

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