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Continuous Gradient Projection Method in Hilbert Spaces

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  • J. Bolte

    (Université Montpellier II)

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.

Suggested Citation

  • J. Bolte, 2003. "Continuous Gradient Projection Method in Hilbert Spaces," Journal of Optimization Theory and Applications, Springer, vol. 119(2), pages 235-259, November.
    • Handle: RePEc:spr:joptap:v:119:y:2003:i:2:d:10.1023_b:jota.0000005445.21095.02
    DOI: 10.1023/B:JOTA.0000005445.21095.02
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    Cited by:

    1. Boţ, Radu Ioan & Kanzler, Laura, 2021. "A forward-backward dynamical approach for nonsmooth problems with block structure coupled by a smooth function," Applied Mathematics and Computation, Elsevier, vol. 394(C).
    2. B. Abbas & H. Attouch & Benar F. Svaiter, 2014. "Newton-Like Dynamics and Forward-Backward Methods for Structured Monotone Inclusions in Hilbert Spaces," Journal of Optimization Theory and Applications, Springer, vol. 161(2), pages 331-360, May.
    3. Sylvain Sorin, 2023. "Continuous Time Learning Algorithms in Optimization and Game Theory," Dynamic Games and Applications, Springer, vol. 13(1), pages 3-24, March.
    4. Wenjie Wang & Chunyan Wang & Mengzhen Li, 2024. "A Second-Order Continuous-Time Dynamical System for Solving Sparse Image Restoration Problems," Mathematics, MDPI, vol. 12(15), pages 1-15, July.
    5. P. Nistri & M. Quincampoix, 2005. "On the Dynamics of a Differential Inclusion Built upon a Nonconvex Constrained Minimization Problem," Journal of Optimization Theory and Applications, Springer, vol. 124(3), pages 659-672, March.
    6. Haixin Ren & Bin Ge & Xiangwu Zhuge, 2023. "Fast Convergence of Inertial Gradient Dynamics with Multiscale Aspects," Journal of Optimization Theory and Applications, Springer, vol. 196(2), pages 461-489, February.

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