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Second Order Dynamics Featuring Tikhonov Regularization and Time Scaling

Author

Listed:
  • Ernö Robert Csetnek

    (University of Vienna)

  • Mikhail A. Karapetyants

    (University of Vienna)

Abstract

In a Hilbert setting we aim to study a second order in time differential equation, combining viscous and Hessian-driven damping, containing a time scaling parameter function and a Tikhonov regularization term. The dynamical system is related to the problem of minimization of a nonsmooth convex function. In the formulation of the problem as well as in our analysis we use the Moreau envelope of the objective function and its gradient and heavily rely on their properties. We show that there is a setting where the newly introduced system preserves and even improves the well-known fast convergence properties of the function and Moreau envelope along the trajectories and also of the gradient of Moreau envelope due to the presence of time scaling. Moreover, in a different setting we prove strong convergence of the trajectories to the element of minimal norm from the set of all minimizers of the objective. The manuscript concludes with various numerical results.

Suggested Citation

  • Ernö Robert Csetnek & Mikhail A. Karapetyants, 2024. "Second Order Dynamics Featuring Tikhonov Regularization and Time Scaling," Journal of Optimization Theory and Applications, Springer, vol. 202(3), pages 1385-1420, September.
    • Handle: RePEc:spr:joptap:v:202:y:2024:i:3:d:10.1007_s10957-024-02500-8
    DOI: 10.1007/s10957-024-02500-8
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    References listed on IDEAS

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    1. 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.
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