Second Order Dynamics Featuring Tikhonov Regularization and Time Scaling

Ernö Robert Csetnek () and Mikhail A. Karapetyants ()
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Ernö Robert Csetnek: University of Vienna
Mikhail A. Karapetyants: University of Vienna

Journal of Optimization Theory and Applications, 2024, vol. 202, issue 3, No 16, 1385-1420

Abstract: 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.

Keywords: Nonsmooth convex optimization; Damped inertial dynamics; Hessian-driven damping; Time scaling; Moreau envelope; Proximal operator; Tikhonov regularization; 37N40; 46N10; 49M99; 65K05; 65K10; 90C25 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s10957-024-02500-8

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