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A fast continuous time approach for non-smooth convex optimization using Tikhonov regularization technique
1Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria.
Computational Optimization and Applications
|February 15, 2024
Summary
This study introduces a novel optimization method using second-order dynamics and Tikhonov regularization. The approach ensures fast convergence to a minimal norm solution for convex functions.
Area of Science:
- Optimization Theory
- Convex Analysis
- Numerical Analysis
Background:
- Classical optimization problems involve minimizing convex, lower semicontinuous functions.
- Existing methods may lack guaranteed convergence rates or strong convergence of trajectories.
- The Moreau envelope and Tikhonov regularization are powerful tools in optimization.
Purpose of the Study:
- To develop a second-order in time dynamics approach for minimizing convex functions.
- To combine viscous and Hessian-driven damping with Tikhonov regularization.
- To achieve fast convergence of function values and strong convergence to a minimal norm solution.
Main Methods:
- Utilizing the Moreau envelope and its properties for nonsmooth functions.
- Extending Tikhonov regularization to a nonsmooth setting.
- Analyzing second-order in time dynamics with combined damping mechanisms.
Main Results:
- Guaranteed fast convergence of function and Moreau envelope values.
- Demonstrated strong convergence of system trajectories to a minimal norm solution.
- Derived precise convergence rates for specific parameter choices.
Conclusions:
- The proposed dynamical system effectively solves convex optimization problems.
- The method provides both efficiency (fast convergence) and accuracy (minimal norm solution).
- Numerical examples validate the theoretical findings and practical applicability.
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