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Second Order Dynamics Featuring Tikhonov Regularization and Time Scaling
Ernö Robert Csetnek1, Mikhail A Karapetyants1
1Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria.
Journal of Optimization Theory and Applications
|September 9, 2024
Summary
This study introduces a new second-order differential equation for minimizing nonsmooth convex functions. The novel system enhances convergence rates and proves strong convergence to minimal norm solutions.
Area of Science:
- Optimization Theory
- Convex Analysis
- Differential Equations
Background:
- Studying dynamical systems for function minimization is crucial in optimization.
- Second-order differential equations offer advanced convergence properties.
- Nonsmooth convex optimization presents unique challenges.
Purpose of the Study:
- To analyze a second-order differential equation with viscous and Hessian-driven damping.
- To incorporate time scaling and Tikhonov regularization for improved performance.
- To investigate convergence properties for minimizing nonsmooth convex functions.
Main Methods:
- Utilizing the Moreau envelope and its gradient properties.
- Developing a novel second-order differential equation model.
- Employing analysis within a Hilbert space setting.
Main Results:
- Demonstrating preservation and improvement of fast convergence rates with time scaling.
- Proving strong convergence of trajectories to minimal norm solutions.
- Validating findings through numerical simulations.
Conclusions:
- The proposed system effectively minimizes nonsmooth convex functions.
- Time scaling significantly enhances convergence speed.
- The method converges to the minimum norm solution.
Keywords:
Damped inertial dynamicsHessian-driven dampingMoreau envelopeNonsmooth convex optimizationProximal operatorTikhonov regularizationTime scalingMore Related Videos
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