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A fast continuous time approach with time scaling for nonsmooth convex optimization.

Radu Ioan Boţ1, Mikhail A Karapetyants1

  • 1Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria.

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Summary

This study analyzes a second-order dynamical system for minimizing nonsmooth convex functions, demonstrating fast convergence rates for the objective function and its trajectory to a global minimum.

Keywords:
Damped inertial dynamicsHessian-driven dampingMoreau envelopeNonsmooth convex optimizationProximal operatorTime scaling

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Area of Science:

  • Optimization Theory
  • Dynamical Systems
  • Convex Analysis

Background:

  • Minimizing nonsmooth convex functions is crucial in many scientific fields.
  • Second-order dynamical systems offer potential for faster convergence compared to first-order methods.
  • Existing methods may face challenges with nonsmoothness and require efficient damping strategies.

Purpose of the Study:

  • To investigate the convergence properties of a novel second-order dynamical system for nonsmooth convex optimization.
  • To analyze the impact of viscous and Hessian-driven damping combined with time scaling.
  • To establish fast convergence rates for the objective function and its trajectory.

Main Methods:

  • Formulation of a second-order dynamical system using the gradient of the Moreau envelope.
  • Inclusion of viscous and Hessian-driven damping with time scaling.
  • Analysis of convergence rates in a Hilbert space setting.

Main Results:

  • Demonstrated fast convergence rates for the Moreau envelope and its gradient along the trajectory.
  • Established fast convergence rates for the system velocity.
  • Derived fast convergence rates for the objective function via the proximal operator.
  • Proved weak convergence of the system trajectory to a global minimizer.

Conclusions:

  • The proposed second-order dynamical system effectively minimizes nonsmooth convex functions.
  • The combination of damping strategies and time scaling accelerates convergence.
  • Numerical examples validate the theoretical findings, showcasing practical applicability.