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The Forward-Douglas-Rachford splitting method offers sublinear global convergence and local linear convergence for non-smooth optimization problems. This method efficiently identifies smooth manifolds and accelerates convergence in applicable fields.

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Area of Science:

  • Optimization Theory
  • Numerical Analysis

Background:

  • Operator splitting methods are widely used for non-smooth optimization due to their efficiency.
  • The Forward-Douglas-Rachford splitting method is a key technique in this area.

Purpose of the Study:

  • To analyze the global and local convergence rates of the Forward-Douglas-Rachford splitting method.
  • To establish convergence guarantees for non-smooth optimization problems.

Main Methods:

  • Analysis of global convergence rates using Bregman divergence.
  • Investigation of local convergence rates under partial smoothness assumptions.
  • Specialization to the Forward-Backward splitting for enhanced convergence rates.

Main Results:

  • Established a sublinear global convergence rate for the Forward-Douglas-Rachford method.
  • Proved a stronger convergence rate for the objective function value when specializing to Forward-Backward splitting.
  • Demonstrated local linear convergence, characterized by manifold identification and convergence to a smooth manifold.

Conclusions:

  • The Forward-Douglas-Rachford splitting method exhibits robust convergence properties for non-smooth optimization.
  • The theoretical findings are validated through numerical experiments in signal/image processing, inverse problems, and machine learning.