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

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Summary
This summary is machine-generated.

This study introduces novel regularization algorithms for structured convex optimization problems. These methods leverage the Moreau envelope and stochastic oracle calls for efficient, large-scale problem-solving.

Keywords:
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Area of Science:

  • Optimization
  • Applied Mathematics
  • Image Processing

Background:

  • Structured convex optimization problems are common in various scientific fields.
  • Primal-dual methods are typically used for these problems, but can be computationally intensive.
  • Nonsmooth functions composed with linear operators present unique challenges.

Purpose of the Study:

  • To develop novel algorithms for structured convex optimization problems involving nonsmooth functions composed with linear operators.
  • To improve the efficiency and scalability of optimization methods for large-scale problems.
  • To apply these new algorithms to practical applications like image denoising and matrix factorization.

Main Methods:

  • Regularization via the Moreau envelope to derive novel algorithms.
  • Utilizing stochastic oracle calls, akin to stochastic gradient techniques, for large-scale problem-solving.
  • Full splitting schemes are considered, with a focus on Lipschitz continuity assumptions.

Main Results:

  • Derivation of new algorithms based on Moreau envelope regularization.
  • Demonstration of effectiveness for large-scale optimization through stochastic methods.
  • Successful application to total variational denoising, deblurring, and matrix factorization.

Conclusions:

  • The proposed Moreau envelope regularization approach offers a novel and effective way to solve structured convex optimization problems.
  • Stochastic oracle calls enable efficient handling of large-scale instances.
  • The methods are validated through practical applications in image processing and matrix factorization.