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We propose and study a weakly convergent variant of the forward-backward algorithm for solving structured monotone inclusion problems. Our algorithm features a per-iteration deviation vector, providing additional degrees of freedom. The only requirement on the deviation vector to guarantee convergence is that its norm is bounded by a quantity that can be computed online. This approach offers great flexibility and paves the way for the design of new forward-backward-based algorithms, while still retaining global convergence guarantees. These guarantees include linear convergence under a metric subregularity assumption. Choosing suitable monotone operators enables the incorporation of deviations into other algorithms, such as the Chambolle-Pock method and Krasnosel'skii-Mann iterations. We propose a novel inertial primal-dual algorithm by selecting the deviations along a momentum direction and deciding their size by using the norm condition. Numerical experiments validate our convergence claims and demonstrate that even this simple choice of a deviation vector can enhance the performance compared to, for instance, the standard Chambolle-Pock algorithm. Copy: 2024 Applied Set-Valued Analysis and Optimization.

Original languageEnglish
Pages (from-to)113-135
Number of pages23
JournalApplied Set-Valued Analysis and Optimization
Issue number2
Publication statusPublished - 2024 Aug 1

Subject classification (UKÄ)

  • Control Engineering

Free keywords

  • Forward-backward splitting
  • Global convergence
  • Inertial primal-dual algorithm
  • Linear convergence rate
  • Monotone inclusions


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