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An Inexact Inertial Projective Splitting Algorithm with Strong Convergence

Author

Listed:
  • M. Marques Alves

    (Universidade Federal de Santa Catarina)

  • J. E. Navarro Caballero

    (Universidade Federal de Santa Catarina)

  • R. T. Marcavillaca

    (Universidad de Chile)

Abstract

We propose and study a strongly convergent inexact inertial projective splitting (PS) algorithm for finding zeros of composite monotone inclusion problems involving the sum of finitely many maximal monotone operators. Strong convergence of the iterates is ensured by projections onto the intersection of appropriately defined half-spaces, even in the absence of inertial effects. We also establish iteration-complexity results for the proposed PS method, which likewise hold without requiring inertial terms. The algorithm includes two inertial sequences, controlled by parameters satisfying mild conditions, while preserving strong convergence and enabling iteration-complexity analysis. Furthermore, for more structured monotone inclusion problems, we derive two variants of the main algorithm that employ forward-backward and forward-backward-forward steps.

Suggested Citation

  • M. Marques Alves & J. E. Navarro Caballero & R. T. Marcavillaca, 2025. "An Inexact Inertial Projective Splitting Algorithm with Strong Convergence," Journal of Optimization Theory and Applications, Springer, vol. 207(3), pages 1-33, December.
    • Handle: RePEc:spr:joptap:v:207:y:2025:i:3:d:10.1007_s10957-025-02827-w
    DOI: 10.1007/s10957-025-02827-w
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    References listed on IDEAS

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    1. M. Marques Alves & Jonathan Eckstein & Marina Geremia & Jefferson G. Melo, 2020. "Relative-error inertial-relaxed inexact versions of Douglas-Rachford and ADMM splitting algorithms," Computational Optimization and Applications, Springer, vol. 75(2), pages 389-422, March.
    2. Hedy Attouch & Alexandre Cabot, 2020. "Convergence rate of a relaxed inertial proximal algorithm for convex minimization," Post-Print hal-02415789, HAL.
    3. Patrick R. Johnstone & Jonathan Eckstein, 2021. "Single-forward-step projective splitting: exploiting cocoercivity," Computational Optimization and Applications, Springer, vol. 78(1), pages 125-166, January.
    4. Majela Pentón Machado & Mauricio Romero Sicre, 2023. "A Projective Splitting Method for Monotone Inclusions: Iteration-Complexity and Application to Composite Optimization," Journal of Optimization Theory and Applications, Springer, vol. 198(2), pages 552-587, August.
    5. Benar F. Svaiter, 2014. "A Class of Fejér Convergent Algorithms, Approximate Resolvents and the Hybrid Proximal-Extragradient Method," Journal of Optimization Theory and Applications, Springer, vol. 162(1), pages 133-153, July.
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