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Single-forward-step projective splitting: exploiting cocoercivity

Author

Listed:
  • Patrick R. Johnstone

    (Brookhaven National Laboratory)

  • Jonathan Eckstein

    (Rutgers University)

Abstract

This work describes a new variant of projective splitting for solving maximal monotone inclusions and complicated convex optimization problems. In the new version, cocoercive operators can be processed with a single forward step per iteration. In the convex optimization context, cocoercivity is equivalent to Lipschitz differentiability. Prior forward-step versions of projective splitting did not fully exploit cocoercivity and required two forward steps per iteration for such operators. Our new single-forward-step method establishes a symmetry between projective splitting algorithms, the classical forward–backward splitting method (FB), and Tseng’s forward-backward-forward method. The new procedure allows for larger stepsizes for cocoercive operators: the stepsize bound is $$2\beta$$ 2 β for a $$\beta$$ β -cocoercive operator, the same bound as has been established for FB. We show that FB corresponds to an unattainable boundary case of the parameters in the new procedure. Unlike FB, the new method allows for a backtracking procedure when the cocoercivity constant is unknown. Proving convergence of the algorithm requires some departures from the prior proof framework for projective splitting. We close with some computational tests establishing competitive performance for the method.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:coopap:v:78:y:2021:i:1:d:10.1007_s10589-020-00238-3
    DOI: 10.1007/s10589-020-00238-3
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    References listed on IDEAS

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    1. Majela Pentón Machado, 2018. "On the Complexity of the Projective Splitting and Spingarn’s Methods for the Sum of Two Maximal Monotone Operators," Journal of Optimization Theory and Applications, Springer, vol. 178(1), pages 153-190, July.
    2. Majela Pentón Machado, 2019. "Projective method of multipliers for linearly constrained convex minimization," Computational Optimization and Applications, Springer, vol. 73(1), pages 237-273, May.
    3. Laurent Condat, 2013. "A Primal–Dual Splitting Method for Convex Optimization Involving Lipschitzian, Proximable and Linear Composite Terms," Journal of Optimization Theory and Applications, Springer, vol. 158(2), pages 460-479, August.
    4. Jonathan Eckstein, 2017. "A Simplified Form of Block-Iterative Operator Splitting and an Asynchronous Algorithm Resembling the Multi-Block Alternating Direction Method of Multipliers," Journal of Optimization Theory and Applications, Springer, vol. 173(1), pages 155-182, April.
    5. Patrick L. Combettes & Jean-Christophe Pesquet, 2011. "Proximal Splitting Methods in Signal Processing," Springer Optimization and Its Applications, in: Heinz H. Bauschke & Regina S. Burachik & Patrick L. Combettes & Veit Elser & D. Russell Luke & Henry (ed.), Fixed-Point Algorithms for Inverse Problems in Science and Engineering, chapter 0, pages 185-212, Springer.
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    Cited by:

    1. 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.
    2. Adil Salim & Laurent Condat & Konstantin Mishchenko & Peter Richtárik, 2022. "Dualize, Split, Randomize: Toward Fast Nonsmooth Optimization Algorithms," Journal of Optimization Theory and Applications, Springer, vol. 195(1), pages 102-130, October.
    3. Dong, Yunda, 2023. "A new splitting method for systems of monotone inclusions in Hilbert spaces," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 203(C), pages 518-537.
    4. Luis M. Briceño-Arias & Fernando Roldán, 2022. "Four-Operator Splitting via a Forward–Backward–Half-Forward Algorithm with Line Search," Journal of Optimization Theory and Applications, Springer, vol. 195(1), pages 205-225, October.

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