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Projected Subgradient Minimization Versus Superiorization

Author

Listed:
  • Yair Censor

    (University of Haifa)

  • Ran Davidi

    (Stanford University)

  • Gabor T. Herman

    (City University of New York)

  • Reinhard W. Schulte

    (Loma Linda University Medical Center)

  • Luba Tetruashvili

    (University of Haifa)

Abstract

The projected subgradient method for constrained minimization repeatedly interlaces subgradient steps for the objective function with projections onto the feasible region, which is the intersection of closed and convex constraints sets, to regain feasibility. The latter poses a computational difficulty, and, therefore, the projected subgradient method is applicable only when the feasible region is “simple to project onto.” In contrast to this, in the superiorization methodology a feasibility-seeking algorithm leads the overall process, and objective function steps are interlaced into it. This makes a difference because the feasibility-seeking algorithm employs projections onto the individual constraints sets and not onto the entire feasible region. We present the two approaches side-by-side and demonstrate their performance on a problem of computerized tomography image reconstruction, posed as a constrained minimization problem aiming at finding a constraint-compatible solution that has a reduced value of the total variation of the reconstructed image.

Suggested Citation

  • Yair Censor & Ran Davidi & Gabor T. Herman & Reinhard W. Schulte & Luba Tetruashvili, 2014. "Projected Subgradient Minimization Versus Superiorization," Journal of Optimization Theory and Applications, Springer, vol. 160(3), pages 730-747, March.
    • Handle: RePEc:spr:joptap:v:160:y:2014:i:3:d:10.1007_s10957-013-0408-3
    DOI: 10.1007/s10957-013-0408-3
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    References listed on IDEAS

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    1. Yair Censor & Wei Chen & Patrick Combettes & Ran Davidi & Gabor Herman, 2012. "On the effectiveness of projection methods for convex feasibility problems with linear inequality constraints," Computational Optimization and Applications, Springer, vol. 51(3), pages 1065-1088, April.
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    Cited by:

    1. Yanni Guo & Xiaozhi Zhao, 2019. "Bounded Perturbation Resilience and Superiorization of Proximal Scaled Gradient Algorithm with Multi-Parameters," Mathematics, MDPI, vol. 7(6), pages 1-14, June.
    2. Chin How Jeffrey Pang, 2019. "Dykstra’s Splitting and an Approximate Proximal Point Algorithm for Minimizing the Sum of Convex Functions," Journal of Optimization Theory and Applications, Springer, vol. 182(3), pages 1019-1049, September.
    3. Wenma Jin & Yair Censor & Ming Jiang, 2016. "Bounded perturbation resilience of projected scaled gradient methods," Computational Optimization and Applications, Springer, vol. 63(2), pages 365-392, March.
    4. Aragón-Artacho, Francisco J. & Censor, Yair & Gibali, Aviv & Torregrosa-Belén, David, 2023. "The superiorization method with restarted perturbations for split minimization problems with an application to radiotherapy treatment planning," Applied Mathematics and Computation, Elsevier, vol. 440(C).
    5. Yair Censor & Alexander J. Zaslavski, 2015. "Strict Fejér Monotonicity by Superiorization of Feasibility-Seeking Projection Methods," Journal of Optimization Theory and Applications, Springer, vol. 165(1), pages 172-187, April.

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