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Strict Fejér Monotonicity by Superiorization of Feasibility-Seeking Projection Methods

Author

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  • Yair Censor

    (University of Haifa)

  • Alexander J. Zaslavski

    (The Technion – Israel Institute of Technology)

Abstract

We consider the superiorization methodology, which can be thought of as lying between feasibility-seeking and constrained minimization. It is not quite trying to solve the full-fledged constrained minimization problem; rather, the task is to find a feasible point which is superior (with respect to the objective function value) to one returned by a feasibility-seeking only algorithm. Our main result reveals new information about the mathematical behavior of the superiorization methodology. We deal with a constrained minimization problem with a feasible region, which is the intersection of finitely many closed convex constraint sets, and use the dynamic string-averaging projection method, with variable strings and variable weights, as a feasibility-seeking algorithm. We show that any sequence, generated by the superiorized version of a dynamic string-averaging projection algorithm, not only converges to a feasible point but, additionally, also either its limit point solves the constrained minimization problem or the sequence is strictly Fejér monotone with respect to a subset of the solution set of the original problem.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:165:y:2015:i:1:d:10.1007_s10957-014-0591-x
    DOI: 10.1007/s10957-014-0591-x
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    References listed on IDEAS

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    1. 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.
    2. Yair Censor & Alexander Zaslavski, 2013. "Convergence and perturbation resilience of dynamic string-averaging projection methods," Computational Optimization and Applications, Springer, vol. 54(1), pages 65-76, January.
    3. 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. Kaiwen Ma & Nikolaos V. Sahinidis & Sreekanth Rajagopalan & Satyajith Amaran & Scott J Bury, 2021. "Decomposition in derivative-free optimization," Journal of Global Optimization, Springer, vol. 81(2), pages 269-292, October.
    2. 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.
    3. Q. L. Dong & J. Z. Huang & X. H. Li & Y. J. Cho & Th. M. Rassias, 2019. "MiKM: multi-step inertial Krasnosel’skiǐ–Mann algorithm and its applications," Journal of Global Optimization, Springer, vol. 73(4), pages 801-824, April.
    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. Alexander J. Zaslavski, 2023. "Superiorization with a Projected Subgradient Algorithm on the Solution Sets of Common Fixed Point Problems," Mathematics, MDPI, vol. 11(21), pages 1-12, November.

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