Numerical Algorithms

, Volume 80, Issue 4, pp 1219–1240 | Cite as

Derivative-free superiorization with component-wise perturbations

  • Yair CensorEmail author
  • Howard Heaton
  • Reinhard Schulte
Original Paper


Superiorization reduces, not necessarily minimizes, the value of a target function while seeking constraints compatibility. This is done by taking a solely feasibility-seeking algorithm, analyzing its perturbation resilience, and proactively perturbing its iterates accordingly to steer them toward a feasible point with reduced value of the target function. When the perturbation steps are computationally efficient, this enables generation of a superior result with essentially the same computational cost as that of the original feasibility-seeking algorithm. In this work, we refine previous formulations of the superiorization method to create a more general framework, enabling target function reduction steps that do not require partial derivatives of the target function. In perturbations that use partial derivatives, the step-sizes in the perturbation phase of the superiorization method are chosen independently from the choice of the nonascent directions. This is no longer true when component-wise perturbations are employed. In that case, the step-sizes must be linked to the choice of the nonascent direction in every step. Besides presenting and validating these notions, we give a computational demonstration of superiorization with component-wise perturbations for a problem of computerized tomography image reconstruction.


Superiorization Derivative-free Component-wise perturbations Image reconstruction Feasibility-seeking Perturbation resilience 


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We greatly appreciate the constructive comments of two anonymous reviewers which helped us improve the paper.


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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of HaifaHaifaIsrael
  2. 2.Department of MathematicsUniversity of California Los AngelesLos AngelesUSA
  3. 3.Division of Biomedical Engineering Sciences, Department of Basic Sciences, School of MedicineLoma Linda UniversityLoma LindaUSA

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