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Derivative-free superiorization with component-wise perturbations

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Abstract

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.

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Acknowledgements

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

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Authors and Affiliations

  1. Department of Mathematics, University of Haifa, Mt. Carmel, 3498838, Haifa, Israel

    Yair Censor

  2. Department of Mathematics, University of California Los Angeles, Los Angeles, CA, 90095, USA

    Howard Heaton

  3. Division of Biomedical Engineering Sciences, Department of Basic Sciences, School of Medicine, Loma Linda University, Loma Linda, CA, 92350, USA

    Reinhard Schulte

Authors
  1. Yair Censor
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  2. Howard Heaton
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  3. Reinhard Schulte
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Corresponding author

Correspondence to Yair Censor.

Additional information

This project was supported by Research Grant No. 2013003 of the United States-Israel Binational Science Foundation (BSF) and by Award No. 1P20183640-01A1 of the National Cancer Institute (NCI) of the National Institutes of Health (NIH).

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Censor, Y., Heaton, H. & Schulte, R. Derivative-free superiorization with component-wise perturbations. Numer Algor 80, 1219–1240 (2019). https://doi.org/10.1007/s11075-018-0524-0

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