Derivative-free superiorization: principle and algorithm

Yair Censor, Edgar Garduño, Elias S. Helou, Gabor T. Herman

Research output: Contribution to journalArticlepeer-review

Abstract

The superiorization methodology is intended to work with input data of constrained minimization problems, that is, a target function and a set of constraints. However, it is based on an antipodal way of thinking to what leads to constrained minimization methods. Instead of adapting unconstrained minimization algorithms to handling constraints, it adapts feasibility-seeking algorithms to reduce (not necessarily minimize) target function values. This is done by inserting target-function-reducing perturbations into a feasibility-seeking algorithm while retaining its feasibility-seeking ability and without paying a high computational price. A superiorized algorithm that employs component-wise target function reduction steps is presented. This enables derivative-free superiorization (DFS), meaning that superiorization can be applied to target functions that have no calculable partial derivatives or subgradients. The numerical behavior of our derivative-free superiorization algorithm is illustrated on a data set generated by simulating a problem of image reconstruction from projections. We present a tool (we call it a proximity-target curve) for deciding which of two iterative methods is “better” for solving a particular problem. The plots of proximity-target curves of our experiments demonstrate the advantage of the proposed derivative-free superiorization algorithm.

Original languageEnglish
Pages (from-to)227-248
Number of pages22
JournalNumerical Algorithms
Volume88
Issue number1
DOIs
StatePublished - Sep 2021

Bibliographical note

Funding Information:
Edgar Garduño received support from DGAPA-UNAM. The work of Yair Censor is supported by the ISF-NSFC joint research program Grant No. 2874/19. Elias S. Helou was partially supported by CNPq grant no. 310893/2019-4. Acknowledgments

Publisher Copyright:
© 2020, Springer Science+Business Media, LLC, part of Springer Nature.

Keywords

  • Bounded perturbations
  • Component-wise perturbations
  • Constrained minimization
  • Derivative-free
  • Proximity function
  • Regularization
  • Superiorization

ASJC Scopus subject areas

  • Applied Mathematics

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