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General Perturbation Resilient Dynamic String-Averaging for Inconsistent Problems with Superiorization.

Kay Barshad1, Yair Censor1

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

Journal of Optimization Theory and Applications
|July 11, 2025
PubMed
Summary
This summary is machine-generated.

This study introduces a General Dynamic String-Averaging (GDSA) iterative scheme for inconsistent cases where operators lack common fixed points. The new method demonstrates weak and strong convergence, extending prior algorithms with enhanced properties.

Keywords:
Approximately shrinking operatorBounded perturbation resilienceBounded regularityCoherenceCoherent sequence of operatorsCommon fixed point problemConvex feasibility problemDynamic string-averagingFejér monotonicityMetric projectionNonexpansive operatorStrong coherenceStrongly coherent sequence of operatorsSuperiorizationWeak convergenceWeak regularity

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Area of Science:

  • Optimization Theory
  • Applied Mathematics
  • Functional Analysis

Background:

  • The Dynamic String-Averaging Projection (DSAP) algorithm, introduced in 2013, demonstrated strong convergence and bounded perturbation resilience in consistent cases.
  • Prior work in 2015 explored combining DSAP with superiorization methods.
  • The concept of "coherence" for operator sequences was established in 2001, with "strong coherence" introduced in 2019.

Purpose of the Study:

  • Introduce and analyze the General Dynamic String-Averaging (GDSA) iterative scheme.
  • Investigate the weak and strong convergence properties of GDSA in the inconsistent case.
  • Examine the bounded perturbation resilience of the GDSA method.

Main Methods:

  • Development of the General Dynamic String-Averaging (GDSA) iterative scheme.
  • Leveraging the concept of "strong coherence" for operator sequences to prove convergence.
  • Analysis of the GDSA method for a general class of operators in inconsistent scenarios.

Main Results:

  • Established weak convergence for the GDSA method in the inconsistent case, building upon the "strong coherence" property.
  • Demonstrated bounded perturbation resilience for the GDSA method in inconsistent settings.
  • Showcased the application of GDSA within the Superiorization Methodology.

Conclusions:

  • The GDSA method offers a robust approach for solving problems involving inconsistent operators.
  • The "strong coherence" property provides a powerful tool for analyzing iterative methods with infinite operator sequences.
  • GDSA enhances the capabilities of superiorization techniques, particularly in challenging inconsistent scenarios.