General Perturbation Resilient Dynamic String-Averaging for Inconsistent Problems with Superiorization
1Department of Mathematics, University of Haifa, Mt. Carmel, Haifa, 3498838 Israel.
Journal of Optimization Theory and Applications
|July 11, 2025
Summary
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.
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.
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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