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A necessary condition for the guarantee of the superiorization method
Kay Barshad1,2, Yair Censor2, Walaa Moursi1
1Department of Combinatorics and Optimization, Faculty of Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1.
Arxiv
|February 20, 2025
Summary
The superiorization method (SM) perturbs algorithms to find better solutions. We identified a condition where SM may fail, guiding future research and improving practical applications.
Area of Science:
- Optimization Methods
- Convex Feasibility Problems
Background:
- The superiorization method (SM) combines feasibility-seeking algorithms with objective function reduction.
- SM perturbs iterates of convergent algorithms using non-ascent steps.
Purpose of the Study:
- Investigate conditions for SM's asymptotic convergence to superior feasible points.
- Identify when SM fails to achieve a better objective function value than the base algorithm.
Main Methods:
- Analysis of iterative algorithms with perturbations.
- Focus on SM utilizing negative gradient descent steps for perturbations.
Main Results:
- A specific condition is identified under which SM fails to yield a superior outcome.
- This 'negative condition' is crucial for future theoretical guarantees of SM.
Conclusions:
- The discovered negative condition highlights limitations of SM with negative gradient descent.
- Understanding and avoiding this condition can enhance SM's success rate in practice.
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