Strong convergence and bounded perturbation resilience of a modified proximal gradient algorithm.

Area of Science:

• Optimization Theory
• Numerical Analysis
• Functional Analysis

Background:

• The proximal gradient algorithm is a common method for solving non-smooth composite optimization problems.
• Standard proximal gradient methods may exhibit only weak convergence in infinite-dimensional settings.
• Addressing these limitations is crucial for broader applicability in complex optimization tasks.

Purpose of the Study:

• To introduce a modified proximal gradient algorithm capable of strong convergence.
• To analyze the algorithm's behavior in Hilbert spaces with outer perturbations.
• To investigate the bounded perturbation resilience of the proposed iterative scheme.

Main Methods:

• Development of a modified proximal gradient algorithm incorporating outer perturbations.
• Theoretical analysis in Hilbert space to establish convergence properties.
• Examination of bounded perturbation resilience for robustness.

Main Results:

• The modified proximal gradient algorithm achieves strong convergence to a solution.
• Demonstration of enhanced convergence properties compared to standard algorithms.
• Analysis of the algorithm's resilience to bounded perturbations.

Conclusions:

• The modified proximal gradient algorithm offers a robust and effective solution for non-smooth composite optimization in infinite-dimensional spaces.
• The findings advance the theoretical understanding and practical application of proximal gradient methods.
• The algorithm's perturbation resilience is a key feature for real-world applications.