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The modified proximal point algorithm in Hadamard spaces.

Shih-Sen Chang^1,2, Lin Wang², Ching-Feng Wen³

¹1Center for General Education, China Medical University, Taichung, Taiwan.

Journal of Inequalities and Applications

|August 24, 2018

Summary

This summary is machine-generated.

This study introduces a modified proximal point algorithm for minimization problems in Hadamard spaces. The algorithm guarantees strong convergence to a minimizer for convex functions, extending prior work.

Keywords:

Hadamard space Implicit iterative rule Moreau–Yosida resolvent Proximal point algorithm Variational inequality

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Area of Science:

Optimization Theory
Metric Geometry
Convex Analysis

Background:

Minimization problems are fundamental in various scientific fields.
Proximal point algorithms are established methods for solving such problems.
Existing algorithms have limitations in specific geometric spaces.

Purpose of the Study:

To propose a novel modified proximal point algorithm.
To address minimization problems within the framework of Hadamard spaces.
To establish convergence properties of the proposed algorithm.

Main Methods:

Development of a modified proximal point algorithm tailored for Hadamard spaces.
Rigorous mathematical analysis to prove convergence properties.
Utilizing concepts from metric geometry and convex analysis.

Main Results:

The proposed algorithm is shown to converge strongly (in metric).
Convergence is demonstrated for convex objective functions.
The findings generalize existing results from Hilbert spaces and manifolds.

Conclusions:

The modified proximal point algorithm is effective for minimization in Hadamard spaces.
This work expands the applicability of proximal algorithms to broader geometric settings.
The results offer a foundation for future research in geometric optimization.