What the study found
The study proposes inertial and non-inertial Douglas–Rachford algorithms for minimizing the sum of two geodesically convex functions on Hadamard manifolds, which are complete, nonpositively curved spaces. The authors also introduce parallel Douglas–Rachford type algorithms for problems with multiple summands and apply them to generalized Heron problems.
Why the authors say this matters
The authors say the goal is to improve the convergence of the Douglas–Rachford algorithm on Hadamard manifolds. They also present the generalized Heron problem as an application area for the parallel algorithms.
What the researchers tested
The researchers developed two algorithm types, inertial and non-inertial, for minimizing sums of geodesically convex functions on Hadamard manifolds. They analyzed convergence using fixed-point theory for nonexpansive operators and studied convergence rates, then tested the methods with numerical experiments for generalized Heron problems.
What worked and what didn't
The abstract states that convergence analysis was provided for both algorithms under suitable assumptions on algorithmic parameters and geodesic convexity of the objective functions. It also says the convergence rates of the two methods were studied, and numerical experiments were presented to demonstrate effectiveness for generalized Heron problems. No specific failures or negative results are described in the abstract.
What to keep in mind
The abstract does not give the detailed assumptions, theorem statements, or numerical outcomes. It also does not describe any limitations beyond noting that the convergence analysis depends on suitable algorithmic parameters and geodesic convexity.
Key points
- The paper studies Douglas–Rachford algorithms on Hadamard manifolds.
- It proposes both inertial and non-inertial methods.
- The algorithms are analyzed for minimizing sums of geodesically convex functions.
- Parallel Douglas–Rachford type algorithms are introduced for problems with multiple summands.
- The methods are applied to generalized Heron problems, with numerical experiments reported.
Disclosure
- Research title:
- Douglas–Rachford methods are studied on Hadamard manifolds
- Authors:
- D. R. Sahu, Shikher Sharma, Pankaj Gautam
- Institutions:
- Banaras Hindu University, Banaras Hindu University, Indian Institute of Technology Roorkee, Technion – Israel Institute of Technology
- Publication date:
- 2026-04-20
- OpenAlex record:
- View
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