What the study found
The study presents inertial and non-inertial Douglas–Rachford algorithms for minimizing the sum of two geodesically convex functions on Hadamard manifolds. It also introduces parallel Douglas–Rachford-type algorithms for problems with multiple summands, including applications 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, which are spaces of nonpositive curvature. The study suggests this is relevant for minimizing functionals with multiple terms and for generalized Heron problems on these manifolds.
What the researchers tested
The researchers studied two algorithm types, inertial and non-inertial, under suitable assumptions on the algorithmic parameters and the geodesic convexity of the objective functions. They based the convergence analysis on fixed-point theory for nonexpansive operators and also examined convergence rates.
What worked and what didn't
According to the abstract, both algorithm types have convergence analysis under the stated assumptions. The paper also reports convergence rates for the two methods and presents numerical experiments for generalized Heron problems to demonstrate effectiveness.
What to keep in mind
The abstract does not give the detailed assumptions, numerical results, or specific rate values. It also does not describe any failures or compare performance between the inertial and non-inertial methods.
Key points
- The paper develops inertial and non-inertial Douglas–Rachford algorithms on Hadamard manifolds.
- The target problem is minimizing the sum of two geodesically convex functions.
- The authors provide convergence analysis and study convergence rates under suitable assumptions.
- Parallel Douglas–Rachford-type algorithms are introduced for multiple-summand functionals.
- The methods are applied to generalized Heron problems, with numerical experiments reported.
Disclosure
- Research title:
- Douglas–Rachford algorithms converge 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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