Douglas–Rachford algorithms converge on Hadamard manifolds

What the study found: The authors propose inertial and non-inertial Douglas–Rachford algorithms for minimizing the sum of two geodesically convex functions on Hadamard manifolds, which are complete, simply connected manifolds of nonpositive curvature. They also introduce parallel Douglas–Rachford type algorithms for problems with multiple summands, including the generalized Heron problem.
Why the authors say this matters: The study suggests these methods are meant to improve the convergence of Douglas–Rachford algorithms in Hadamard manifolds, and the authors present applications to generalized Heron problems.
What the researchers tested: The paper develops two algorithm types, establishes convergence under suitable assumptions on algorithmic parameters and geodesic convexity, and uses fixed-point theory for nonexpansive operators in the analysis. It also studies convergence rates and includes numerical experiments for generalized Heron problems.
What worked and what didn't: The abstract states that convergence analysis was provided for both the inertial and non-inertial methods under the stated assumptions. It also reports convergence-rate analysis and numerical experiments, but it does not give detailed numerical outcomes in the abstract.
What to keep in mind: The abstract does not specify the exact parameter conditions, the precise convergence rates, or the numerical results. Limitations are not otherwise described in the available summary.

Key points

• The paper proposes inertial and non-inertial Douglas–Rachford algorithms on Hadamard manifolds.
• It also introduces parallel Douglas–Rachford type algorithms for problems with multiple summands.
• The convergence analysis relies on fixed-point theory for nonexpansive operators.
• The abstract says the methods were applied to generalized Heron problems and tested numerically.
• Detailed parameter conditions and numerical outcomes are not given in the abstract.