Abstract:The subgradient method is a classical and foundational approach in non-smooth convex optimization; its simplicity, robustness, and role as a conceptual and algorithmic starting point have made it the backbone of many significant optimization algorithms. Motivated by classical Euclidean results and recent advances in first-order Riemannian optimization, we study the convergence of the subgradient method on Hadamard manifolds with lower bounded curvature. Assuming a nonempty solution set and employing a corresponding non-summable diminishing step-size condition, we establish convergence of the generated sequence $\{x^k\}$ to a minimizer whenever at least one of the following holds: (a) the sequence $\{x^k\}$ is bounded; (b) the solution set $S$ is bounded; or (c) the step-sizes are square-summable ($\sum_{k=1}^{\infty}\lambda_k^2<\infty$). Additionally, we prove that if $\operatorname{int}(S)\neq\emptyset$, the method achieves finite termination. Our main contribution provides a Riemannian counterpart to Shepilov's Euclidean analysis [Cybernetics, 12 (1976), pp. 544-548], thus complementing existing literature on convex minimization over manifolds with lower bounded curvature.
| Comments: | 21 pages |
| Subjects: | Optimization and Control (math.OC) |
| MSC classes: | 49M37, 90C25, 53C23, 49J52 |
| Cite as: | arXiv:2605.24780 [math.OC] |
| (or arXiv:2605.24780v1 [math.OC] for this version) | |
| https://doi.org/10.48550/arXiv.2605.24780 arXiv-issued DOI via DataCite (pending registration) |
From: Glaydston Bento Carvalho [view email]
[v1]
Sat, 23 May 2026 23:45:37 UTC (42 KB)
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