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| Main Authors: | , , , |
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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2605.24780 |
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| _version_ | 1866918520029184000 |
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| author | Bento, G. C. Neto, J. X. Cruz Lopes, J. O. Melo, I. D. L. |
| author_facet | Bento, G. C. Neto, J. X. Cruz Lopes, J. O. Melo, I. D. L. |
| contents | 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 ($\sum_{k=1}^{\infty}λ_k^2<\infty$). 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_24780 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Subgradient Methods on Manifolds with Lower Bounded Curvature Bento, G. C. Neto, J. X. Cruz Lopes, J. O. Melo, I. D. L. Optimization and Control 49M37, 90C25, 53C23, 49J52 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 ($\sum_{k=1}^{\infty}λ_k^2<\infty$). 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. |
| title | Subgradient Methods on Manifolds with Lower Bounded Curvature |
| topic | Optimization and Control 49M37, 90C25, 53C23, 49J52 |
| url | https://arxiv.org/abs/2605.24780 |