Subgradient Methods on Manifolds with Lower Bounded Curvature
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.